Maharashtra Board 12th Maths Solutions Chapter 3 Trigonometric Functions Ex 3.2

Balbharti 12th Maharashtra State Board Maths Solutions Book Pdf Chapter 3 Trigonometric Functions Ex 3.2 Questions and Answers.

Maharashtra State Board 12th Maths Solutions Chapter 3 Trigonometric Functions Ex 3.2

Question 1.
Find the Cartesian co-ordinates of the point whose polar co-ordinates are:
(i) \(\left(\sqrt{2}, \frac{\pi}{4}\right)\)
Solution:
Here, r = \(\sqrt {2}\) and θ = \(\frac{\pi}{4}\)
Let the cartesian coordinates be (x, y)
Then, x = rcosθ = \(\sqrt {2}\)cos\(\frac{\pi}{4}\) = \(\sqrt{2}\left(\frac{1}{\sqrt{2}}\right)\) = 1
y = rsinθ = \(\sqrt {2}\)sin\(\frac{\pi}{4}\) = \(\sqrt{2}\left(\frac{1}{\sqrt{2}}\right)\) = 1
∴ the cartesian coordinates of the given point are (1, 1).

(ii) \(\left(4, \frac{\pi}{2}\right)\)
Solution:

(iii) \(\left(\frac{3}{4}, \frac{3 \pi}{4}\right)\)
Solution:
Here, r = \(\frac{3}{4}\) and θ = \(\frac{3 \pi}{4}\)
Let the cartesian coordinates be (x, y)
Maharashtra Board 12th Maths Solutions Chapter 3 Trigonometric Functions Ex 3.2 1

Maharashtra Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

(iv) \(\left(\frac{1}{2}, \frac{7 \pi}{3}\right)\)
Solution:
Here, r = \(\frac{1}{2}\) and θ = \(\frac{7 \pi}{4}\)
Let the cartesian coordinates be (x, y)
Maharashtra Board 12th Maths Solutions Chapter 3 Trigonometric Functions Ex 3.2 2
Maharashtra Board 12th Maths Solutions Chapter 3 Trigonometric Functions Ex 3.2 3
∴ the cartesian coordinates of the given point are \(\left(\frac{1}{4}, \frac{\sqrt{3}}{4}\right)\)

Question 2.
Find the of the polar co-ordinates point whose Cartesian co-ordinates are.
(i) \((\sqrt{2}, \sqrt{2})\)
Solution:
Here x = \(\sqrt {2}\) and y = \(\sqrt {2}\)
∴ the point lies in the first quadrant.
Let the polar coordinates be (r, θ)
Then, r2 = x2 + y2 = (\(\sqrt {2}\) )2 + (\(\sqrt {2}\) )2 = 2 + 2 = 4
∴ r = 2 … [∵ r > 0]
cos θ = \(\frac{x}{r}=\frac{\sqrt{2}}{2}=\frac{1}{\sqrt{2}}\)
and sin θ = \(\frac{y}{r}=\frac{\sqrt{2}}{2}=\frac{1}{\sqrt{2}}\)
∴ tan θ = 1
Since the point lies in the first quadrant and
0 ≤ θ ≤ 2π, tan θ = 1 = tan\(\frac{\pi}{4}\)
∴ θ = \(\frac{\pi}{4}\)
∴ the polar coordinates of the given point are \(\left(2, \frac{\pi}{4}\right)\).

Maharashtra Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

(ii) \(\left(0, \frac{1}{2}\right)\)
Solution:
Here x = 0 and y = \(\frac{1}{2}\)
the point lies on the positive side of Y-axis. Let the polar coordinates be (r, θ)
Then, r2 = x2 + y2 = (0)2 + \(\left(\frac{1}{2}\right)^{2}=0+\frac{1}{4}=\frac{1}{4}\)
∴ r = \(\frac{1}{2}\) …[∵ r > 0]
cosθ = \(\frac{x}{r}=\frac{0}{(1 / 2)}\) = 0
and sin θ = \(\frac{y}{r}=\frac{(1 / 2)}{(1 / 2)}\) = 1
Since, the point lies on the positive side of Y-axis and 0 ≤ θ ≤ 2π
cosθ = 0 = cos\(\frac{\pi}{2}\) and sinθ = 1 = sin\(\frac{\pi}{2}\)
∴ θ = \(\frac{\pi}{2}\)
∴ the polar coordinates of the given point are \(\left(\frac{1}{2}, \frac{\pi}{2}\right)\).

(iii) \((1,-\sqrt{3})\)
Solution:
Here x = 1 and y = \(-\sqrt{3}\)
∴ the point lies in the fourth quadrant.
Let the polar coordinates be (r, θ).
Then, r2 = x2 + y2 = (1)2 + (\(-\sqrt {3}\) )2 = 1 + 3 = 4
∴ r = 2 … [∵ r > 0]
Maharashtra Board 12th Maths Solutions Chapter 3 Trigonometric Functions Ex 3.2 4
∴ the polar coordinates of the given point are \(\left(2, \frac{5 \pi}{3}\right)\).

Maharashtra Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

(iv) \(\left(\frac{3}{2}, \frac{3 \sqrt{3}}{2}\right)\)
Solution:

Question 3.
In ∆ABC, if ∠A = 45º, ∠B = 60º then find the ratio of its sides.
Solution:
By the sine rule,
\(\frac{a}{\sin \mathrm{A}}\) = \(\frac{b}{\sin \mathrm{B}}\) = \(\frac{c}{\sin \mathrm{C}}\)
∴ \(\frac{a}{b}=\frac{\sin A}{\sin B}\) and \(\frac{b}{c}=\frac{\sin B}{\sin C}\)
∴ a : b : c = sinA : sinB : sinC
Given ∠A = 45° and ∠B = 60°
∵ ∠A + ∠B + ∠C = 180°
∴ 45° + 60° + ∠C = 180°
∴ ∠C = 180° – 105° = 75°
Maharashtra Board 12th Maths Solutions Chapter 3 Trigonometric Functions Ex 3.2 5

Maharashtra Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

Question 4.
In ∆ABC, prove that sin \(\left(\frac{\mathbf{B}-\mathbf{C}}{2}\right)=\left(\frac{\boldsymbol{b}-\boldsymbol{c}}{a}\right)\) cos \(\frac{A}{2}\).
Solution:
By the sine rule,
Maharashtra Board 12th Maths Solutions Chapter 3 Trigonometric Functions Ex 3.2 6
Maharashtra Board 12th Maths Solutions Chapter 3 Trigonometric Functions Ex 3.2 7

Question 5.
With usual notations prove that 2 \(\left\{a \sin ^{2} \frac{C}{2}+c \sin ^{2} \frac{A}{2}\right\}\) = a – b + c.
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Trigonometric Functions Ex 3.2 8

Maharashtra Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

Question 6.
In ∆ABC, prove that a3sin(B – C) + b3sin(C – A) + c3sin(A – B) = 0
Solution:
By the sine rule,
\(\frac{a}{\sin A}\) = \(\frac{b}{\sin B}\) = \(\frac{c}{\sin C}\) = k
∴ a = k sin A, b = k sin B, c = k sin C
LHS = a3sin (B – C) + b3sin (C – A) + c3sin (A – B)
= a3(sin B cos C – cos B sin C) + b3(sinCcos A – cos C sin A) + c3(sinAcosB – cos A sin B)
Maharashtra Board 12th Maths Solutions Chapter 3 Trigonometric Functions Ex 3.2 9
= \(\frac{1}{2 k}\) [a2(a2 + b2 – c2) – a2(a2 + c2 – b2) + b2(b2 + c2 – a2) – b2(a2 + b2 – c2) + c2(c2 + a2 – b2) – c2(b2 + c2 – a2)]
= \(\frac{1}{2 k}\) [a4 + a2b2 – a2c2 – a4 – a2c2 + a2b2 + b4 + b2c2 – a2b2 – a2b2 – b4 + b2c2 + c4 + a2c2 – b2c2 – b2c2 – c4 + a2c2]
= \(\frac{1}{2 k}\)(0) = 0 = RHS.

Question 7.
In ∆ABC, if cot A, cot B, cot C are in A.P. then show that a2, b2, c2 are also in A.P
Solution:
By the sine rule,
\(\frac{\sin \mathrm{A}}{a}\) = \(\frac{\sin \mathrm{B}}{b}\) =\(\frac{\sin \mathrm{C}}{c}\) = k
∴ sin A = ka, sin B = kb, sin C = kc …(1)
Now, cot A, cotB, cotC are in A.P.
∴ cotC – cotB = cotB – cot A
∴ cotA + cotC = 2cotB
Maharashtra Board 12th Maths Solutions Chapter 3 Trigonometric Functions Ex 3.2 10
Maharashtra Board 12th Maths Solutions Chapter 3 Trigonometric Functions Ex 3.2 11

Maharashtra Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

Question 8.
In ∆ABC, if a cos A = b cos B then prove that the triangle is right angled or an isosceles traingle.
Solution:
By the sine rule,
\(\frac{a}{\sin \mathrm{A}}\) = \(\frac{b}{\sin \mathrm{B}}\) = k
a = k sin A and b = k sin B
∴ a cos A = b cos B gives
k sin A cos A = k sin B cos B
∴ 2 sin A cos A = 2 sin B cos B
∴ sin 2A = sin 2B ∴ sin 2A – sin 2B = 0
∴ 2 cos (A + B)∙sin (A -B) = 0
∴ 2cos (π – C)∙sin(A – B) = 0 … [∵ A + B + C = π]
∴ -2 cos C∙sin (A – B) = 0
∴ cos C = 0 OR sin(A -B) = 0
∴ C = 90° OR A – B = 0
∴ C = 90° OR A = B
∴ the triangle is either rightangled or an isosceles triangle.

Question 9.
With usual notations prove that 2(bc cos A + ac cos B + ab cos C) = a2 + b2 + c2.
Solution:
LHS = 2 (bc cos A + ac cos B + ab cos C)
= 2bc cos A + 2ac cos B + 2ab cos C
= 2bc \(\left(\frac{b^{2}+c^{2}-a^{2}}{2 b c}\right)\) + 2ac\(\left(\frac{c^{2}+a^{2}-b^{2}}{2 c a}\right)\) + 2ab\(\left(\frac{a^{2}+b^{2}-c^{2}}{2 a b}\right)\) …(By cosine rule]
= b2 + c2 – a2 + c2 + a2 – b2 + a2 + b2 – c2 = a2 + b2 + c2 = RHS.

Maharashtra Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

Question 10.
In △ABC, if a = 18, b = 24, c = 30 then find the values of
(i) cos A
Solution:
Given : a = 18, b = 24 and c = 30
∴ 2s = a + b + c = 18 + 24 + 30 = 72 ∴ s = 36
Maharashtra Board 12th Maths Solutions Chapter 3 Trigonometric Functions Ex 3.2 12

(ii) sin\(\frac{A}{2}\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Trigonometric Functions Ex 3.2 13

(iii) cos\(\frac{A}{2}\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Trigonometric Functions Ex 3.2 14

Maharashtra Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

(iv) tan\(\frac{A}{2}\)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Trigonometric Functions Ex 3.2 15

(v) A(△ABC)
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Trigonometric Functions Ex 3.2 16

(iv) sin A.
Solution:
Maharashtra Board 12th Maths Solutions Chapter 3 Trigonometric Functions Ex 3.2 17

Question 11.
In △ABC prove that (b + c – a) tan \(\frac{A}{2}\) = (c + a – b) tan\(\frac{B}{2}\) = (a + b – c) tan\(\frac{C}{2}\).
Solution:
(b + c – a) tan \(\frac{A}{2}\)
Maharashtra Board 12th Maths Solutions Chapter 3 Trigonometric Functions Ex 3.2 18
Maharashtra Board 12th Maths Solutions Chapter 3 Trigonometric Functions Ex 3.2 19

Maharashtra Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

Question 12.
In △ABC prove that sin \(\frac{A}{2}\)∙sin \(\frac{A}{2}\)∙sin \(\frac{A}{2}\) = \(\frac{[A(\triangle A B C)]^{2}}{a b c s}\)
Solution:
LHS = sin \(\frac{A}{2}\)∙sin \(\frac{B}{2}\)∙sin \(\frac{C}{2}\)
Maharashtra Board 12th Maths Solutions Chapter 3 Trigonometric Functions Ex 3.2 20

Maharashtra Board 12th Maths Solutions Chapter 3 Trigonometric Functions Ex 3.1

Balbharti 12th Maharashtra State Board Maths Solutions Book Pdf Chapter 3 Trigonometric Functions Ex 3.1 Questions and Answers.

Maharashtra State Board 12th Maths Solutions Chapter 3 Trigonometric Functions Ex 3.1

Question 1.
Find the principal solutions of the following equations :
(i) cos θ= \(\frac{1}{2}\)
Solution:
We know that, cos\(\frac{\pi}{3}\) = \(\frac{1}{2}\) and cos (2π – θ) = cos θ
∴ cos\(\frac{\pi}{3}\) = cos(2π – \(\frac{\pi}{3}\)) = cos\(\frac{5 \pi}{3}\)
∴ cos\(\frac{\pi}{3}\) = cos\(\frac{5 \pi}{3}\) = \(\frac{1}{2}\), where
0 < \(\frac{\pi}{3}\) < 2π and 0 < \(\frac{5 \pi}{3}\) < 2π
∴ cos θ = \(\frac{1}{2}\) gives cos θ = cos\(\frac{\pi}{3}\) = cos\(\frac{5 \pi}{3}\)
∴ θ = \(\frac{\pi}{3}\) and θ = \(\frac{5 \pi}{3}\)
Hence, the required principal solutions are
θ = \(\frac{\pi}{3}\) and θ = \(\frac{5 \pi}{3}\)

(ii) sec θ = \(\frac{2}{\sqrt{3}}\)
Solution:

(iii) cot θ = \(\sqrt {3}\)
Solution:
The given equation is cot θ = \(\sqrt {3}\) which is same as tan θ = \(\frac{1}{\sqrt{3}}\).
We know that,
Maharashtra Board 12th Maths Solutions Chapter 3 Trigonometric Functions Ex 3.1 1
Hence, the required principal solution are
θ = \(\frac{\pi}{6}\) and θ = \(\frac{7 \pi}{6}\).

Maharashtra Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

(iv) cot θ = 0.
Solution:

Question 2.
Find the principal solutions of the following equations:
(i) sinθ = \(-\frac{1}{2}\)
Solution:
We know that,
sin\(\frac{\pi}{6}\) = \(\frac{1}{2}\) and sin (π + θ) = -sinθ,
sin(2π – θ) = -sinθ
Maharashtra Board 12th Maths Solutions Chapter 3 Trigonometric Functions Ex 3.1 2
Hence, the required principal solutions are
θ = \(\frac{7\pi}{6}\) and θ = \(\frac{11 \pi}{6}\).

(ii) tanθ = -1
Solution:
We know that,
tan\(\frac{\pi}{4}\) = 1 and tan(π – θ) = -tanθ,
tan(2π – θ) = -tanθ
Maharashtra Board 12th Maths Solutions Chapter 3 Trigonometric Functions Ex 3.1 3
Hence, the required principal solutions are
θ = \(\frac{3\pi}{4}\) and θ = \(\frac{7 \pi}{4}\).

Maharashtra Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

(iii) \(\sqrt {3}\) cosecθ + 2 = 0.
Solution:

Question 3.
Find the general solutions of the following equations :
(i) sinθ = \(\frac{1}{2}\)
Solution:
(i) The general solution of sin θ = sin ∝ is
θ = nπ + (-1 )n∝, n ∈ Z
Now, sinθ = \(\frac{1}{2}\) = sin\(\frac{\pi}{6}\) …[∵ sin\(\frac{\pi}{6}\) = \(\frac{1}{2}\)]
∴ the required general solution is
θ = nπ + (-1)n\(\frac{\pi}{6}\), n ∈ Z.

(ii) cosθ = \(\frac{\sqrt{3}}{2}\)
Solution:
The general solution of cos θ = cos ∝ is
θ = 2nπ ± ∝, n ∈ Z
Now, cosθ = \(\frac{\sqrt{3}}{2}\) = cos\(\frac{\pi}{6}\) …[∵ cos\(\frac{\pi}{6}\) = \(\frac{\sqrt{3}}{2}\)]
∴ the required general solution is
θ = 2nπ ± \(\frac{\pi}{6}\), n ∈ Z.

Maharashtra Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

(iii) tanθ = \(\frac{1}{\sqrt{3}}\)
Solution:
The general solution of tan θ = tan ∝ is
θ = nπ + ∝, n ∈ Z
Now, tan θ = \(\frac{1}{\sqrt{3}}\) = tan\(\frac{\pi}{6}\) …[tan\(\frac{\pi}{6}\) = \(\frac{1}{\sqrt{3}}\)]
∴ the required general solution is
θ = nπ + \(\frac{\pi}{6}\) , n ∈ Z.

(iv) cotθ = 0.
Solution:
The general solution of tan θ = tan ∝ is
θ = nπ + ∝, n ∈ Z
Now, cot θ = 0 ∴ tan θ does not exist
∴ tanθ = tan\(\frac{\pi}{2}\) [∵ tan\(\frac{\pi}{2}\) does not exist]
∴ the required general solution is
θ = nπ + \(\frac{\pi}{2}\), n ∈ Z.

Maharashtra Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

Question 4.
Find the general solutions of the following equations:
(i) secθ = \(\sqrt {2}\)
Solution:
The general solution of cos θ = cos ∝ is
θ = nπ ± ∝, n ∈ Z.
Now, secθ = \(\sqrt {2}\) ∴ cosθ = \(\frac{1}{\sqrt{2}}\)
∴ cosθ = cos\(\frac{\pi}{4}\) ….[cos\(\frac{\pi}{4}\) = \(\frac{1}{\sqrt{2}}\)]
∴ the required general solution is
θ = 2nπ ± \(\frac{\pi}{4}\), n ∈ Z.

(ii) cosecθ = –\(\sqrt {2}\)
Solution:
The general solution of sinθ = sin∝ is
Maharashtra Board 12th Maths Solutions Chapter 3 Trigonometric Functions Ex 3.1 4

(iii) tanθ = -1
Solution:
The general solution of tanθ = tan∝ is
Maharashtra Board 12th Maths Solutions Chapter 3 Trigonometric Functions Ex 3.1 5

Maharashtra Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

Question 5.
Find the general solutions of the following equations :
(i) sin 2θ = \(\frac{1}{2}\)
Solution:
The general solution of sin θ = sin ∝ is
θ = nπ + (-1)n∝, n ∈ Z
Maharashtra Board 12th Maths Solutions Chapter 3 Trigonometric Functions Ex 3.1 6

(ii) tan \(\frac{2 \theta}{3}\) = \(\sqrt {3}\)
Solution:
The general solution of tan θ = tan ∝ is
Maharashtra Board 12th Maths Solutions Chapter 3 Trigonometric Functions Ex 3.1 7

Maharashtra Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

(iii) cot 4θ = -1
Solution:
The general solution of tan θ = tan ∝ is
Maharashtra Board 12th Maths Solutions Chapter 3 Trigonometric Functions Ex 3.1 8

Question 6.
Find the general solutions of the following equations :
(i) 4 cos2θ = 3
Solution:
The general solution of cos2θ = cos2 ∝ is
θ = nπ ± ∝, n ∈ Z
Now, 4 cos2θ = 3
Maharashtra Board 12th Maths Solutions Chapter 3 Trigonometric Functions Ex 3.1 9

(ii) 4 sin2θ = 1
Solution:
The general solution of sin2θ = sin2 ∝ is
θ = nπ ± ∝, n ∈ Z
Now, 4 sin2θ = 3
Maharashtra Board 12th Maths Solutions Chapter 3 Trigonometric Functions Ex 3.1 10

Maharashtra Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

(iii) cos 4θ = cos 2θ
Solution:
The general solution of cos θ = cos ∝ is
θ = 2nπ ± ∝, n ∈ Z
∴ the general solution of cos 4θ = cos 2θ is given by
4θ = 2nπ ± 2θ, n ∈ Z
Taking positive sign, we get
4θ = 2nπ + 2θ, n ∈ Z
∴ 2θ = 2nπ, n ∈ Z
∴ θ = nπ, n ∈ Z
Taking negative sign, we get
4θ = 2nπ – 2θ, n ∈ Z
∴ 6θ = 2nπ, n ∈ Z
∴ θ = \(\frac{n \pi}{3}\), n ∈ Z
Hence, the required general solution is
θ = \(\frac{n \pi}{3}\), n ∈ Z or ∴ θ = nπ, n ∈ Z.
Alternative Method:
cos 4θ = cos 2θ
∴ cos4θ – cos 20 = 0
∴ -2sin\(\left(\frac{4 \theta+2 \theta}{2}\right)\)∙sin\(\left(\frac{4 \theta-2 \theta}{2}\right)\) = 0
∴ sin3θ∙sinθ = 0
∴ either sin3θ = 0 or sin θ = 0
The general solution of sin θ = 0 is
θ = nπ, n ∈ Z.
∴ the required general solution is given by
3θ = nπ, n ∈ Z or θ = nπ, n ∈ Z
i.e. θ = \(\frac{n \pi}{3}\), n ∈ Z or θ = nπ, n ∈ Z.

Question 7.
Find the general solutions of the following equations :
(i) sinθ = tanθ
Solution:
sin θ = tan θ
∴ sin θ = \(\frac{\sin \theta}{\cos \theta}\)
∴ sin θ cos θ = sin θ
∴ sin θ cos θ – sinθ = 0
∴ sin θ (cos θ – 1) = θ
∴ either sinθ = 0 or cosθ – 1 = 0
∴ either sin θ = 0 or cos θ = 1
∴ either sinθ = 0 or cosθ = cosθ …[∵ cos0 = 1]
The general solution of sinθ = 0 is θ = nπ, n ∈ Z and cos θ = cos ∝ is θ = 2nπ ± ∝, where n ∈ Z.
∴ the required general solution is given by
θ = nπ, n ∈ Z or θ = 2nπ ± 0, n ∈ Z
∴ θ = nπ, n ∈ Z or θ = 2nπ, n ∈ Z.

Maharashtra Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

(ii) tan3θ = 3tanθ
Solution:
tan3θ = 3tanθ
∴ tan3θ – 3tanθ = 0
∴ tan θ (tan2θ – 3) = 0
∴ either tan θ = 0 or tan2θ – 3 = 0
∴ either tanθ = 0 or tan2θ = 3
∴ either tan θ = 0 or tan2θ = (\(\sqrt {3}\) )3
∴ either tan θ = 0 or tan2θ = (tan\(\frac{\pi}{3}\))3 …[tan\(\frac{\pi}{3}\) = \(\sqrt {3}\)]
∴ either tanθ = 0 or tan2θ = tan2\(\frac{\pi}{3}\)
The general solution of
tanθ = 0 is θ = nπ, n ∈ Z and
tan2θ = tan2∝ is θ = nπ ± ∝, n ∈ Z.
∴ the required general solution is given by
θ = nπ, n ∈ Z or θ = nπ ± \(\frac{\pi}{3}\), n ∈ Z.

(iii) cosθ + sinθ = 1.
Solution:
cosθ + sinθ = 1
Maharashtra Board 12th Maths Solutions Chapter 3 Trigonometric Functions Ex 3.1 11
Maharashtra Board 12th Maths Solutions Chapter 3 Trigonometric Functions Ex 3.1 12
Maharashtra Board 12th Maths Solutions Chapter 3 Trigonometric Functions Ex 3.1 13
Maharashtra Board 12th Maths Solutions Chapter 3 Trigonometric Functions Ex 3.1 14

Question 8.
Which of the following equations have solutions ?
(i) cos 2θ = -1
Solution:
cos 2θ = -1
Since -1 ≤ cos θ ≤ 1 for any θ,
cos 2θ = -1 has solution.

Maharashtra Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

(ii) cos2θ = -1
Solution:
cos2θ = -1
This is not possible because cos2θ ≥ 0 for any θ.
∴ cos2θ = -1 does not have any solution.

(iii) 2 sinθ = 3
Solution:
2 sin θ = 3 ∴ sin θ = \(\frac{3}{2}\)
This is not possible because -1 ≤ sin θ ≤ 1 for any θ.
∴ 2 sin θ = 3 does not have any solution.

Maharashtra Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

(iv) 3 tanθ = 5
Solution:
3tanθ = 5 ∴ tanθ = \(\frac{5}{3}\)
This is possible because tan θ is any real number.
∴ 3tanθ = 5 has solution.

Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2B

Balbharti 12th Maharashtra State Board Maths Solutions Book Pdf Chapter 2 Matrices Miscellaneous Exercise 2B Questions and Answers.

Maharashtra State Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2B

I. Choose the correct answer from the given alternatives in each of the following questions:

Question 1.
If A = \(\left(\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right)\), adj = \(\left(\begin{array}{ll}
4 & a \\
-3 & b
\end{array}\right)\) then the values of a and b are,
(a) a = – 2, b = 1
(b) a = 2, b = 4
(c) a = 2, b = –1
(d) a = 1, b = –2
Solution:
(a) a = – 2, b = 1

Question 2.
The inverse of \(\left(\begin{array}{ll}
0 & 1 \\
1 & 0
\end{array}\right)\) is
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2B 1
Solution:
\(\left(\begin{array}{ll}
0 & 1 \\
1 & 0
\end{array}\right)\)

Question 3.
If A = \(\left(\begin{array}{ll}
1 & 2 \\
2 & 1
\end{array}\right)\) and A(adj A) = k 1, then the value of k is
(a) 1
(b) -1
(c) 0
(d) -3
Solution:
(d) -3 [Hint : A(adj A) = |A| ∙ I]

Maharashtra Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

Question 4.
If A = \(\left(\begin{array}{ll}
2 & -4 \\
3 & 1
\end{array}\right)\), then the adjoint of matrix A is
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2B 2
Solution:\(\left(\begin{array}{ll}
1 & 4 \\
-3 & 2
\end{array}\right)\)

Question 5.
If A = \(\left(\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right)\) and A(adj A) = kI, then the value of k is
(a) 2
(b) -2
(c) 10
(d) -10
Solution:
(b) -2

Question 6.
If A = \(\left(\begin{array}{rr}
\lambda & 1 \\
-1 & -\lambda
\end{array}\right)\), then A-1 does not exist if λ = ………..
(a) 0
(b) ± 1
(c) 2
(d) 3
Solution:
(b) ± 1

Maharashtra Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

Question 7.
If A = \(\left[\begin{array}{ll}
\cos \alpha & -\sin \alpha \\
\sin \alpha & \cos \alpha
\end{array}\right]\) then A-1 = ….
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2B 3
Solution:
\(\left[\begin{array}{rr}
\cos \alpha & -\sin \alpha \\
-\sin \alpha & \cos \alpha
\end{array}\right]\)

Question 8.
If F (∝) = \(\left[\begin{array}{ccc}
\cos \alpha & -\sin \alpha & 0 \\
\sin \alpha & \cos \alpha & 0 \\
0 & 0 & 1
\end{array}\right]\) where ∝ ∈ R then [F(∝)]-1 is =
(a) F(-∝)
(b) F(∝-1)
(c) F(2∝)
(d) None of these
Solution:
(a) F(-∝)

Question 9.
The inverse of A = \(\left[\begin{array}{lll}
0 & 1 & 0 \\
1 & 0 & 0 \\
0 & 0 & 1
\end{array}\right]\)
(a) I
(b) A
(c) A’
(d) -I
Solution:
(b) A

Maharashtra Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

Question 10.
The inverse of a symmetric matrix is
(a) Symmetric
(b) Non-symmetric
(c) Null matrix
(d) Diagonal matrix
Solution:
(a) Symmetric

Question 11.
For a 2 × 2 matrix A, if A(adjA) = \(\left(\begin{array}{ll}
10 & 0 \\
0 & 10
\end{array}\right)\) then determinant A equals
(a) 20
(b) 10
(c) 30
(d) 40
Solution:
(b) 10

Question 12.
If A2 = \(-\frac{1}{2}\left[\begin{array}{cc}
1 & -4 \\
-1 & 2
\end{array}\right]\) then A =
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2B 4
Solution:
\(-\frac{1}{2}\left[\begin{array}{cc}
2 & 4 \\
1 & 1
\end{array}\right]\)

II. Solve the following equations by the methods of inversion.

(i) 2x – y = -2 , 3x + 4y = 5
Solution:
The given equations can be written in the matrix form as :
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2B 5
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2B 6
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2B 7
By equality of matrices,
x = \(-\frac{5}{11}\), y = \(\frac{12}{11}\) is the required solution.

Maharashtra Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

(ii) x + y + z = 1, 2x + 3y + 2z = 2 and ax + ay + 2az = 4, a ≠ 0.
Solution:
The given equations can be written in the matrix form as :
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2B 8
= 1(6a – 2a) – 1(4a – 2a) + 1(2a – 3a)
= 4a – 2a – a = a ≠ 0 ∴ A-1 exists.
Consider AA-1 = I
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2B 9
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2B 10
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2B 11
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2B 12

(iii) 5x – y +4z = 5, 2x + 3y + 5z = 2 and 5x – 2y + 6z = -1
Solution:
The given equations can be written in the matrix form as :
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2B 13
= 5(18 + 10) + 1 (12 – 25) + 4( -4 – 15)
= 140 – 13 – 76 = 51 #0
∴ A-1 exists.
Now, we have to find the cofactor matrix
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2B 14
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2B 15
Now, premultiply AX = B by A-1, we get,
A-1(AX) = A-1B
∴ (A-1A)X = A-1B
∴ IX = A-1B
∴ X = \(\frac{1}{51}\left[\begin{array}{rrr}
28 & -2 & -17 \\
13 & 10 & -17 \\
-19 & 5 & 17
\end{array}\right]\left[\begin{array}{r}
5 \\
2 \\
-1
\end{array}\right]\)
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2B 16
By equality of matrices,
x = 3, y = 2, z = -2 is the required solution.

Maharashtra Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

(iv) 2x + 3y = -5, 3x + y = 3
Solution:

(v) x + y + z = -1, y + z = 2 and x + y – z = 3
Solution:
The given equations can be written in the matrix form as :
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2B 17
= 1(-1 – 1) – 1 (0 – 1) + 1(0 – 1)
= -2 + 1 – 1 = -2 ≠ 0 ∴ A-1 exists.
Consider AA-1 = I
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2B 18
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2B 19
Now, premultiply AX = B by A-1, we get,
A-1(AX) = A-1B
∴ (A-1A)X = A-1B
∴ IX = A-1B
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2B 20
∴ by equality of the matrices, x= -3, y = 4, z = -2 is the required solution.

Question 2.
Express the following equation in matrix from and solve them by the method of reduction.
(i) x – y + z = 1, 2x – y = 1, 3x + 3y – 4z = 2
Solution:
The given equations can be written in the matrix form as :
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2B 21
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2B 22
By equality of matrices,
x – y + z = 1 ……(1)
y – 2z = -1 …..(2)
5z = 5 ….(3)
From (3), z = 1
Substituting z = 1 in (2), we get,
y – 2 = -1 ∴ y = 1
Substituting y = 1, z = 1 in (1), we get,
x – 1 + 1 = 1
∴ x = 1
Hence, x = 1, y = 1, z = 1 is the required solution.

Maharashtra Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

(ii) x + y = 1, y + z = \(\frac{5}{3}\), z + x = \(\frac{4}{3}\).
Solution:
The given equations can be written in the matrix form as :
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2B 23
By equality of matrices,
x + y = 1 ……(1)
y + z = \(\frac{5}{3}\) …(2)
2z = 2 ……..(3)
From (3), z = 1
Substituting z = 1 in (2), we get,
y + 1 = \(\frac{5}{3}\) ∴ y = \(\frac{2}{3}\)
Substituting y = \(\frac{2}{3}\) in (1), we get,
x + \(\frac{2}{3}\) = 1 ∴ x = \(\frac{1}{3}\)
Hence, x = \(\frac{1}{3}\), y = \(\frac{2}{3}\), z = 1 is the required solution.

(iii) 2x – y + z = 1, x + 2y + 3z = 8 and 3x + y – 4z = 1
Solution:
The given equations can be written in the matrix form as :
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2B 24
∴ \(\left[\begin{array}{r}
x+2 y+3 z \\
0-5 y-5 z \\
0+0-8 z
\end{array}\right]\) = \(\left[\begin{array}{r}
8 \\
-15 \\
-8
\end{array}\right]\)
By equality of matrices,
x + 2y + 3z = 8 …..(1)
-5y – 5z = -15 ….(2)
-8z = -8 …..(3)
From (3), z = 1
Substituting z = 1 in (2), we get,
-5y – 5 = -15
-5y = -10
∴ y = 2
Substituting y = 2, z = 1 in (1), we get,
x + 4 + 3 = 8 ∴ x = 1
Hence, x = 1, y = 2, z = 1 is the required solution.

Maharashtra Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

(iv) x + y + z = 6, 3x – y + 3z =10 and 5x + 5y – 4z = 3.
Solution:

(v) x + 2y + z = 8, 2x + 3y – z =11 and 3x – y – 2z = 5
Solution:
The given equations can be written in the matrix form as :
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2B 25
By equality of matrices,
x + 2y + z = 8 … (1)
-y – 3z = -5 … (2)
16z = 16 … (3)
From (3), z = 1
Substituting z = 1 in (2), we get,
-y – 3 = -5, ∴ y = 2
Substituting y = 2, z = 1 in (1), we get,
x + 4 + 1 = 8 ∴ x = 3
Hence, x = 3, y = 2, z = 1 is the required solution.

(vi) x + 3y + 2z = 6, 3x – 2y + 5z =5 and 2x – 3y + 6z = 7.
Solution:
The given equations can be written in the matrix form as :
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2B 26
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2B 27
By equality of matrices,
x + 3y + 2z = 6 …(1)
y + \(\frac{3}{2}\) z = 4 …(2)
\(\frac{31}{2}\)z = 31 …..(3)
From (3), z = 2
Substituting z = 2 in (2), we get,
y + \(\frac{3}{2}\)z = 4
y + \(\frac{3}{2}\)(2) = 4
y + 3 = 4
y = 1
Substituting y = 1, z = 2 in (2), we get,
x + 3y + 2z = 6
x + 3(1) + 2(2) = 6
x + 3 + 4 = 6
x = -1
Hence, x = -1, y = 1, z = 2 is the required solution.

Maharashtra Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

Question 3.
The sum of three numbers is 6. If we multiply third number by 3 and add it to the second number we get 11. By adding first and the third numbers we get a number which is double the second number. Use this information and find a system of linear equations. Find the three numbers using matrices.
Solution:
Let the three numbers be x, y and z. According to the given conditions,
x + y + z = 6.
3z + y = 11, i.e., y + 3z = 11 and x + z = 2y,
i.e., x – 2y + z = 0
Hence, the system of the linear equations is
x + y + z = 6
y + 3z = 11
x – 2y + z = 0
These equations can be written in the matrix form as :
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2B 28
By equality of matrices,
x + y + z = 6 …(1)
y + 3z = 11 …(2)
-3y = -6 …(3)
From (3), y = 2
Substituting y = 2 in (2), we get,
2 + 3z = 11
∴ 3z = 9 ∴ z = 3
Put y = 2, z — 3 in (1), we get,
x + 2 + 3 = 6 ∴ x = 1
∴ x = 1, y = 2, z = 3
Hence, the required numbers are 1, 2 and 3.

Question 4.
The cost of 4 pencils, 3 pens and 2 books is ₹ 150. The cost of 1 pencil, 2 pens and 3 books is ₹ 125. The cos of 6 pencils, 2 pens and 3 books is ₹ 175. Fild the cost of each item by using Matrices.
Solution:
Let the cost of 1 pencil, 1 pen and 1 book be ₹x, ₹ y, ₹ z respectively.
According to the given conditions,
4x + 3y + 2z = 150
x + 2y + 3z = 125
6x + 2y + 3z = 175
The equations can be written in matrix form as :
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2B 29
By equality of matrices,
x + 2y + 3z = 125 …(1)
-5y – 10z = -350 …(2)
5z = 125 …(3)
From (3), z = 25
Substituting z = 25 in (2), we get
-5y – 10(25) = -350
∴ -5y = -350 + 250 = -100
∴ y = 20
Substituting y = 20, z = 25 in (1), we get
x + 2(20) + 3(25) = 125
∴ x = 125 – 40 – 75 = 10
∴ x = 10, y = 20, z = 25
Hence, the cost of 1 pencil is ₹ 10, 1 pen is ₹ 20 and 1 book is ₹ 25.

Maharashtra Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

Question 5.
The sum of three numbers is 6. Thrice the third number when added to the first number, gives 7. On adding three times first number to the sum of second and third number, we get 12. Find the three numbers by using Matrices.
Solution:
Let the numbers be x, y and z.
According to the given conditions,
x + y + z = 6
3z + x = 7, i.e., x + 3z = 7
and 3x + y + z = 12
Hence, the system of linear equations is
x + y + z = 6
x + 3z = 7
3x + y + z = 12
These equations can be written in matrix form as :
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2B 30
By equality of matrices,
x + y + z = 6 …(1)
-y + 2z = 1 …(2)
-3y = -5 …(3)
From (3), y = \(\frac{5}{3}\)
Substituting y = \(\frac{5}{3}\) in (2), we get,
–\(\frac{5}{3}\) + 2z = 1
∴ 2z = 1 + \(\frac{5}{3}\) = \(\frac{8}{3}\)
∴ z = \(\frac{4}{3}\)
Substituting y =\(\frac{5}{3}\), z = \(\frac{5}{3}\) in (1), we get,
x + \(\frac{5}{3}+\frac{4}{3}\) = 6
∴ x = 3
∴ x = 3, y = \(\frac{5}{3}\), z = \(\frac{4}{3}\)
Hence, the required numbers are 3, \(\frac{5}{3}\) and \(\frac{4}{3}\).

Question 6.
The sum of three numbers is 2. If twice the second number is added to the sum of first and third number, we get 1 adding five times the first number to the sum of second and third we get 6. Find the three numbers by using matrices.
Solution:
Let the three numbers be x, y and z.
According to the question,
x + y + 2
x + 2y + z = 1
5x + y + z = 6
The given system of equations can be written in matrix form as follows:
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2B 33
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2B 34
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2B 35

Maharashtra Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

Question 7.
An amount of ₹ 5000 is invested in three types of investments, at interest rates 6%, 7%, 8% per annum respectively. The total annual income from this investment is ₹ 350. If the total annual income from the first two investments is ₹ 70 more than the income from the third, find the amount of each investment using the matrix method.
Solution:
Let the amounts in three investments by ₹ x, ₹ y, and ₹ z respectively.
Then x + y + z = 5000
Since the rate of interest in these investments are 6%, 7% and 8% respectively, the annual income of the three investments are \(\frac{6 x}{100}\), \(\frac{7 y}{100}\) and \(\frac{8 z}{100}\) respectively.
According to the given conditions,
\(\frac{6 x}{100}+\frac{7 y}{100}+\frac{8 z}{100}\) = 350
i.e. 6x + 7y + 8z = 35000
Also, \(\frac{6 x}{100}+\frac{7 y}{100}\) = \(\frac{8 z}{100}\) + 70
i.e. 6x + 7y – 8z = 7000
Hence, the system of linear equation is
x + y + z = 5000
6x + 7y + 8z = 35000
6x + 7y – 8z = 7000
These equations can be written in matrix form as :
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2B 31
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2B 32
By equality of matrices,
x + y + z = 5000 …(1)
y + 2z = 5000 …(2)
-16z = -28000 ….(3)
From (3), z = 1750
Substituting z = 1750 in (2), we get,
y + 2(1750) = 5000
∴ y = 5000 – 3500 = 1500
Substituting y = 1500, z = 1750 in (1), we get,
x + 1500 + 1750 = 5000
∴ x = 5000 – 3250 = 1750
∴ x = 1750, y = 1500, z = 1750
Hence, the amounts of the three investments are ₹ 1750, ₹ 1500 and ₹ 1750 respectively.

Maharashtra Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

Question 8.
The sum of the costs of one ook each of Mathematics, Physics and Chemistry is ₹ 210. Total cost of a mathematics book, 2 physics books, and a chemistry book is ₹ 240 Also the total cost of a Mathematics book, 3 physics book and chemistry books is Rs. 300/-. Find the cost of each book, using Matrices.

Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A

Balbharti 12th Maharashtra State Board Maths Solutions Book Pdf Chapter 2 Matrices Miscellaneous Exercise 2A Questions and Answers.

Maharashtra State Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A

Question 1.
If A = \(\left[\begin{array}{lll}
1 & 0 & 0 \\
2 & 1 & 0 \\
3 & 3 & 1
\end{array}\right]\) then reduce it to I3 by using column transformations.
Solution:
|A| = \(\left|\begin{array}{lll}
1 & 0 & 0 \\
2 & 1 & 0 \\
3 & 3 & 1
\end{array}\right|\)
= 1(1 – 0) – 0 + 0 = 1 ≠ 0
∴ A is a non-singular matrix.
Hence, the required transformation is possible.
Now, A = \(\left[\begin{array}{lll}
1 & 0 & 0 \\
2 & 1 & 0 \\
3 & 3 & 1
\end{array}\right]\)
By C1 – 2C2, we get, A ~ \(\left[\begin{array}{lll}
1 & 0 & 0 \\
0 & 1 & 0 \\
-3 & 3 & 1
\end{array}\right]\)
By C1 + 3C3 and C2 – 3C3, we get,
A ~ \(\left[\begin{array}{lll}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right]\) = I3.

Question 2.
If A = \(\left[\begin{array}{lll}
2 & 1 & 3 \\
1 & 0 & 1 \\
1 & 1 & 1
\end{array}\right]\), then reduce it to I3 by using row transformations.
Solution:
|A| = \(\left|\begin{array}{lll}
2 & 1 & 3 \\
1 & 0 & 1 \\
1 & 1 & 1
\end{array}\right|\)
= 2 (0 – 1) – 1(1 – 1) + 3 (1 – 0)
= -2 – 0 + 3 = 1 ≠ 0
∴ A is a non-singular matrix.
Hence, the required transformation is possible.
Now, A = \(\left[\begin{array}{lll}
2 & 1 & 3 \\
1 & 0 & 1 \\
1 & 1 & 1
\end{array}\right]\)
By R1 – R2, we get,
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 1
By R1 – R3 and By R2 – R3, we get
A ~ \(\left[\begin{array}{lll}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right]\) = I3.

Maharashtra Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

Question 3.
Check whether the following matrices are invertible or not:
(i) \(\left[\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right]\)
Solution:
Let A = \(\left[\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right]\)
Then, |A| = \(\left|\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right|\) = 1 – 0 = 1 ≠ 0.
∴ A is a non-singular matrix.
Hence, A-1 exists.

(ii) \(\left[\begin{array}{ll}
1 & 1 \\
1 & 1
\end{array}\right]\)
Solution:
Let A = \(\left[\begin{array}{ll}
1 & 1 \\
1 & 1
\end{array}\right]\)
Then, |A| = \(\left|\begin{array}{ll}
1 & 1 \\
1 & 1
\end{array}\right|\) = 1 – 1 = 0.
∴ A is a singular matrix.
Hence, A-1 does not exist.

(iii) \(\left[\begin{array}{ll}
1 & 2 \\
3 & 3
\end{array}\right]\)
Solution:
Let A = \(\left[\begin{array}{ll}
1 & 2 \\
3 & 3
\end{array}\right]\)
Then, |A| = \(\left|\begin{array}{ll}
1 & 2 \\
3 & 3
\end{array}\right|\) = 3 – 6 = -3 ≠ 0.
∴ A is a non-singular matrix.
Hence, A-1 exist.

Maharashtra Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

(iv) \(\left[\begin{array}{ll}
2 & 3 \\
10 & 15
\end{array}\right]\)
Solution:
Let A = \(\left[\begin{array}{ll}
2 & 3 \\
10 & 15
\end{array}\right]\)
Then, |A| = \(\left|\begin{array}{ll}
2 & 3 \\
10 & 15
\end{array}\right|\) = 30 – 30 = 0.
∴ A is a singular matrix.
Hence, A-1 does not exist.

(v) \(\left[\begin{array}{rr}
\cos \theta & \sin \theta \\
-\sin \theta & \cos \theta
\end{array}\right]\)
Solution:
Let A = \(\left[\begin{array}{rr}
\cos \theta & \sin \theta \\
-\sin \theta & \cos \theta
\end{array}\right]\)
Then, |A| = \(\left|\begin{array}{cc}
\sec \theta & \tan \theta \\
\tan \theta & \sec \theta
\end{array}\right|\)
= sec2θ – tan2θ = 1 ≠ 0.
∴ A is a non-singular matrix.
Hence, A-1 exist.

(vii) \(\left[\begin{array}{lll}
3 & 4 & 3 \\
1 & 1 & 0 \\
1 & 4 & 5
\end{array}\right]\)
Solution:
let A = \(\left[\begin{array}{lll}
3 & 4 & 3 \\
1 & 1 & 0 \\
1 & 4 & 5
\end{array}\right]\)
Then, |A| = \(\left|\begin{array}{lll}
3 & 4 & 3 \\
1 & 1 & 0 \\
1 & 4 & 5
\end{array}\right|\)
= 3(5 – 0) – 4(5 – 0) + 3(4 – 1)
= 15 – 20 + 9 = 4 ≠ 0
∴ A is a non-singular matrix.
Hence, A-1 exist.

Maharashtra Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

(viii) \(\left[\begin{array}{lll}
1 & 2 & 3 \\
2 & -1 & 3 \\
1 & 2 & 3
\end{array}\right]\)
Solution:
Let A = \(\left[\begin{array}{lll}
1 & 2 & 3 \\
2 & -1 & 3 \\
1 & 2 & 3
\end{array}\right]\)
Then, |A| = \(\left|\begin{array}{lll}
1 & 2 & 3 \\
2 & -1 & 3 \\
1 & 2 & 3
\end{array}\right|\)
= 1 (-3 -6) – 2 (6 – 3) + 3 (4 + 1)
= -9 – 6 + 15 = 0
∴ A is a singular matrix.
Hence, A-1 does not exist.

(ix) \(\left[\begin{array}{lll}
1 & 2 & 3 \\
3 & 4 & 5 \\
4 & 6 & 8
\end{array}\right]\)
Solution:
Let A = \(\left[\begin{array}{lll}
1 & 2 & 3 \\
3 & 4 & 5 \\
4 & 6 & 8
\end{array}\right]\)
Then, |A| = \(\left|\begin{array}{lll}
1 & 2 & 3 \\
3 & 4 & 5 \\
4 & 6 & 8
\end{array}\right|\)
= 1(32 – 30) – 2(24 – 20) + 3(18 – 16)
= 2 – 8 + 6 = 0
∴ A is a singular matrix.
Hence, A-1 does not exist.

Maharashtra Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

Question 4.
Find AB, if A = \(\left[\begin{array}{ccc}
1 & 2 & 3 \\
1 & -2 & -3
\end{array}\right]\) and B = \(\left[\begin{array}{cc}
1 & -1 \\
1 & 2 \\
1 & -2
\end{array}\right]\) Examine whether AB has inverse or not.
Solution:
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 2
∴ A is a non-singular matrix.
Hence, (AB)-1 exist.

Question 5.
If A = \(\left[\begin{array}{lll}
x & 0 & 0 \\
0 & y & 0 \\
0 & 0 & z
\end{array}\right]\) is a nonsingular matrix then find A-1 by elementary row transformations.
Hence, find the inverse of \(\left[\begin{array}{lll}
2 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & -1
\end{array}\right]\)
Solution:
Since A is a non-singular matrix, then find A-1 by using elementary row transformations.
We write AA-1 = I
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 3
Comparing \(\left[\begin{array}{lll}
2 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & -1
\end{array}\right]\) with \(\left[\begin{array}{lll}
x & 0 & 0 \\
0 & y & 0 \\
0 & 0 & z
\end{array}\right]\),
we get, x = 2, y = 1, z = -1
∴ \(\frac{1}{x}\) = \(\frac{1}{2}\), \(\frac{1}{y}\) = \(\frac{1}{1}\) = 1, \(\frac{1}{z}\) = \(\frac{1}{-1}\) = -1
\(\left[\begin{array}{lll}
2 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & -1
\end{array}\right]\) is \(\left(\begin{array}{rrr}
\frac{1}{2} & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & -1
\end{array}\right)\).

Maharashtra Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

Question 6.
if A = \(\left[\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right]\) and X is a 2 × 2 matrix such that AX = I , then find X.
Solution:
We will reduce the matrix A to the identity matrix by using row transformations. During this pro¬cess, I will be converted to the matrix X.
We have AX = I.
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 4

Question 7.
Find the inverse of each of the following matrices (if they exist).
(i) \(\left[\begin{array}{ll}
1 & -1 \\
2 & 3
\end{array}\right]\)
Solution:
Let A = \(\left[\begin{array}{ll}
1 & -1 \\
2 & 3
\end{array}\right]\)
∴ |A| = \(\left|\begin{array}{ll}
1 & -1 \\
2 & 3
\end{array}\right|\) = 3 + 2 = 5 ≠ 0
∴ A-1 exists.
Consider AA-1 = I
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 5
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 6

(ii) \(\left[\begin{array}{ll}
2 & 1 \\
1 & -1
\end{array}\right]\)
Solution:
Let A = \(\left[\begin{array}{ll}
2 & 1 \\
1 & -1
\end{array}\right]\)
∴ |A| = \(\left|\begin{array}{ll}
2 & 1 \\
1 & -1
\end{array}\right|\) = -2 – 1 = -3 ≠ 0
∴ A-1 exists.
Consider AA-1 = I
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 7

Maharashtra Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

(iii) \(\left[\begin{array}{ll}
1 & 3 \\
2 & 7
\end{array}\right]\)
Solution:
Let A = \(\left[\begin{array}{ll}
1 & 3 \\
2 & 7
\end{array}\right]\)
∴ |A| = \(\left|\begin{array}{ll}
1 & 3 \\
2 & 7
\end{array}\right|\) = 7 – 6 = 1 ≠ 0
∴ A-1 exists.
Consider AA-1 = I
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 8

(iv) \(\left[\begin{array}{ll}
2 & -3 \\
5 & 7
\end{array}\right]\)
Solution:
Let A = \(\left[\begin{array}{ll}
2 & -3 \\
5 & 7
\end{array}\right]\)
∴ |A| = \(\left|\begin{array}{ll}
2 & -3 \\
5 & 7
\end{array}\right|\) = 14 + 15 = 29 ≠ 0
∴ A-1 exists.
Consider AA-1 = I
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 9
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 10

Maharashtra Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

(v) \(\left[\begin{array}{ll}
2 & 1 \\
7 & 4
\end{array}\right]\)
Solution:
Let A = \(\left[\begin{array}{ll}
2 & 1 \\
7 & 4
\end{array}\right]\)
∴ |A| = \(\left|\begin{array}{ll}
2 & 1 \\
7 & 4
\end{array}\right|\) = 8 – 7 = 1 ≠ 0
∴ A-1 exists.
Consider AA-1 = I
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 11

(vi) \(\left[\begin{array}{ll}
3 & -10 \\
2 & -7
\end{array}\right]\)
Solution:
Let A = \(\left[\begin{array}{ll}
3 & -10 \\
2 & -7
\end{array}\right]\)
∴ |A| = \(\left|\begin{array}{ll}
3 & -10 \\
2 & -7
\end{array}\right|\) = -21 + 20 = -1 ≠ 0
∴ A-1 exists.
Consider AA-1 = I
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 12
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 13

Maharashtra Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

(vii) \(\left[\begin{array}{lll}
2 & -3 & 3 \\
2 & 2 & 3 \\
3 & -2 & 2
\end{array}\right]\)
Solution:
Let A = \(\left[\begin{array}{lll}
2 & -3 & 3 \\
2 & 2 & 3 \\
3 & -2 & 2
\end{array}\right]\)
∴ |A| = \(\left|\begin{array}{lll}
2 & -3 & 3 \\
2 & 2 & 3 \\
3 & -2 & 2
\end{array}\right|\)
= 2(4 + 6) +3(4 – 9) + 3(-4 – 6)
= 20 – 15 – 30 = -25 ≠ 0
∴ A-1 exists.
Consider AA-1 = I
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 14
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 15
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 16
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 17

(viii) \(\left[\begin{array}{lll}
1 & 3 & -2 \\
-3 & 0 & -5 \\
2 & 5 & 0
\end{array}\right]\)
Solution:
Let A = \(\left[\begin{array}{lll}
1 & 3 & -2 \\
-3 & 0 & -5 \\
2 & 5 & 0
\end{array}\right]\)
∴ |A| = \(\left|\begin{array}{lll}
1 & 3 & -2 \\
-3 & 0 & -5 \\
2 & 5 & 0
\end{array}\right|\)
= 1(0 + 25) + 3(0 + 10) + 2(-15 – 0)
= 25 + 30 -30
= 25 ≠ 0
∴ A-1 exists.
Consider AA-1 = I
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 18
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 19
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 20
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 21

(ix) \(\left[\begin{array}{lll}
2 & 0 & -1 \\
5 & 1 & 0 \\
0 & 1 & 3
\end{array}\right]\)
Solution:
Let A =\(\left[\begin{array}{lll}
2 & 0 & -1 \\
5 & 1 & 0 \\
0 & 1 & 3
\end{array}\right]\)
∴ |A| = \(\left|\begin{array}{lll}
2 & 0 & -1 \\
5 & 1 & 0 \\
0 & 1 & 3
\end{array}\right|\)
= 2(3 – 0) – 0 – 1(5 – 0)
= 6 – 0 – 5 = 1 ≠ 0
∴ A-1 exists.
Consider AA-1 = I
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 22
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 23
∴ A-1 = \(\left[\begin{array}{lll}
3 & -1 & 1 \\
-15 & 6 & -5 \\
5 & -2 & 2
\end{array}\right]\)

Maharashtra Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

(x) \(\left[\begin{array}{lll}
1 & 2 & -2 \\
0 & -2 & 1 \\
-1 & 3 & 0
\end{array}\right]\)
Solution:
Let A = \(\left[\begin{array}{lll}
1 & 2 & -2 \\
0 & -2 & 1 \\
-1 & 3 & 0
\end{array}\right]\)
∴ A-1 = \(\left[\begin{array}{lll}
1 & 2 & -2 \\
0 & -2 & 1 \\
-1 & 3 & 0
\end{array}\right]\)
= 1\(\left|\begin{array}{ll}
-2 & 1 \\
3 & 0
\end{array}\right|\) – 2\(\left|\begin{array}{ll}
0 & 1 \\
-1 & 1
\end{array}\right|\) – 2\(\left|\begin{array}{ll}
0 & -2 \\
-1 & 3
\end{array}\right|\)
|A| = 1(0 – 3) – 2(0 + 1) – 2(0 – 2)
= -3 – 2 + 4
= -1 ≠ 0
∴ A-1 exists.
We have
AA-1 = I
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 24
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 25
∴ A-1 = \(\left[\begin{array}{lll}
3 & 6 & 2 \\
1 & 2 & 1 \\
2 & 5 & 2
\end{array}\right]\)

Question 8.
Find the inverse of A = \(\left[\begin{array}{ccc}
\cos \theta & -\sin \theta & 0 \\
\sin \theta & \cos \theta & 0 \\
0 & 0 & 1
\end{array}\right]\) by
(i) elementary row transformations
Solution:
|A| = \(\left|\begin{array}{ccc}
\cos \theta & -\sin \theta & 0 \\
\sin \theta & \cos \theta & 0 \\
0 & 0 & 1
\end{array}\right|\)
= cosθ (cosθ – 0) + sinθ (sinθ – 0) + 0
= cos2θ + sin2θ = 1 ≠ 0
∴ A-1 exists.
(i) Consider AA-1 = I
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 26
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 27

Maharashtra Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

(ii) elementary column transformations
Solution:
Consider A-1A = I
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 28
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 29

Question 9.
If A = \(\left[\begin{array}{ll}
2 & 3 \\
1 & 2
\end{array}\right]\), B = \(\left[\begin{array}{ll}
1 & 0 \\
3 & 1
\end{array}\right]\) find AB and (AB)-1. Verify that (AB)-1 = B-1A-1
Solution:
AB = \(\left[\begin{array}{ll}
2 & 3 \\
1 & 2
\end{array}\right]\) \(\left[\begin{array}{ll}
1 & 0 \\
3 & 1
\end{array}\right]\)
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 30
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 31
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 32
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 33
From (1) and (2), (AB)-1 = B-1 ∙ A-1.

Question 10.
If A = \(\left[\begin{array}{ll}
4 & 5 \\
2 & 1
\end{array}\right]\), then show that A-1 = \(\frac{1}{6}\)(A – 5I)
Solution:
|A| = \(\left|\begin{array}{ll}
4 & 5 \\
2 & 1
\end{array}\right|\) = 4 – 10 = -6 ≠ 0
∴ A-1 exists.
Consider AA-1 = I
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 34
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 35
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 36

Maharashtra Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

Question 11.
Find matrix X such that AX = B, where A = \(\left[\begin{array}{ll}
1 & 2 \\
-1 & 3
\end{array}\right]\) and B = \(\left[\begin{array}{ll}
0 & 1 \\
2 & 4
\end{array}\right]\)
Solution:
AX = B
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 37

Question 12.
Find X, if AX = B where A = \(\left[\begin{array}{lll}
1 & 2 & 3 \\
-1 & 1 & 2 \\
1 & 2 & 4
\end{array}\right]\) and B = \(\left[\begin{array}{l}
1 \\
2 \\
3
\end{array}\right]\).
Solution:
AX = B
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 38
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 39

Question 13.
If A = \(\left[\begin{array}{ll}
1 & 1 \\
1 & 2
\end{array}\right]\), B = \(\left[\begin{array}{ll}
4 & 1 \\
3 & 1
\end{array}\right]\) and C = \(\left[\begin{array}{ll}
24 & 7 \\
31 & 9
\end{array}\right]\) then find matrix X such that AXB = C.
Solution:
AXB = C
∴ \(\left(\begin{array}{ll}
1 & 1 \\
1 & 2
\end{array}\right)(\mathrm{XB})\) =\(\left[\begin{array}{ll}
24 & 7 \\
31 & 9
\end{array}\right]\)
First we perform the row transformations.
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 40
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 41

Maharashtra Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

Question 14.
Find the inverse of \(\left[\begin{array}{lll}
1 & 2 & 3 \\
1 & 1 & 5 \\
2 & 4 & 7
\end{array}\right]\) by adjoint method.
Solution:
Let A = \(\left[\begin{array}{lll}
1 & 2 & 3 \\
1 & 1 & 5 \\
2 & 4 & 7
\end{array}\right]\)
∴ |A| = \(\left|\begin{array}{lll}
1 & 2 & 3 \\
1 & 1 & 5 \\
2 & 4 & 7
\end{array}\right|\)
= 1(7 – 20) – 2(7 – 10) + 3(4 – 2)
= -13 + 6 + 6 = -1 ≠ 0
∴ A-1 exists.
First we have to find the cofactor matrix
= [Aij]3×3 where Aij = (-1)i+jMij
Now, A11 = (-1)1+1M11 = \(\left|\begin{array}{ll}
1 & 5 \\
4 & 7
\end{array}\right|\) = 7 – 20 = -13
A12 = (-1)1+2M12 = \(\left|\begin{array}{ll}
1 & 5 \\
2 & 7
\end{array}\right|\) = -(7 – 10) = 3
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 42
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 43

Question 15.
Find the inverse of \(\left[\begin{array}{lll}
1 & 0 & 1 \\
0 & 2 & 3 \\
1 & 2 & 1
\end{array}\right]\) by adjoint method.
Solution:
where A = \(\left[\begin{array}{lll}
1 & 0 & 1 \\
0 & 2 & 3 \\
1 & 2 & 1
\end{array}\right]\)
|A| = 1(2 – 6) – 0(0 – 3) + 1(0 – 2)
|A| = -4 – 2
|A| = -6 ≠ 0
∴ A-1 exists.
First we have to find the cofactor matrix
= [Aij]3×3, where Aij = (-1)i+jMij
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 44
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 45

Maharashtra Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

Question 16.
Find A-1 by adjoint method and by elementary transformations if A = \(\left[\begin{array}{lll}
1 & 2 & 3 \\
-1 & 1 & 2 \\
1 & 2 & 4
\end{array}\right]\)
Solution:
|A| = \(\left|\begin{array}{lll}
1 & 2 & 3 \\
-1 & 1 & 2 \\
1 & 2 & 4
\end{array}\right|\)
= 1(4 – 4) – 2(-4 – 2) + 3(-2 – 1)
= 0 + 12 – 9 = 3 ≠ 0
∴ A-1 exists.
A-1by adjoint method :
We have to find the cofactor matrix
= [Aij]3×3, where Aij = (-1)i+j Mij
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 46
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 47
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 48
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 49

Question 17.
Find the inverse of A = \(\left[\begin{array}{lll}
1 & 0 & 1 \\
0 & 2 & 3 \\
1 & 2 & 1
\end{array}\right]\) by elementary column transformations.
Solution:
|A| = \(\left|\begin{array}{lll}
1 & 0 & 1 \\
0 & 2 & 3 \\
1 & 2 & 1
\end{array}\right|\)
= 1 (2 – 6) – 0 + 1 (0 – 2)
= -4 – 2= -6 ≠ 0
∴ A-1 exists.
Consider A-1A = I
∴ A-1\(\left[\begin{array}{lll}
1 & 0 & 1 \\
0 & 2 & 3 \\
1 & 2 & 1
\end{array}\right]\) = \(\left[\begin{array}{lll}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right]\)
By C3 – C1, we get,
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 50
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 51

Maharashtra Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

Question 18.
Find the inverse of \(\left[\begin{array}{lll}
1 & 2 & 3 \\
1 & 1 & 5 \\
2 & 4 & 7
\end{array}\right]\) by elementary row transformations.
Solution:
Let A = \(\left[\begin{array}{lll}
1 & 2 & 3 \\
1 & 1 & 5 \\
2 & 4 & 7
\end{array}\right]\)
∴ |A| = \(\left|\begin{array}{lll}
1 & 2 & 3 \\
1 & 1 & 5 \\
2 & 4 & 7
\end{array}\right|\)
= 1(7 – 20) – 2(7 – 10) + 3(4 – 2)
= -13 + 6 + 6 = -1 ≠ 0
∴ A-1 exists.
Consider AA-1 = I
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 52
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 53

Question 19.
Show with usual notations that for any matrix A = [aij]3×3
(i) a11A21 + a12A22 + a13A23 = 0
Solution:
A = [aij]3×3 = \(\left[\begin{array}{lll}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{array}\right]\)
(i) A21 = (-1)2+1M21 = \(-\left|\begin{array}{ll}
a_{12} & a_{13} \\
a_{32} & a_{33}
\end{array}\right|\)
= -(a12a33 – a13a32)
= -a12a33 + a13a32
A22 = (-1)2+2M22 = \(\left|\begin{array}{ll}
a_{11} & a_{13} \\
a_{31} & a_{33}
\end{array}\right|\)
= a11a33 – a13a31
A23 = (-1)2+3M23 = \(-\left|\begin{array}{ll}
a_{11} & a_{12} \\
a_{31} & a_{32}
\end{array}\right|\)
= -(a11a32 – a12a31)
= -a11a32+ a12a31
∴ a11A21 + a12A22 + a13A23
= a11(-a1233 + a13a32) + a12(a11a33 – a13a31) + a13(-a11a32 + a12a31)
= -a11a12a33 + a11a13a32 + a11a12a33 – a12a13a31 – a11a13a32 + a12a13a31
= 0

Maharashtra Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

(ii) a11A11 + a12A12 + a13A13 = |A|
Solution:
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 54

Question 20.
If A = \(\left[\begin{array}{lll}
1 & 0 & 1 \\
0 & 2 & 3 \\
1 & 2 & 1
\end{array}\right]\) and B = \(\left[\begin{array}{lll}
1 & 2 & 3 \\
1 & 1 & 5 \\
2 & 4 & 7
\end{array}\right]\), then find a matrix X such that XA= B.
Solution:
Consider XA = B
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 55
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Miscellaneous Exercise 2A 56

Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Ex 2.3

Balbharti 12th Maharashtra State Board Maths Solutions Book Pdf Chapter 2 Matrices Ex 2.3 Questions and Answers.

Maharashtra State Board 12th Maths Solutions Chapter 2 Matrices Ex 2.3

Question 1.
Solve the following equations by the inversion method.
(i) x + 2y = 2, 2x + 3y = 3
Solution:
The given equations can be written in the matrix form as :
\(\left[\begin{array}{ll}
1 & 2 \\
2 & 3
\end{array}\right]\left[\begin{array}{l}
x \\
y
\end{array}\right]\) = \(\left[\begin{array}{l}
2 \\
3
\end{array}\right]\)
This is of the form AX = B, where
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Ex 2.3 1
∴ A-1 = \(\left[\begin{array}{rr}
-3 & 2 \\
2 & -1
\end{array}\right]\)
Now, premultiply AX = B by A-1, we get,
A-1(AX) = A-1B
∴ (A-1A)X = A-1B
∴ IX = A-1B
∴ X = \(=\left[\begin{array}{rr}
-3 & 2 \\
2 & -1
\end{array}\right]\left[\begin{array}{l}
2 \\
3
\end{array}\right]\)
∴ \(\left[\begin{array}{l}
x \\
y
\end{array}\right]\) = \(=\left[\begin{array}{r}
-6+6 \\
4-3
\end{array}\right]\) = \(=\left[\begin{array}{l}
0 \\
1
\end{array}\right]\)
By equality of matrices,
x = 0, y = 1 is the required solution.

Maharashtra Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

(ii) x + y = 4, 2x – y = 5
Solution:
x + y = 4, 2x – y = 5
The given equations can be written in the matrix form as:
\(\left[\begin{array}{cc}
1 & 1 \\
2 & -1
\end{array}\right]\left[\begin{array}{l}
x \\
y
\end{array}\right]\) = \(\left[\begin{array}{l}
4 \\
5
\end{array}\right]\)
This is of the form AX = B ⇒ X ⇒ A-1B
A = \(\left[\begin{array}{cc}
1 & 1 \\
2 & -1
\end{array}\right]\)
|A| = -1 – 2 = -3 ≠ 0
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Ex 2.3 5
By equality of matrices.
x = 3, y = 1

(iii) 2x + 6y = 8, x + 3y = 5
Solution:
The given equations can be written in the matrix form as :
\(\left[\begin{array}{ll}
2 & 6 \\
1 & 3
\end{array}\right]\left[\begin{array}{l}
x \\
y
\end{array}\right]=\left[\begin{array}{l}
8 \\
5
\end{array}\right]\)
This is of the form AX = B, where
A = \(\left[\begin{array}{ll}
2 & 6 \\
1 & 3
\end{array}\right]\), X = \(\left[\begin{array}{l}
x \\
y
\end{array}\right]\) and B = \(\left[\begin{array}{l}
8 \\
5
\end{array}\right]\)
Let us find A-1.
|A| = \(\left|\begin{array}{ll}
2 & 6 \\
1 & 3
\end{array}\right|\) = 6 – 6 = 0
∴ A-1 does not exist.
Hence, x and y do not exist.

Maharashtra Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

Question 2.
Solve the following equations by reduction method.
(i) 2x + y = 5, 3x + 5y = -3
Solution:
The given equations can be written in the matrix form as :
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Ex 2.3 2
By equality of matrices,
2x + y = 5 …(1)
7y = -21 …(2)
From (2), y = -3
Substituting y = -3 in (1), we get,
2x – 3 = 5
∴ 2x = 8 ∴ x = 4
Hence, x = 4, y = -3 is the required solution.

(ii) x + 3y = 2, 3x + 5y = 4.
Solution:
The given equations can be written in the matrix form as :
\(\left[\begin{array}{ll}
1 & 3 \\
3 & 5
\end{array}\right]\left[\begin{array}{l}
x \\
y
\end{array}\right]\) = \(\left[\begin{array}{l}
2 \\
4
\end{array}\right]\)
By R2 – 3R1, we get
\(\left[\begin{array}{rr}
1 & 3 \\
0 & -4
\end{array}\right]\left[\begin{array}{l}
x \\
y
\end{array}\right]\) = \(\left(\begin{array}{r}
2 \\
-2
\end{array}\right)\)
∴ \(\left[\begin{array}{l}
x+3 \\
0-4 y
\end{array}\right]\) = \(\left[\begin{array}{r}
2 \\
-2
\end{array}\right]\)
By equality of matrices,
x + 3y = 2 …(1)
-4y = -2
From (2), y = \(\frac{1}{2}\)
Substituting y = \(\frac{1}{2}\) in (1), we get,
x + \(\frac{3}{2}\) = 2
∴ x = 2 – \(\frac{3}{2}=\frac{1}{2}\)
Hence, x = \(\frac{1}{2}\), y = \(\frac{1}{2}\) is the required solution.

Maharashtra Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

(iii) 3x – y = 1, 4x + y = 6
Solution:
The given equations can be written in the matrix form as :
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Ex 2.3 3
By equality of matrices,
12x – 4y = 4 … (1)
7y = 14 … (2)
From (2), y = 2
Substituting y = 2 in (1), we get,
12x – 8 = 4
∴ 12x = 12 ∴ x = 1
Hence, x = 1, y = 2 is the required solution.

(iv) 5x + 2y = 4, 7x + 3y = 5
Solution:
5x + 2y = 4 ………..(1)
7x + 3y = 5 …………(2)
Multiplying Eq. (1) with 7 and Eq. (2) with 5
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Ex 2.3 6
Put y = -3 into Eq. (1)
5x + 2y = 4
5x + 2(-3) = 4
5x – 6 = 4
5x = 4 + 6
5x = 10
x = \(\frac{10}{5}\)
x = 2
Hence, x = 2, y = -3 is the required solution.

Maharashtra Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

Question 3.
The cost of 4 pencils, 3 pens and 2 erasers is ₹ 60. The cost of 2 pencils, 4 pens and 6 erasers is ₹ 90, whereas the cost of 6 pencils, 2 pens and 3 erasers is ₹ 70. Find the cost of each item by using matrices.
Solution:
Let the cost of 1 pencil, 1 pen and 1 eraser be ₹ x, ₹ y and ₹ z respectively.
Then, from the given conditions,
4x + 3y + 2z = 60
2x + 4y + 6z = 90, i.e., x + 2y + 3z = 45
6x + 2y + 3z = 70
These equations can be written in the matrix form as :
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Ex 2.3 4
By equality of matrices,
x + 2y + 3z = 45 …….(1)
– 5y – 10z = – 120 …….(2)
5z = 40
From (3), z = 8
Substituting z = 8 in (2), we get,
– 5y – 80 = -120
∴ – 5y = -40 ∴ y = 8
Substituting y = 8, z = 8 in (1), we get,
x + 16 + 24 = 45
∴ x + 40 = 45 ∴ x = 5
∴ x = 5, y = 8, z = 8
Hence, the cost is ₹ 5 for a pencil, ₹ 8 for a pen and ₹ 8 for an eraser.

Maharashtra Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

Question 4.
If three numbers are added, their sum is 2. If 2 times the second number is subtracted from the sum of first and third numbers, we get 8 and if three times the first number is added to the sum of second and third numbers, we get 4. Find the numbers using matrices.
Solution:
Let the three numbers be x, y and z. According to the given conditions,
x + y + z = 2
x + z – 2y = 8, i.e., x – 2y + 2 = 8
and y + z + 3x = 4, i.e., 3x + y + z = 4
Hence, the system of linear equations is
x + y + z = 2
x – 2y + z = 8
3x + y + z = 4
These equations can be written in the matrix form as :
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Ex 2.3 7
By equality of matrices,
x + y + z = 2 ……(1)
-3y = 6 ……(2)
– 2y – 2z = -2 ……..(3)
From (2), y = -2
Substituting y = -2 in (3), we get,
-2(-2) – 2z = -2
∴ -2z = -6 ∴ z = 3
Substituting y = -2, z = 3 in (1), we get,
x – 2 + 3 = 2 ∴ x = 1
Hence, the required numbers are 1, -2 and 3.

Maharashtra Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

Question 5.
The total cost of 3 T.V. sets and 2 V.C.R.s is ₹ 35000. The shop-keeper wants profit of ₹ 1000 per television and ₹ 500 per V.C.R. He can sell 2 T. V. sets and 1 V.C.R. and get the total revenue as ₹ 21,500. Find the cost price and the selling price of a T.V. sets and a V.C.R.
Solution:
Let the cost of each T.V. set be ₹ x and each V.C.R. be ₹ y. Then the total cost of 3 T.V. sets and 2 V.C.R.’s is ₹ (3x + 2y) which is given to be ₹ 35,000.
∴ 3x + 2y = 35000
The shopkeeper wants profit of ₹ 1000 per T.V. set and of ₹ 500 per V.C.R.
∴ the selling price of each T.V. set is ₹ (x + 1000) and of each V.C.R. is ₹ (y + 500).
∴ selling price of 2 T.V. set and 1 V.C.R. is
₹ [2(x + 1000) + (y + 500)] which is given to be ₹ 21,500.
∴ 2(x + 1000) + (y + 500) = 21500
∴ 2x + 2000 + y + 500 = 21500
∴ 2x + y = 19000
Hence, the system of linear equations is
3x + 2y = 35000
2x + y = 19000
These equations can be written in the matrix form as :
Maharashtra Board 12th Maths Solutions Chapter 2 Matrices Ex 2.3 8
By equality of matrices,
2x + y = 19000 ……….(1)
-x = -3000 ……….(2)
From (2), x = 3000
Substituting x = 3000 in (1), we get,
2(3000) + y = 19000
∴ y = 13000
∴ the cost price of one T.V. set is ₹ 3000 and of one V.C.R. is ₹ 13000 and the selling price of one T.V. set is ₹ 4000 and of one V.C.R. is ₹ 13500.

Maharashtra Board Class 11 Chemistry Important Questions Chapter 13 Nuclear Chemistry and Radioactivity

Balbharti Maharashtra State Board 11th Chemistry Important Questions Chapter 13 Nuclear Chemistry and Radioactivity Important Questions and Answers.

Maharashtra State Board 11th Chemistry Important Questions Chapter 13 Nuclear Chemistry and Radioactivity

Question 1.
Explain the term nuclear chemistry. Give few examples of nuclear reactions.
Answer:
Nuclear chemistry is a branch of physical chemistry and it deals with the study of reactions involving changes in atomic nuclei. This branch started with the discovery of natural radioactivity by physicist Antoine Henri Becquerel.

Examples of nuclear reactions are as follows:

  • Radioactive decay
  • Artificial transmutation
  • Nuclear fission
  • Nuclear fusion

Question 2.
Write a short note on similarity between the solar system and structure of atom.
Answer:
Solar system: It consists of the Sun and planets in which Sun is at the centre of solar system and planets move around it under the force of gravity.

Atomic system: It consists of tiny central core called as nucleus at the centre of atom around which electrons are present. Like in solar system, electrostatic attractions hold subatomic particles in a structure of atom. The nucleus consists of protons and neutrons.

Question 3.
Answer the following.
i. Give the symbolic representation for calcium, (no. of protons = 20, mass number = 40)
ii. Calculate the number of neutrons for calcium.
Answer:
i. \({ }_{20}^{40} \mathrm{Ca}\), in which Z = 20 and A = 40.
ii. Number of neutrons: It can be calculated from formula (A = Z + N).
For calcium, N = A – Z = 40 – 20 = 20
Nucleus of the calcium atom contains 20 neutrons.

Question 4.
Explain the term nucleons with examples.
Answer:
The term nucleon refers to the sum of protons (p) and neutrons (n) present in atom, e.g. Number of nucleons present in \({ }_{20}^{40} \mathrm{Ca}\) are 40 (i.e., 20 protons and 20 neutrons). Number of nucleons present in \({ }_{11}^{23} \mathrm{Na}\) are 23 (i.e., 11 protons and 12 neutrons).

Maharashtra Board Class 11 Chemistry Important Questions Chapter 13 Nuclear Chemistry and Radioactivity

Question 5.
Define: Nuclide
Answer:
The nucleus of a specific isotope is called as nuclide.

Question 6.
Atom as a whole is electrically neutral. Justify.
Answer:

  • The magnitude of electronic charge (e) on the nucleus is +Ze and that of outer sphere is -Ze. Number of protons and number of electrons are always equal in an atom.
  • As a result of this, the charges get nullified, therefore, the atom as a whole is electrically neutral.

Question 7.
Define:
i. Isotopes
ii. Isobars
Answer:
i. Isotopes: Nuclides which contain same number of protons but different number of neutrons in their nuclei are called as isotopes. e.g. \({ }_{11}^{22} \mathrm{Na}\), \({ }_{11}^{23} \mathrm{Na}\) and \({ }_{11}^{24} \mathrm{Na}\)
ii. Isobars: Nuclides (of different element) which have same mass number but have different number of protons and neutrons in their nuclei are called as isobars.
OR
The atoms of different elements having the same mass number but different atomic numbers are called isobars.
e.g. \({ }_{6}^{14} \mathrm{C}\) and \({ }_{7}^{14} \mathrm{N}\)

Question 8.
Define mirror nuclei and isotones.
Answer:

  • Isobars in which the number of protons and neutrons differ by 1 unit and are interchanged are called as mirror nuclei.
  • Isotones are defined as nuclides having the same number of neutrons but different number of protons and hence, different mass numbers.

Question 9.
Name the following.
i. Nuclides in which number of protons and neutrons differ by 1 and are interchanged.
ii. Nuclides having the same number of neutrons but different number of protons.
iii. Nuclides with the same mass number which differ in energy states.
Answer:
i. Mirror nuclei
ii. Isotones
iii. Nuclear isomers

Maharashtra Board Class 11 Chemistry Important Questions Chapter 13 Nuclear Chemistry and Radioactivity

Question 10.
Explain the term nuclear isomers?
Answer:

  • The nuclides with the same number of protons (Z) and neutrons (N) or the same mass number (A) which differ in energy states are called nuclear isomers.
  • In this, the isomer of higher energy is said to be in the metastable state which is represented by writing “m” after the mass number.
    e.g. Nuclear isomers of cobalt can be represented as, 60mCo and 60Co.

Question 11.
State true or false. Correct the false statement.
i. The number of nucleons in C-12 atom is 6.
ii. N-13 and C-13 are mirror nuclei.
iii. Nuclear isomers have same number of protons and neutrons.
Answer:
i. False,
The number of nucleons in C-12 atom is 12.
ii. True
iii. True

Question 12.
Give classification of nuclides on the basis of nuclear stability.
Answer:
Nuclides can be classified into stable and unstable/radioactive nuclides on the basis of nuclear stability.

  • Stable nuclides: In this type of nuclides, the number of electrons and the location of nuclei may change in outer sphere but the number of protons and neutrons remain unchanged.
  • Radioactive (unstable) nuclides: These nuclides undergo spontaneous change forming new nuclides.

Question 13.

Number of protons (Z) Number of neutrons (N) Number of such nuclides
i. Even Even 165
ii. Even Odd 55

What conclusion can be drawn from the above given data?
Answer:

  • Number of nuclides with even ‘Z’ and even ‘N’ are higher in number as compared to nuclides with even ‘Z’ and odd ‘N’
  • Nuclides with even number of ‘Z’ and odd number of ‘N’ are about 1/3rd of nuclides where both ‘Z’ and ‘N’ are even.
  • Nuclides with even number of protons (Z) and even number of neutrons (N) are most stable. These nuclides tend to fonn proton-proton and neutron-neutron pairs. This impart stability to the nucleus.

Question 14.
Write a note on naturally occurring nuclides with either odd number of protons or odd number of neutrons.
Answer:
i. The number of stable nuclides with either Z or N odd is about one third of nuclides where both are even.
ii. These nuclides are less stable than those having even number of protons and neutrons.
iii. In these nuclides one nucleon has no partner and therefore, these nuclides are less stable.
iv. Further the number of nuclides with odd A are nearly the same, irrespective of Z or N is odd. This indicates that protons and neutrons behave similarly in the respect of stability.
v. Following table gives the estimate of such nuclides occurring in nature.

Number of protons (Z) Number of neutrons (N) Number of such nuclides
i. Even Odd 55
ii. Odd Even 50

Maharashtra Board Class 11 Chemistry Important Questions Chapter 13 Nuclear Chemistry and Radioactivity

Question 15.
State true or false. Correct the false statements.
i. The nuclides with even Z and even N constitute 85% of earth crust.
ii. Nuclides with either ‘Z’ or ‘N’ odd are more stable than those having even number of both ‘Z’ and ‘N’
iii. The number of nuclides with odd number of ‘Z’ and odd number of ‘N’ are only four.
Answer:
i. True
ii. False
Nuclides with either ‘Z’ or ‘N’ odd are less stable than nuclides having even number of both ‘Z’ and ‘N’.
iii. True

Question 16.
Heavier nuclides require greater number of neutrons (than protons) to attain stability. Justify.
Answer:

  • The heavier nuclides with the increasing number of protons lead to large coulombic repulsions.
  • Increased number of neutrons will separate the protons within the nuclei, which will impart stability. Thus, in order to attain stability heavier nuclide need more number of neutrons.

Question 17.
Consider the graph of neutron (N) plotted against proton number (Z). How will you identify radioactive nuclides from the graph?
Answer:
Nuclides which fall outside the belt or stability zone are radioactive nuclides.

Question 18.
Write a note Magic numbers.
Answer:
Magic numbers: The nuclei with 2, 8, 20, 28, 50, 82 and 126 neutrons or protons are particularly stable and abundant in nature. These numbers are known as magic numbers.
e.g. Lead (\({ }_{82}^{208} \mathrm{~Pb}\)) has two magic numbers, 82 protons and 126 neutrons.

Question 19.
What is the order of distance between two protons present in the nucleus?
Answer:
The order of distance between two protons present in the nucleus is typically of order of 10-15 m.

Question 20.
Which factor is responsible for nuclear stability?
Answer:
Nuclear forces of attractions exist within nuclei. These are attractions between proton-proton (p-p), neutron-neutron(n-n) and proton-neutron(p-n). They constitute or give rise to nuclear potential which is responsible for nuclear stability.

Question 21.
Write short notes on: nuclear potential.
Answer:

  • Nuclear potential is the attraction between p-p, n-n and p-n.
  • These attractive forces are independent of the charge on nucleons or attraction between p-p, n-n and p-n are equal.
  • These attractive forces operate over short range within the nucleus.
  • Nuclear potential is responsible for the nuclear stability.

Maharashtra Board Class 11 Chemistry Important Questions Chapter 13 Nuclear Chemistry and Radioactivity

Question 22.
State true or false. Correct the false statement.
i. The nuclear forces of attractions are dependent on the charge on the nucleons.
ii. The actual mass of an atom is observed to be more than sum of the masses of its constituents.
Answer:
i. False
The nuclear forces of attractions are independent of the charge on the nucleons.
ii. False
The actual mass of an atom is observed to be less than sum of the masses of its constituents.

Question 23.
Define: Nuclear binding energy
Answer:
An energy equivalent to the mass lost is released during the formation of nucleus. This is called the nuclear binding energy.
OR
The energy requiredfor holding the nucleons together within the nucleus of an atom is called as the nuclear binding energy.

Question 24.
Explain the term: mass defect.
Answer:
During the formation of nucleus, certain mass is lost. This phenomenon is known as mass defect (Δm).
The exact mass of nucleus is slightly less than sum of the exact masses of the constituent nucleons. This difference is called as mass defect. It is represented by symbol Δm.
Formulae: Δm = calculated mass – observed mass

Question 25.
Explain the relation between nuclear mass and energy? Also give the energy released in the conversion of one atomic mass unit into energy.
Answer:
i. The nuclear mass is expressed in atomic mass unit (u) which is exactly 1/12th of the mass of 12C atom. Thus, u = 1/12th mass of C-12 atom = 1.66 × 10-2 kg.
ii. The conversion of mass into energy is established through Einstein’s equation, E = mc2.
Where m is the mass of matter converted into energy (E) and velocity of light (c).
iii. The energy released in the conversion of one u mass into energy is given by:
E = mc2 = (1.66 × 10-27kg) × (3 × 108 m s-1)2

Maharashtra Board Class 11 Chemistry Important Questions Chapter 13 Nuclear Chemistry and Radioactivity

Question 26.
Derive the expression for nuclear binding energy for a nuclide.
Answer:
Expression for nuclear binding energy:
i. Consider a nuclide \({ }_{z}^{A} X\) that contains Z protons and (A – Z) neutrons. Suppose the mass of the nuclide is m. The mass of proton is mp and that of neutron is mn.

ii. Total mass = (A – Z)mn + Zmp + Zme …..(1)
Δm = [(A – Z)mn + Zmp + Zme] – m
= [(A – Z)mn + Z(mp + me] – m
= [(A – Z)mn + ZmH] – m …..(2)
Where (mp + me) = mH = mass of H atom.
Thus, (Δm) = [Zmp + (A – Z)mn] – m
Where Z = atomic number
A = mass number
(A – Z) = neutron number
mp and mn = masses of proton and neutron, respectively
m = mass of nuclide

iii. The mass defect, Δm is related to binding energy of nucleus by Einstein’s equation,
ΔE = Δm × c2
Where, ΔE = Binding energy, Δm = mass defect.
iv. Nuclear energy is measured in million electro volt (MeV).
v. The total binding energy is then given by,
B.E. = Δm (u) × 931.4
Where 1.00 u = 931.4 MeV
B.E. = 931.4 [ZmH + (A – Z)mn – m] ……(3)
Total binding energy of nucleus containing A nucleons is the B.E.
vi. The binding energy per nucleon is then given by,
\(\bar{B}\) = B.E./A

Question 27.
Calculate the mean binding energy per nucleon for the formation of \({ }_{8}^{16} \mathrm{O}\) nucleus. The mass of oxygen atom is 15.994 u. The masses of H atom and neutron are 1.0078 u and 1.0087 u, respectively.
Solution:
Given: mH = 1.0078 u
mn= 1.0087 u
m= 15.994 u
Z = 8, A= 16
To find: Mean binding energy per nucleon (\(\bar{B}\))
Formulae: i. Δm = ZmH + (A – Z)mn – m
ii. B.E. = Δm × 931.4 MeV
iii. \(\overline{\mathrm{B}}=\frac{\mathrm{B} . \mathrm{E} .}{\mathrm{A}}\)
Calculation: i. The mass defect, Δm = ZmH + (A – Z)mn – m
Δm = 8 × 1.0078 u + 8 × 1.0087 u – 15.994 u = 0.138 u
ii. Total binding energy, B.E. (MeV) = Δm (amu) × 931.4
Hence, B.E. = 0.138 × 931.4 = 128.533 MeV
iii. Binding energy per nucleon, \(\overline{\mathrm{B}}=\frac{\mathrm{B} . \mathrm{E} .}{\mathrm{A}}\)
Hence, \(\bar{B}\) = \(\frac{128.533}{16}\) = 8.033 MeV/nucleon
Ans: Binding energy per nucleon for the formation of \({ }_{8}^{16} \mathrm{O}\) nucleus = 8.033 MeV/nucleon

Question 28.
Calculate the binding energy per nucleon for the formation of \({ }_{2}^{4} \mathrm{He}\) nucleus. Mass of \({ }_{2}^{4} \mathrm{He}\) atom = 4.0026 u.
Solution:
Given: m = 4.0026 u
Z = 2, A = 4
To find: Binding energy per nucleon (\(\bar{B}\))
Formulae: i. Δm = ZmH + (A – Z)mn – m
ii. B.E. = Δm × 931.4 MeV
iii. \(\overline{\mathrm{B}}=\frac{\mathrm{B} . \mathrm{E} .}{\mathrm{A}}\)
The mass defect, Δm = [ZmH + (A – Z)mn] – m
Δm = [(2 × 1.0078) + (2 × 1.0087)] – 4.0026 = 0.0304 u
Total binding energy, B.E. (MeV) = Δm (amu) × 931.4
= 0.0304 × 931.4
= 28.315 MeV
iii. B.E. per nucleon, \(\overline{\mathrm{B}}=\frac{\mathrm{B} . \mathrm{E} .}{\mathrm{A}}\)
\(\bar{B}\) = \(\frac{28.315}{4}\) = 7.079 Mey/nucleon
Ans: Binding energy per nucleon for formation of \({ }_{2}^{4} \mathrm{He}\) nucleus = 7.079 MeV/nucleon

Question 29.
Define radioactivity and give examples of two radioactive elements.
Answer:
Radioactivity is a phenomenon in which the nuclei spontaneously emit a nuclear particle and gamma radiation transforming to a different nuclide. e.g. Uranium and radium
[Note: Radioactivity is the phenomenon related to the nucleus.]

Question 30.
What is the criteria for an element to be known as radioactive element?
Answer:

  • An element is considered to be radioactive if the nuclei of its atoms are unstable.
  • That is, when element undergoes nuclear changes (i.e., emission of nuclear particles and gamma radiation), it is said to be radioactive.

Maharashtra Board Class 11 Chemistry Important Questions Chapter 13 Nuclear Chemistry and Radioactivity

Question 31.
What are the different types of radiations emitted by radioactive element?
Answer:
The radiations emitted by radioactive elements are as follows:

  • Alpha (α) radiations
  • Beta (β) radiations
  • Gamma (γ) radiations

Question 32.
Write the unit of rate of decay.
Answer:
The rate of decay is expressed in the form of disintegrations per second (dps).

Question 33.
Derive the equation λ = \(\frac{\left(-\frac{\mathbf{d} \mathbf{N}}{\mathbf{d} \mathbf{t}}\right)}{\mathbf{N}}\) and write what does λ denotes.
Answer:
The rate of decay of a radioelement at any instant is proportional to the number of nuclei (atoms) present at that instant. It can be represented as,
\(-\frac{\mathrm{d} \mathrm{N}}{\mathrm{dt}} \propto \mathrm{N} \quad \text { or }-\frac{\mathrm{d} \mathrm{N}}{\mathrm{dt}}=\lambda \mathrm{N}\) …….(i)
Where, \(-\frac{\mathrm{d} \mathrm{N}}{\mathrm{dt}}\) = Rate of decay at any time, t
λ = Decay constant
N = Number of nuclei (atoms) present at time, t
From equation (i),
Maharashtra Board Class 11 Chemistry Important Questions Chapter 13 Nuclear Chemistry and Radioactivity 1
Decay constant (λ) is the fraction of nuclei decaying in unit time.
OR
It is the ratio of the amount of substance disintegrated per unit time to the amount of substance present at that time.

Question 34.
Derive the expression for decay constant.
Answer:
Decay constant (λ) is the fraction of nuclei decaying in unit time.
Thus,
λ = \(-\frac{d N}{d t} \times \frac{1}{N}\) …(i)
Rearranging equation (i) we get,
\(\frac{\mathrm{d} \mathrm{N}}{\mathrm{N}}\) = -λ dt
On integrating above equation, we get
∫\(\frac{\mathrm{d} \mathrm{N}}{\mathrm{N}}\) = -∫ λ dt …(ii)
On performing the integration, we get lnN = -λt + C ……(iii)
where C is the constant of integration whose value is obtained as follows:
Let N0 be the number of nuclei present at some arbitrary zero time. At time t, the number of nuclei is N. So, at t = 0, N = N0, substituting in equation (iii), we get
lnN0 = C
With this value of C, equation (iii) becomes
lnN = -λt + lnN0
or λt = lnN0 – InN = ln \(\frac{\mathrm{N}_{0}}{\mathrm{~N}}\) ……(iv)
Hence, λ = \(\frac{1}{t} \ln \frac{N_{0}}{N}\) …….(v)
Converting natural logarithm (ln) to logarithm to the base 10, equation (v) becomes
λ = \(\frac{2.303}{t} \log _{10} \frac{N_{0}}{N}\) ………(vi)
The equation (iv) can be expressed as ln \(\frac{\mathrm{N}_{0}}{\mathrm{~N}}\) = -λt. Taking antilog of both sides, we get
\(\frac{\mathrm{N}}{\mathrm{N}_{0}}=\mathrm{e}^{-\lambda \mathrm{t}} \text { or } \mathrm{N}=\mathrm{N}_{0} \mathrm{e}^{-\lambda t}\) …….(vii)
The equation (vi) and equation (vii) give the decay constant.

Maharashtra Board Class 11 Chemistry Important Questions Chapter 13 Nuclear Chemistry and Radioactivity

Question 35.
Write a note on half-life of a radioelement.
Answer:

  • Half-life of a radioelement (t1/2): It is the time needed for a given number of nuclei (atoms) of radioelement to decay exactly to half of its initial value.
  • Each radio isotope has its own half-life.
  • Half-life of a radioelement can be expressed in seconds, minutes, hours, days or years.
  • Mathematical expression for half-life of a radioelement can be given as,
    \(t_{1 / 2}=\frac{0.693}{\lambda}\)

Question 36.
Complete the following statements based on the given graph.
i. As decay progresses, the number of radioactive atoms will ……….. with time.
ii. As decay progresses, the rate of decay will …………..
iii. Rate of radioactive decay at any instant is ………… to the number of atoms of the radioactive element present at that instant.
Answer:
i. decrease
ii. decrease
iii. proportional

Question 37.
218Po decays initially at a rate of 816 dps. The rate falls to 408 dps after 24 min. Calculate the decay constant.
Solution:
Maharashtra Board Class 11 Chemistry Important Questions Chapter 13 Nuclear Chemistry and Radioactivity 2
Maharashtra Board Class 11 Chemistry Important Questions Chapter 13 Nuclear Chemistry and Radioactivity 3

Question 38.
After how many seconds will the concentration of radioactive element X will be halved, if the decay constant is 1.155 × 10-3 s-1?
Solution:
Maharashtra Board Class 11 Chemistry Important Questions Chapter 13 Nuclear Chemistry and Radioactivity 4
Ans: Concentration of radioactive element (X) will be halved in 600 s.

Question 39.
41Ar decays initially at a rate of 575 Bq. The rate falls to 358 dps after 75 minutes. What is the half-life of 41Ar?
Solution:
Maharashtra Board Class 11 Chemistry Important Questions Chapter 13 Nuclear Chemistry and Radioactivity 5
Maharashtra Board Class 11 Chemistry Important Questions Chapter 13 Nuclear Chemistry and Radioactivity 6
Ans: The half-life of Ar is 109.7 min.

Maharashtra Board Class 11 Chemistry Important Questions Chapter 13 Nuclear Chemistry and Radioactivity

Question 40.
The half-life of 32P is 14.26 d. What percentage of 32P sample will remain after 40 d?
Solution:
Given: t1/2 = 14.26 d,
N0 = 100,
t = 40 d
To find: Percentage of 32P sample remaining after 40 d
Maharashtra Board Class 11 Chemistry Important Questions Chapter 13 Nuclear Chemistry and Radioactivity 7
Maharashtra Board Class 11 Chemistry Important Questions Chapter 13 Nuclear Chemistry and Radioactivity 8

Question 41.
The half-life of 34Cl is 1.53 s. How long does it take for 99.9 % of sample of 34Cl to decay?
Solution:
Given: t1/2 = = 1.53 s,
N0 = 100,
N = 100 – 99.9 = 0.1,
To find: Time (t)
Maharashtra Board Class 11 Chemistry Important Questions Chapter 13 Nuclear Chemistry and Radioactivity 9

Question 42.
The half-life of 209Po is 102 y. How much of 1 mg sample of polonium decays in 62 y?
Solution:
Given: t1/2 = 102y,
t = 62 y,
N0 = 1 mg
To find: Amount of polonium that decayed in 62 y
Maharashtra Board Class 11 Chemistry Important Questions Chapter 13 Nuclear Chemistry and Radioactivity 10
Taking antilog of both sides we get,
\(\frac{\mathrm{N}_{0}}{\mathrm{~N}}\) = antilog (0.1829) = 1.524
N = \(\frac{\mathrm{N}_{0}}{1.524}=\frac{1 \mathrm{mg}}{1.524}\) = 0.656 mg
N is the amount that remains after 62 y.
Hence, the amount decayed in 62 y = 1 mg – 0.656 mg = 0.344 mg
Ans: The amount decayed in 62 y is 0.344 mg

Question 43.
What will be the approximate time taken for 90 % decay of 174Ir in terms of its half-life?
Solution:
Given: N0 = 100
N = 100 – 90 = 10
To find: Time (t)
Maharashtra Board Class 11 Chemistry Important Questions Chapter 13 Nuclear Chemistry and Radioactivity 11
Ans: Thus, the approximate time required for 90 % decay of 174Ir in terms of its half-life is 3.3t1/2.

Maharashtra Board Class 11 Chemistry Important Questions Chapter 13 Nuclear Chemistry and Radioactivity

Question 44.
A radioactive decay of element X (Z = 35) is 30 % complete in 2 hours. Calculate its half-life period.
Solution:
Given: t = 2 hrs,
N0 = 100
N= 100 – 30 = 70
To find: t1/2
Maharashtra Board Class 11 Chemistry Important Questions Chapter 13 Nuclear Chemistry and Radioactivity 12

Question 45.
What are the different modes by which radio elements decay?
Answer:
There are 3 modes by which radio elements decay: α-decay, β-decay and γ-emission.

Question 46.
What is α-decay?
Answer:
Radioactive isotope/radioelement when undergoes decay by the emission of α-particle from the nuclei then the process involved is referred to as α-decay.

Question 47.
Give equation for radium-222 when it undergoes decay by emission of an α-particle.
\({ }_{88}^{226} \mathrm{Ra} \longrightarrow{ }_{86}^{222} \mathrm{X}+{ }_{2}^{4} \mathrm{He}\)
Answer:
\({ }_{88}^{226} \mathrm{Ra} \longrightarrow{ }_{86}^{222} \mathrm{X}+{ }_{2}^{4} \mathrm{He}\)
Thus, atomic number of element ‘X’ will be 86 and atomic mass number will be 222.

Question 48.
Identify the mode of decay and state whether following equation is CORRECT or NOT. Justify.
\({ }_{92}^{238} \mathbf{U} \longrightarrow{ }_{90}^{234} \mathrm{Th}+{ }_{2}^{4} \mathbf{H e}\)
Ans:
i. It involves α-decay process.
ii. As uranium undergoes decay by emission of an α-particle (i.e., \({ }_{2}^{4} \mathrm{He}\)), daughter nuclei (in this case thorium) ‘will observe the decrease in atomic number by 2 units and decrease in atomic mass number by 4 units.
Hence, the given equation is correct.

Maharashtra Board Class 11 Chemistry Important Questions Chapter 13 Nuclear Chemistry and Radioactivity

Question 49.
If radioactive element ‘X’ undergoes α-emission then what will be the position of daughter nuclei in the periodic table with respect to element ‘X’.
Answer:
If radioactive element ‘X’ undergoes α-emission, then corresponding daughter nuclei formed will occupy two places to the left of the periodic table with respect to element ‘X’.

Question 50.
What is β – decay? Also explain the changes that occur in the parent nuclei due to β-emission with one example.
Answer:
β – decay: The emission of negatively charged stream of β particles from the nucleus is called β – decay.
i. β – Particles are electrons with a charge and mass of an electron, mass being negligible as compared to the nuclei.
ii. When a nucleus decays by emitting a high-speed electron called a beta particle (β), a new nucleus is formed with the same mass number as the original nucleus and with an atomic number that is one unit greater than the parent nuclei.
General equation:
Maharashtra Board Class 11 Chemistry Important Questions Chapter 13 Nuclear Chemistry and Radioactivity 13
Note: The mass number A does not change, the atomic number changes when a nuclei undergoes β-decay. e.g. Neptunium-238 decays to form plutonium-238:
Maharashtra Board Class 11 Chemistry Important Questions Chapter 13 Nuclear Chemistry and Radioactivity 14

Question 51.
Mention the atomic number and atomic mass number of the parent radioelement ‘X’ in the following case if parent nuclei undergo β-emission.
i. \(\mathrm{X} \longrightarrow{ }_{94}^{238} \mathrm{Pu}\)
ii. \(\mathrm{X} \longrightarrow{ }_{95}^{241} \mathrm{Am}\)
Answer:
Maharashtra Board Class 11 Chemistry Important Questions Chapter 13 Nuclear Chemistry and Radioactivity 15

Question 52.
How many α and β-particles are emitted in the following?
\({ }_{93}^{237} \mathrm{~Np} \longrightarrow{ }_{83}^{209} \mathrm{Bi}\)
Answer:
The emission of one α-particle decreases the mass number by 4 whereas the emission of β particles has no effect on mass number.
Net decrease in mass number = 237 – 209 = 28. This decrease is only due to α- particles. Hence, number of α- particles emitted = \(\frac {28}{4}\) = 7
Now, the emission of one α-particle decreases the atomic number by 2 and one β-particle emission increases it by 1.
The net decrease in atomic number = 93 – 83 = 10
The emission of 7 α-particles causes decrease in atomic number by 14. However, the actual decrease is only 10. It means atomic number increases by 4. This increase is due to emission of 4 β-particles.
Thus, 7 α and 4 β- particles are emitted.

Maharashtra Board Class 11 Chemistry Important Questions Chapter 13 Nuclear Chemistry and Radioactivity

Question 53.
Explain the process of γ-decay in detail with a suitable example.
Answer:
γ-decay:
i. γ-Radiation is always accompanied with α and β decay processes.
ii. During γ-radiation, the daughter nucleus is left in energetically excited state which decays to the ground state of product with emission of γ-rays.
For example, \({ }_{92}^{238} \mathrm{U} \longrightarrow{ }_{90}^{234} \mathrm{Th}+{ }_{2}^{4} \mathrm{He}+\gamma\)
iii. \({ }_{92}^{238} \mathrm{U}\) emits α-particles of two different energies, 4.147 MeV (23%) and 4.195 MeV (77%).
iv. When α-particles of energy 4.147 MeV are emitted, 234Th is left in an excited state which de-excites to the ground state with emission of γ-ray photons with energy 0.0048 MeV.

Question 54.
Half-life of 209Po is 102 y. How many α-particles are emitted in 1 s from 2 mg sample of Po?
Solution:
Given: t1/2 = 102 y,
t = 1 s,
Amount of sample = 2 mg
To find: Number of α-particles emitted
Maharashtra Board Class 11 Chemistry Important Questions Chapter 13 Nuclear Chemistry and Radioactivity 16

Question 55.
Nuclear transmutation is a spontaneous or non-spontaneous process?
Answer:
Nuclear transmutation is a non-spontaneous (man-made) process.

Question 56.
What is nuclear transmutation?
Answer:
Nuclear transmutation:

  • It is the process of transformation of a stable nucleus into another nucleus which can be stable or unstable.
  • It can occur by the radioactive decay of a nucleus or the reaction of a nucleus with another particle.

Question 57.
Differentiate between chemical reactions and nuclear reactions.
Answer:
Chemical reactions:

  • Rearrangement of atoms by breaking and forming of chemical bonds.
  • Different isotopes of an element have same behaviour.
  • Only outer shell electrons take part in the chemical reaction.
  • The chemical reaction is accompanied by relatively small amounts of energy.
    e.g. chemical combustion of 1.0 g methane releases only 56 kJ energy.
  • The rates of reaction are influenced by the temperature, pressure, concentration and catalyst.

Nuclear reactions:

  • Elements or isotopes of one element are converted into another element in a nuclear reaction.
  • Isotopes of an element behave differently.
  • In addition to electrons, protons, neutrons, other elementary particles may be involved.
  • The nuclear reaction is accompanied by a large amount of energy change, e.g. The nuclear transformation of 1 g of Uranium – 235 release 8.2 × 107 kJ
  • The rate of nuclear reactions is unaffected by temperature, pressure and catalyst.

Question 58.
What will happen when a nucleus of J’B is bombarded with α-particle? Identify the process involved.
Answer:
i. When a stable nucleus of \({ }_{5}^{10} \mathrm{~B}\) is is bombarded with α-particle, it transforms into \({ }_{7}^{13} \mathrm{~N}\), which is radioactive and spontaneously emits positrons to produces \({ }_{6}^{13} \mathrm{C}\).
This can be represented as,
Maharashtra Board Class 11 Chemistry Important Questions Chapter 13 Nuclear Chemistry and Radioactivity 17
ii. The process involved is known as induced radioactivity or artificial radioactivity.

Maharashtra Board Class 11 Chemistry Important Questions Chapter 13 Nuclear Chemistry and Radioactivity

Question 59.
Define: Nuclear fission
Answer:
Nuclear fission is defined as a process which involves splitting of the heavy nucleus of an atom into two nearly equal fragments accompanied by release of the large amount of energy.

Question 60.
Nuclear fission of 235U is a chain process. Justify.
Answer:

  • Nuclear fission of 235U occurs when nucleus absorbs neutron. When a uranium nucleus absorbs neutron, it breaks into two lighter fragments and releases energy (heat), more neutrons, and other radiation.
  • When one uranium 235 nucleus undergoes fission, three neutrons are emitted.
  • These neutrons emitted in fission cause more fission of the uranium nuclei which yield more neutrons. These neutrons again bring forth fission producing further neutrons.
  • The process continues indefinitely leading to chain reaction which continues even after the removal of bombarding neutrons.

Question 61.
Explain the term: Nuclear fusion and give one example.
Answer:
Nuclear fusion: In this process, the lighter nuclei combine (fuse) together and form a heavy nucleus which is accompanied by an enormous amount of energy.
e. g. The energy received by earth from the sun is due to the nuclear fusion reactions.

Question 62.
Which will produce more energy: Nuclear fission or fusion?
Answer:
Nuclear fusion will produce relatively more energy per given mass of fuel.

Question 63.
What is the range of temperature required to carry out nuclear fusion reaction?
Answer:
Nuclear fusion reaction requires extremely high temperature typically of the order of 108 K.

Question 64.
Distinguish between nuclear fission and nuclear fusion.
Answer:
Nuclear fission:

  • It is the process in which a heavy nucleus splits up into two lighter nuclei of nearly equal masses.
  • About 200 MeV of energy is available per fission in case of \({ }_{92}^{235} \mathrm{U}\).
  • The products of nuclear fission are, in general, radioactive.

Nuclear fusion:

  • It is the process in which two lighter nuclei combine together to form a heavy nucleus.
  • Energy available per fusion is much less but the energy per unit mass of material is much greater than that for fission of heavy nuclei.
  • The products of fusion are, in general, non-radioactive.

Maharashtra Board Class 11 Chemistry Important Questions Chapter 13 Nuclear Chemistry and Radioactivity

Question 65.
Estimate the energy released in the fusion reaction.
\({ }_{1}^{2} \mathbf{H}+{ }_{2}^{3} \mathbf{H e} \longrightarrow{ }_{2}^{4} \mathbf{H e}+{ }_{1}^{1} \mathbf{H}\)
(Given atomic masses: 2H = 2.0141 u. 3He = 3.0160 u, 4He = 4.0026 u, 1H = 1.0078 u)
Answer:
Maharashtra Board Class 11 Chemistry Important Questions Chapter 13 Nuclear Chemistry and Radioactivity 18

Question 66.
Explain the term: Radiocarbon dating in detail.
Answer:
Radiocarbon dating: The technique is used to find the age of historic and archaeological organic samples such as old wood samples and animal or human fossils.
Radioisotope used for carbon dating is 14C.
i. Radioactive 14C is formed in the upper atmosphere by bombardment of neutrons from cosmic ray on 14N.
\({ }_{7}^{14} \mathrm{~N}+{ }_{0}^{1} \mathrm{n} \longrightarrow{ }_{6}^{14} \mathrm{C}+{ }_{1}^{1} \mathrm{H}\)
ii. 14C combines with atmospheric oxygen to form 14CO2 which mixes with ordinary 12CO2.
iii. This carbon dioxide is absorbed by plants during photosynthesis.
iv. Animals eat plants which have absorbed a carbon dioxide (14CO2 + 12CO2). Hence, 14C becomes a part of plant and animal bodies.
v. As long as the plant is alive, the ratio 14C/12C remains constant.
vi. When the plant dies, photosynthesis will not occur and the ratio 14C/12C decreases with the decay of radioactive 14C which has a half-life 5730 years.
vii. The decay process of 14C is given below:
\({ }_{6}^{14} \mathrm{C} \longrightarrow{ }_{7}^{14} \mathrm{~N}+{ }_{-1}^{0} \mathrm{e}\)
viii. The activity (N) of given wood sample and that of fresh sample of live plant (N0) is measured, where, N0 denotes the activity of the given sample at the time of death.
ix. The age of the given wood sample, can be determined by applying following Formulae:
Maharashtra Board Class 11 Chemistry Important Questions Chapter 13 Nuclear Chemistry and Radioactivity 19

Question 67.
What is nuclear power?
Answer:
Nuclear power is the electricity generated from the fission of uranium and plutonium.

Question 68.
Nuclear power is a clean source of energy. Justify.
Answer:
Nuclear power offers huge environmental benefits in producing electricity because,

  • it releases zero carbon dioxide.
  • it releases zero sulphur and nitrogen oxides.
  • these are atmospheric pollutants which pollute the air.

Thus, nuclear power is a clean source of energy.

Question 69.
How much energy will be produced by fission of 1 gram of 235U?
Answer:
Fission of 1 gram of uranium-235 produces about 24,000 kW/h of energy.

Maharashtra Board Class 11 Chemistry Important Questions Chapter 13 Nuclear Chemistry and Radioactivity

Question 70.
Nuclear fission is an alternative energy source. Explain.
Answer:

  • Fission of 1 gram of uranium-235 produces about 24,000 kW/h of energy.
  • This is the same amount of energy produced by burning 3 tons of coal or 12 barrels of oil, or nearly 5000 m3 of natural gas.
  • The sources like coal, oil, natural gas are depleting very fast.
  • Also, the costs of petrol and other products from petroleum industry is increasing.
  • Thus, we need to depend on the nuclear fission as an alternative source of energy for electricity.

Question 71.
Label the follow ing diagram of simplified nuclear reactor.
Maharashtra Board Class 11 Chemistry Important Questions Chapter 13 Nuclear Chemistry and Radioactivity 20
Answer:
Maharashtra Board Class 11 Chemistry Important Questions Chapter 13 Nuclear Chemistry and Radioactivity 21

Question 72.
Explain in brief: Nuclear reactor
Answer:
Nuclear reactor: Nuclear reactor is a device for using atomic energy in controlled manner for peaceful purposes. During nuclear fission energy is released. The released energy can be utilized to generate electricity in a nuclear reactor.

Working of a nuclear reactor:

  • In a nuclear reactor, U235 or U239, a fissionable material is stacked with heavy water (D2O deuterium oxide) or graphite called moderator.
  • The neutrons produced in the fission pass through the moderator and lose a part of their energy. The slow neutrons produced during the process are captured which initiate new fission.
  • Cadmium rods are inserted in the moderator as they have ability to absorb neutrons. This controls the rate of chain reaction.
  • The energy released during the reaction appears as heat and removed by circulating a liquid (coolant). The coolant which has absorbed excess of heat from the reactor is passed over a heat exchanger for producing steam.
  • Steam is then passed through the turbines to produce electricity. Thus, the atomic energy produced with the use of fission reaction can be controlled in the nuclear reactor.
  • This process can be explored for peaceful purpose such as conversion of atomic energy into electrical energy which can be used for civilian purposes, ships, submarines, etc.

Note: Schematic diagram of nuclear power plant:
Maharashtra Board Class 11 Chemistry Important Questions Chapter 13 Nuclear Chemistry and Radioactivity 22

Question 73.
Why cadmium rods are used in nuclear reactor?
Answer:
Cadmium rods are inserted in the moderator as they have ability to absorb neutrons which help to control the rate of chain reaction.

Question 74.
Why short-lived isotopes are used for diagnostic purposes?
Answer:
For diagnostic purpose, short-lived isotopes are used in order to limit the exposure time to radiation. Note: Diagnostic Radioisotopes are listed below:
Maharashtra Board Class 11 Chemistry Important Questions Chapter 13 Nuclear Chemistry and Radioactivity 23

Question 75.
Give one application of therapeutic radioisotopes.
Answer:
Therapeutic radioisotopes are used to destroy abnormal cell growth in the body, e.g. cancerous cells.
Note: Therapeutic Radioisotopes are listed below:
Maharashtra Board Class 11 Chemistry Important Questions Chapter 13 Nuclear Chemistry and Radioactivity 24

Maharashtra Board Class 11 Chemistry Important Questions Chapter 13 Nuclear Chemistry and Radioactivity

Question 76.
Give example of isotopes used in following.
i. Isotope used in the treatment of leukaemia.
ii. Isotopes used in the preservation of agricultural products by irradiation.
Answer:
i. Isotope of phosphorus, \({ }_{15}^{35} \mathrm{P}\).
ii. 60Co or 137Cs

Question 77.
At which places has BARC Mumbai set up irradiation plants for preservation of agricultural produce?
Answer:
Bhabha Atomic Research Centre (BARC) Mumbai has set up irradiation plants for preservation of agricultural produce such as mangoes, onion and potatoes at Vashi (Navi Mumbai) and Lasalgaon (Nashik).

Question 78.
Why radiotracer technique is used in chemistry?
Answer:
Radiotracer technique is used to trace the path/mechanism followed by a reaction in the system.

Question 79.
The half-life for radioactive decay of an element X is 140 days. Complete the following flow chart showing decay of 1 g of X.
Maharashtra Board Class 11 Chemistry Important Questions Chapter 13 Nuclear Chemistry and Radioactivity 25
Answer:
Maharashtra Board Class 11 Chemistry Important Questions Chapter 13 Nuclear Chemistry and Radioactivity 26
Shortcut method:
Amount of the element X left after n half-lives is given as [X] = \(\frac{[\mathrm{X}]_{0}}{2^{n}}\)
e.g. \(\frac{1}{2^{4}} \mathrm{~g}=\frac{1}{16} \mathrm{~g}\)

Question 80.
A sample of 35S complete its 10% decay in 20 min, then calculate the time required to complete decay by 19%.
Answer:
When decay is 10 % complete, if N0 = 100 , then N = 100 – 10 = 90 and t = 20 minutes
When decay is 19 % complete, N = 100 – 19 = 81
Substituting these values in formula we get,
Maharashtra Board Class 11 Chemistry Important Questions Chapter 13 Nuclear Chemistry and Radioactivity 27

Multiple Choice Questions

1. Radius of the nucleus is related to the mass number A by ………….
(A) R = R0A1/2
(B) R = R0A
(C) R = R0A2
(D) R = R0A1/3
Answer:
(D) R = R0A1/3

2. Which of the following nuclides has the magic number of both protons and neutrons?
(A) \({ }_{50}^{115} \mathrm{Sn}\)
(B) \({ }_{81}^{206} \mathrm{Pb}\)
(C) \({ }_{82}^{208} \mathrm{Pb}\)
(D) \({ }_{50}^{118} \mathrm{Pb}\)
Answer:
(C) \({ }_{82}^{208} \mathrm{Pb}\)

Maharashtra Board Class 11 Chemistry Important Questions Chapter 13 Nuclear Chemistry and Radioactivity

3. The probability of decay of a radioactive element depends on …………..
i. the age of nucleus
ii. the presence of catalyst
iii. pressure
iv. temperature
(A) only i. and iv
(B) all of these
(C) only ii. And iii.
(D) none of these
Answer:
(D) none of these

4. The decay constant for 67Ga is 7.0 × 10-4 s-1. If initial concentration of is 0.07 g, what is the half-life of 67Ga?
(A) 990 s
(B) 79.2 s
(C) 12375 s
(D) 10.10 × 10-4 s
Answer:
(A) 990 s

5. The half-life of radioactive element X having decay constant of 1.7 × 10-5 s-1 is …………
(A) 21.5 h
(B) 19.7 h
(C) 11.3 h
(D) 2.8 h
Answer:
(C) 11.3 h

6. A radioactive decay of element X (Z = 90) is 30 % complete in 30 minutes. It has a half-life period of ……………
(A) 24.3 min
(B) 58.3 min
(C) 102.3 min
(D) 120.3 min
Answer:
(B) 58.3 min

Maharashtra Board Class 11 Chemistry Important Questions Chapter 13 Nuclear Chemistry and Radioactivity

7. The half-life of radium is 1600 years. The fraction of a sample of radium that would remain after 6400 year is ……….
(A) \(\frac {1}{2}\)
(B) \(\frac {1}{4}\)
(C) \(\frac {1}{8}\)
(D) \(\frac {1}{16}\)
Answer:
(D) \(\frac {1}{16}\)

8. The half-life of an element is 5 d. How much time is required for the decay of 7/8th of the sample?
(A) 5 d
(B) 10 d
(C) 15 d
(D) 35/8 d
Answer:
(C) 15 d

9. The composition of an α-particle can be expressed as ……………….
(A) 1p + 1n
(B) 1p + 2n
(C) 2p + 1n
(D) 2p + 2n
Answer:
(D) 2p + 2n

10. If a radioactive nuclide of group 15 element undergoes β-particle emission, the daughter element will be found in ………………..
(A) 16 group
(B) 14 group
(C) 13 group
(D) same group
Answer:
(A) 16 group

Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15

Balbharti Maharashtra State Board 11th Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15 Important Questions and Answers.

Maharashtra State Board 11th Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15

Question 1.
What are p-block elements?
Answer:

  • Elements in which the differentiating electron (the last filling electron) enters the outermost p orbital are p-block elements.
  • Since maximum six electrons can be accommodated in p-subshell i.e., three p-orbitals, the p-block contains six groups numbered from 13 to 18 in the modem periodic table.
  • The p-block elements show greater variation in the properties than s-block elements.

Question 2.
Write the names of the elements present in groups 13, 14 and 15.
Answer:

Group Name of family Name of the elements
13 Boron family Boron (5B), aluminium (13Al), gallium (31Ga), indium (49In), thallium (81Tl)
14 Carbon family Carbon (6C), silicon (14Si), germanium (32Ge), tin (50Sn), lead (82Pb)
15 Nitrogen family Nitrogen (7N), phosphorus (15P), arsenic (33AS), antimony (51Sb), bismuth (83Bi)

Question 3.
i. Write the general outer electronic configuration of the elements of group 13, group 14 and group 15.
ii. By how many electrons do their outer electronic configurations differ from their nearest inert gas?
Answer:
i.

Group General outer electronic configuration
13 ns2 np1
14 ns2 np2
15 ns2 np3

ii. The outer electronic configurations of the elements group 13, group 14 and group 15 differ from their nearest inert gas by 5, 4 and 3 electrons, respectively.

Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15

Question 4.
In which form do the elements of groups 13,14 and 15 occur in nature?
Answer:

  • The elements of groups 13, 14 and 15 do not occur in free monoatomic state and are found as compounds with other elements.
  • They also occur in the form of polyatomic molecules (such as N2, P4, C60) or polyatomic covalent arrays (such as graphite, diamond).

Question 5.
Write condensed electronic configurations of the following elements.
13Al, 49In, 14Si, 50Sn, 15P, 33As
Answer:
Condensed electronic configurations of
i. 13Al: [Ne]3s2 3p1
ii. 49In: [Kr]4d105s25p1
iii. 14Si: [Ne]3s23p2
iv. 50Sn: [Kr]4d105s25p2
v. 15P: [Ne]3s23p3
vi. 33As: [Ar]3d104s24p3

Note: Condensed electronic configurations of elements of groups 13, 14 and 15 are given in the table below.
Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15, 1

Question 6.
Name the following.
i. A metalloid present in group 13.
ii. A group 13 element which is the third most abundant element in the earth’s crust.
Answer:
i. Boron
ii. Aluminium

Question 7.
Why boron is classified as a metalloid?
Answer:
Boron is glossy and hard solid like metals but a poor conductor of electricity like nonmetals. Since it exhibits properties of both metals and nonmetals, boron is classified as a metalloid.

Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15

Question 8.
Describe the variation in the electronegativity of group 13 elements.
Answer:

  • In group 13, on moving down the group, the electronegativity decreases from B to Al.
  • However, there is a marginal increase in the electronegativity from Al to Tl.
  • This trend is a result of the irregularities observed in atomic size of elements.

Question 9.
Atomic numbers of the group 13 elements are in the order B < Al < Ga < In < Tl. Arrange these elements in increasing order of ionic radii of M3+.
Answer:

  • The given elements are in an increasing order of their atomic number.
  • The general outer electronic configuration of group 13 elements is ns2np1.
  • M3+ ion is formed by the removal of three electrons from the outermost shell ‘n’.
  • In the M3+ ions, the ‘n-1’ shell becomes the outermost shell. Size of the ‘n-1’ shell increases down the group.

Therefore, the ionic radii of M3+ ion increases down the group in the following order:
B3+ < Al3+ < Ga3+ < In3+ < Tl3+

Question 10.
Why the atomic radius of Gallium is less than that of aluminium?
Answer:

  • Atomic radius of the elements increases down the group due to addition of new shells.
  • Electronic configuration of Al is [Ne]3s23p1 while that of Ga is [Ar]3d104s24p1.
  • As Al does not have d-electrons, it offers an exception to this trend.
  • As we go from Al down to Ga the nuclear charge increases by 18 units. Out of the 18 electrons added, 10 electrons are in the inner 3d subshell of Ga. These d-electrons offer poor shielding effect.
  • Therefore, the effect of attraction due to increased nuclear charge is experienced prominently by the outer electrons of Ga and thus, its atomic radius becomes smaller than that of Al.

Hence, the atomic radius of gallium is less than that of aluminium.

Question 11.
The values of the first ionization enthalpy of Al, Si and P are 577, 786 and 1012 kJ mol-1 respectively. Explain the observed trend.
Answer:

  • The trend shows increasing first ionization enthalpy from Al to Si to P.
  • Al, Si and P belong to the third period in the periodic table and hence, they have same valence shell.
  • As we move across a period from left to right, the nuclear charge increases. Due to this, electrons in the valence shell are held more tightly by the nucleus as we go from Al to Si to P.
  • Therefore, more energy is required to remove an electron from its outermost shell.

Hence, the value of first ionization enthalpy increases from Al to Si to P.

Note: Atomic and physical properties of group 13 elements.
Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15, 2

Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15

Question 12.
Name metal(s), nonmetal(s) and metalloid(s) of group 14.
Answer:
i. Metal: Tin, lead
ii. Nonmetal: Carbon
iii. Metalloid: Silicon, germanium

Question 13.
Explain the variation in the following properties of the group 14 elements,
i. Atomic radii
ii. Ionization enthalpy
iii. Electronegativity
Answer:
i. Atomic radii (Covalent radii):

  • In the periodic table as we move down the group 14 from C to Pb, the atomic radii increases due to the addition of new shell at each succeeding element.
  • However, the increase is comparatively less after silicon due to poor shielding by inner d- and f-electrons in the atoms.

ii. Ionization enthalpy:

  • Due to increased effective nuclear charge, group 14 elements have higher value of ionization enthalpy than corresponding group 13 elements.
  • In the periodic table, as we move down the group 14 from C to Sn, the ionization enthalpy decreases.
  • From Si to Sn, the ionization enthalpy decreases slightly.
  • However, from Sn to Pb, the ionization enthalpy increases slightly. It is due to the poor shielding effect of intervening d and f orbitals and increase in the size of the atoms.

iii. Electronegativity:

  • Due to small atomic size, group 14 elements are slightly more electronegative than the corresponding group 13 elements.
  • As we move down the group 14 from C to Si in the periodic table, the electronegativity decreases.
  • The electronegativity values for elements from Si to Pb are almost the same.
  • Among group 14 elements, carbon is the most electronegative with electronegativity of 2.5.

Question 14.
Explain why there is a phenomenal decrease in ionization enthalpy from carbon to silicon.
Answer:

  • Carbon is the first element of group 14 and thus, it has the smallest atomic size.
  • The ionization enthalpy of carbon (1086 kJ mol-1) is very high due to its small atomic size (77 pm) and high electronegativity (2.5).
  • However, the ionization enthalpy of silicon (786 kJ mol-1) decreases phenomenally due to the increase in its atomic size (118 pm) and low electronegativity (1.8).

Note: Atomic and physical properties of group 14 elements.
Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15, 3

Question 15.
Which type of elements are present in group 15? Mention their physical state.
Answer:

  • Group 15 includes all the three traditional types of elements i.e., metals, nonmetals and metalloids.
  • Nitrogen is a gas whereas the remaining group 15 elements are solids.
  • The gaseous nitrogen and brittle phosphorus are nonmetals.
  • Arsenic and antimony are metalloids while bismuth is moderately reactive metal.

Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15

Question 16.
Explain the trends in physical properties of group 15 elements.
i. Atomic and ionic radii
ii. Ionization enthalpy
iii. Electronegativity
Answer:
i. Atomic and ionic radii:

  • Atomic size increases down the group with increasing atomic number.
  • The effective nuclear charge in case of group 15 elements is larger than that of group 14 elements. Due to the increased effective nuclear charge, electrons are strongly attracted by the nucleus. Thus, the atomic and ionic radii of group 15 elements are smaller than the atomic and ionic radii of the corresponding group 14 elements.
  • On moving down the group, number of shells increases which leads to increased shielding effect and as a result atomic radii and ionic radii increases.

ii. Ionization enthalpy:

  • Due to extra stability of half-filled p-orbitals and relatively smaller size of group 15 elements, ionization enthalpy of group 15 elements is much greater than that of the group 14 elements in the corresponding periods.
  • On moving down the group, increase in atomic size and screening effect overcome the effective nuclear charge and thus, ionization enthalpy decreases.

iii. Electronegativity:

  • Due to smaller size and greater effective nuclear charge of atoms, group 15 elements have higher electronegativity values than group 14 elements.
  • On moving down the group, electronegativity values decreases due to increase in the size of the atoms and shielding effect.
  • Nitrogen is the most electronegative element among group 15 elements. However, there is not much of a difference between the electronegativity values of other elements of group 15.

Note: Atomic and physical properties of group 15 elements.
Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15, 4

Question 17.
Write a note on the oxidation state of p-block elements with respect to groups 13, 14 and 15 elements.
Answer:

  • Oxidation state is the primary chemical property of all elements.
  • The highest oxidation state exhibited by the p-block elements is equal to the total number of valence electrons i.e., the sum of s- and p-electrons present in the valence shell. This is sometimes called the group oxidation state.
  • In boron, carbon and nitrogen families, the group oxidation state is the most stable oxidation state for the lighter elements.
  • Besides, the elements of groups 13, 14 and 15 exhibit other oxidation states which are lower than the group oxidation state by two units.
  • The lower oxidation states become increasingly stable as we move down to heavier elements in the groups.

Note: Group oxidation states and common oxidation states with examples for groups 13, 14 and 15.
Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15, 5

Question 18.
What are general oxidation states of group 13 elements? Explain.
Answer:

  • The general oxidation states of group 13 are +1 and +3.
  • The group 13 elements have the outermost electronic configuration ns2 np1.
  • If only np1 electron takes part in bonding, the oxidation state is +1 and if all the three electrons i.e., ns2 np1 take part in bonding, the oxidation state is +3. Hence, the expected oxidation states are +1 and +3.

Question 19.
Give reason: The increased stability of the oxidation state is lowered by 2 units than the group oxidation state in heavier p-block elements.
Answer:

  • The increased stability of the oxidation state lowered by 2 units than the group oxidation state in heavier p-block elements is due to inert pair effect.
  • In these elements, the two s-electrons are involved less readily in chemical reactions.
  • This is because, in heavier p-block elements, the s-electrons of valence shell experience poor shielding than valence p-electrons due to ten inner d-electrons.

Hence, the increased stability of the oxidation state is lowered by 2 units than the group oxidation state in heavier p-block elements.

Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15

Question 20.
Why Tl1+ ion is more stable than Tl3+?
Answer:

  • Tl is a heavy element which belongs to group 13 of the p-block.
  • The common oxidation state for this group is +3.
  • In p-block, the lower oxidation state is more stable for heavier elements due to inert pair effect.

Hence, Tl1+ ion is more stable than Tl3+ ion.

Question 21.
How can you explain the higher stability of BCl3 as compared to TlCl3?
Answer:

  • Boron is a light element in group 13 and has outermost electronic configuration 2s2 2p1 whereas thallium is a heavy element in group 13 and has outermost electronic configuration 6s2 6p1.
  • Because of its small ionic radius, boron forms stable compounds in +3 oxidation state.
  • Thallium has a large atomic size and due to the inert pair effect forms more stable compounds with lower oxidation state +1 than compounds with +3 oxidation state.

Therefore, BCl3 has higher stability than TlCl3.

Question 22.
State the oxidation state for the following:
i. The group oxidation state of group 14 elements.
ii. The stable oxidation state for lead.
iii. Oxidation state of carbon in CH4.
Answer:
i. +4
ii. +2
iii. -4

Question 23.
GeCl4 is more stable than GeCl2 while PbCl2 is more stable than PbCl4. Explain.
Answer:

  • Elements Ge and Pb belong to 4th and 6th period in the group 14.
  • The group oxidation state of group 14 elements is +4.
  • However, the stability of other oxidation state which is lower by 2 units i.e., +2, increases down the group due to inert pair effect.
  • Therefore, the stability of the oxidation state +4 is more in Ge than in Pb while the stability of the oxidation state +2 is more in Pb than in Ge.

Hence, GeCl4 is more stable than GeCl2 while PbCl2 is more stable than PbCl4.

Question 24.
Name the elements of group 14 which are generally occur in +2 oxidation state.
Answer:
The elements of group 14 that are generally occur in +2 oxidation state are tin (Sn) and lead (Pb).

Question 25.
Discuss the nature of bonding in compounds of group 13, 14 and 15 elements.
Answer:

  • The lighter elements in groups 13, 14 and 15 have small atomic radii and high ionization enthalpy values. They form covalent bonds with other atoms by overlapping of valence shell orbitals.
  • As we move down the group, the value of ionization enthalpy decreases. The atomic radius increases since the valence shell orbitals are more diffused.
  • The heavier elements in these groups tend to form ionic bonds. The first member of these groups belongs to second period and do not have d orbitals and hence, B, C and N cannot expand their octet.
  • The subsequent elements in the group possess vacant d orbital in their valence shell, which can expand their octet forming a variety of compounds.

Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15

Question 26.
Explain the reactivity of groups 13, 14 and 15 elements towards air.
Answer:
i. Group 13 elements:
a. On heating with air or oxygen, group 13 elements form oxide of the type E2O3.
\(4 \mathrm{E}_{(\mathrm{s})}+3 \mathrm{O}_{2(\mathrm{~g})} \stackrel{\Delta}{\longrightarrow} 2 \mathrm{E}_{2} \mathrm{O}_{3(\mathrm{~s})}\) (where, E = B, Al, Ga, In, Tl)

b. At high temperature, group 13 elements also react with nitrogen present in the air to form corresponding nitrides.
\(2 \mathrm{E}_{(\mathrm{s})}+\mathrm{N}_{2(\mathrm{~g})} \stackrel{\Delta}{\longrightarrow} 2 \mathrm{EN}_{(\mathrm{s})}\) (where, E = B, Al, Ga, In, Tl)

ii. Group 14 elements: The elements of group 14 on heating in air or oxygen form oxide of the type EO and EO2 in accordance with the stable oxidation state and availability of oxygen.
Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15, 6

iii. Group 15 elements: The elements of group 15 on heating in air or oxygen forms two types of oxide i.e., E2O3 and E2O5.
Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15, 7
Due to increase in metallic character down the groups 13, 14 and 15, the nature of their oxides gradually varies from acidic through amphoteric to basic.
[Note: The temperature required for the reaction of nitrogen with oxygen is very high. This is produced by striking an electric arc.]

Question 27.
Classify the following oxides into acidic, basic or amphoteric.
B2O3, Ga2O3, Tl2O3, In2O3, Al2O3
Answer:

Acidic oxide B2O3
Basic oxides In2O3, Tl2O3
Amphoteric oxides Al2O3, Ga2O3

Question 28.
Match the following.

Column A Column B
i. N2O5 a. Amphoteric
ii. Bi2O3 b. Acidic
iii. Sb2O3 c. Basic

Answer:
i – b,
ii – c,
iii – a

Note: Nature of stable oxides of groups 13, 14 and 15 elements
Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15, 8

Question 29.
State TRUE or FALSE. Correct the false statement.
i. Sb is more stable in +3 oxidation state.
ii. Oxides of the type E2O5 are formed by group 15 elements.
iii. As4O6 is an acidic oxide.
Answer:
i. True
ii. True
iii. False
As4O6 is an amphoteric oxide.

Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15

Question 30.
What happens when the elements of groups 13, 14 and 15 react with water?
Answer:
i. Most of the elements of groups 13, 14 and 15 are unaffected by water.
ii. Aluminium reacts with water on heating and forms hydroxide while tin reacts with steam to form oxide.
Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15, 9
iii. Lead is unaffected by water due to the formation of a protective film of oxide.

Question 31.
Why is phosphorus stored under water?
Answer:
Phosphorus is highly reactive and hence, it is stored under water to prevent its reaction with air as it catches fire on being exposed to air.

Question 32.
Explain the reactivity of group 13 elements towards halogens.
Answer:
i. All the elements of group 13 react directly with halogens to form trihalides (EX3).
2E(S) + 3X2(g) → 2EX3(s) (where, E = B, Al, Ga, In and X = F, Cl, Br, I)
ii. Thallium is an exception as it forms monohalides (TlX).

Question 33.
Describe the reactivity of group 14 elements with halogens.
Answer:

  • All the elements of group 14 (except carbon) react directly with halogens to form tetrahalides (EX4).
  • The heavy elements Ge and Pb form dihalides as well.
  • Stability of dihalides increases down the group due to inert pair effect.
  • The ionic character of halides also increases steadily down the group.

Question 34.
Discuss the reactivity of group 15 elements with halogens.
Answer:

  • Elements of the group 15 reacts with halogens to form two series of halides i.e., trihalides (EX3) and pentahalides (EX5).
  • The pentahalides possess more covalent character due to availability of vacant d orbitals of the valence shell for bonding.
  • Nitrogen being second period element, does not have d orbitals in its valence shell, and therefore, does not form pentahalides.
  • Trihalides of the group 15 elements are predominantly covalent except BiF3. The only stable trihalide of nitrogen is NF3.

Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15

Question 35.
Nitrogen does not form pentahalides. Give reason.
Answer:

  • The electronic configuration of 7N is 1s2 2s2 2p3. It has 3 unpaired electrons which can form 3 covalent bonds, thus forming NX3 molecule.
  • Valence shell of nitrogen (n = 2) contains only s and p orbitals.
  • Thus, due to the absence of d orbitals in the valence shell, nitrogen cannot expand its octet, therefore, it cannot form compounds like NCl5 and NF5.

Hence, nitrogen does not form pentahalides.

Question 36.
Nitrogen does not form NCl5 or NF5 but phosphorus can. Explain.
Answer:

  • The electronic configuration of 7N is 1s2 2s2 2p3 while that of 15P is 1s2 2s2 2p6 3s2 3p3.
  • As phosphorus contains d orbitals, it can expand its octet to form MX3 as well MX5 compounds.
  • However, due to absence of d orbitals, nitrogen cannot form MX3 or MX5.

Hence, Nitrogen does not form NCl5 or NF5 but phosphoms can form compounds like PCl5 or PF5.

Question 37.
Define catenation.
Answer:
The property of self-linking of atoms of an element by covalent bonds to form chains and rings is called catenation.

Question 38.
Explain catenation of group 14 elements.
Answer:
i. The property of self-linking of atoms of an element by covalent bonds to form chains and rings is called catenation.
ii. The strength of the element-element bond determines the tendency of an element to form a chain.
iii. Among the elements of group 14, the bond strength is maximum for C-C bond (348 kJ mol-1). Hence, carbon has maximum tendency for catenation.

Bond Bond strength (Bond enthalpy kJ mol-1)
C-C 348
Si-Si 297
Ge-Ge 260
Sn-Sn 240

iv. From the values of bond enthalpy, it can be concluded that the tendency to form chains is maximum for carbon and much lesser for silicon. Germanium has still lesser tendency and tin has hardly any tendency for catenation. Lead does not show catenation.
Therefore, the order of catenation of group 14 elements is C >> Si > Ge = Sn.

Question 39.
State TRUE or FALSE. Correct the false statement.
i. Among the group 14 elements, Ge does not show the property of catenation.
Answer:
i. False
Among the group 14 elements, Pb (lead) does not shows the property of catenation.

Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15

Question 40.
Define allotropy.
Answer:
When a solid element exists in different crystalline forms with different physical properties such as colour, density, melting point, etc. the phenomenon is called allotropy.

Question 41.
i. What are allotropes?
ii. Name various allotropes of carbon.
Answer:
i. When a solid element exists in different crystalline forms with different physical properties such as colour, density, melting point, etc. the phenomenon is called allotropy and the individual crystalline forms are called allotropes.
ii. Diamond, graphite, fiillerenes, graphene and carbon nanotubes are various allotropes of carbon.

Question 42.
Explain the structure of diamond.
Answer:
Structure of diamond:

  • In diamond, each carbon atom undergoes sp3 hybridization and is linked to four other carbon atoms in tetrahedral manner.
  • The C – C bond length is 154 pm.
  • The tetrahedra are linked together forming a three-dimensional network structure involving strong C-C single bonds which makes diamond the hardest natural substance.

Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15, 10

Question 43.
Write physical properties of diamond. Also, state its uses.
Answer:
i. Physical properties

  • Diamond is the hardest natural substance.
  • It has abnormally high melting point (3930 °C).
  • It is a bad conductor of electricity.

ii. Uses: Diamond is used

  • for cutting glass and in drilling tools.
  • for making dies for drawing thin wire from metal.
  • for making jewellery.

Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15

Question 44.
Describe the structure of graphite.
Answer:

  • Graphite is composed of layers of two-dimensional sheets of carbon atoms.
  • Each sheet is made up of hexagonal net of sp2 carbons bonded to three neighbours forming three bonds.
  • The fourth electron in the unhybrid p-orbital of each carbon is shared by all carbon atoms resulting in a π bond. These it electrons are delocalized over the whole layer.
  • The C – C bond length in graphite is 141.5 pm.
  • The individual layers are held by weak van der Waals forces and separated by 335 pm.
  • Graphite is soft and slippery and is thermodynamically most stable allotrope of carbon.

Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15, 11

Question 45.
Diamond is very hard whereas graphite is soft. Explain.
Answer:

  • Diamond has three-dimensional network of sp3 hybridized carbon atoms joined by extended covalent bonds which are difficult to break. Therefore, diamond is hard.
  • Graphite has two-dimensional sheet like structure, like layers of hexagonal rings formed from sp2 hybridized carbon atoms. These layers are held by weak van der Waals forces, which can be broken easily. Therefore, graphite is soft and slippery.

Hence, diamond is very hard whereas graphite is soft.

Question 46.
i. What are fullerenes?
ii. How are they prepared?
Answer:
i. Fullerenes are allotropes of carbon in which carbon molecules are linked by a definite numbers of carbon atoms, for example as in C60.
ii. Fullerenes are produced when an electric arc is struck between the graphite electrodes in an inert atmosphere of argon or helium. The soot formed contains significant amount of C60 fullerene and smaller amounts of other fullerenes C32, C50, C70 and C84.

Question 47.
Discuss the structure and properties of fullerene (C6o).
Answer:

  • C60 has a shape like soccer ball and called Buckminsterfullerene or bucky ball.
  • It contains 20 hexagonal and 12 pentagonal fused rings of carbon.
  • The C60 fullerene structure exhibit separations between the neighbouring carbons as 143.5 pm and 138.3 pm.
  • Fullerenes are covalent and soluble in organic solvents.
  • Fullerene C60 reacts with group 1 metals forming solids such as K3C60.
  • The compound K3C60 behaves as a superconductor below 18 K, which means that its carries electric current with zero resistance.

Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15, 12

Question 48.
Explain the structure of carbon nanotubes.
Answer:

  • Carbon nanotubes are cylindrical in shape consisting of rolled-up graphite sheet.
  • Nanotubes can be single-walled (SWNTs) with a diameter of less than 1 nm or multi-walled (MWNTs) with diameter reaching more than 100 nm.
  • Their lengths range from several micrometres to millimetres.

Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15

Question 49.
Describe the properties of carbon nanotubes.
Answer:

  • Carbon nanotubes are robust. They can be bent, and when released, they will spring back to the original shape.
  • Carbon nanotubes have high electrical or heat conductivities and highest strength-to-weight ratio for any known material to date.
  • The researchers of NASA are combining carbon nanotubes with other materials into composites that can be used to build lightweight spacecraft.

Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15, 13

Question 50.
What is graphene?
Answer:

  • Isolated layer of graphite is called graphene.
  • Graphene sheet is a two dimensional solid.
  • It has unique electronic properties.

Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15, 14

Question 51.
Explain the structure of various allotropes of phosphorus.
Answer:
Phosphorus is found in different allotropic forms. White and red phosphorus are important allotropes of phosphorus.
i. White (yellow) phosphorus:

  1. White (yellow) phosphorus consists of discrete tetrahedral P4 molecules.
  2. The P – P – P bond angle is 60°.
  3. White phosphorus is less stable and hence more reactive, because of angular strain in the P4 molecule.

Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15, 15

ii. Red phosphorus:

  • Red phosphorus consists of chains of P4 linked together by covalent bonds.
  • Thus, it is polymeric in nature.

Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15, 16

Question 52.
Enlist properties of
i. white phosphorus.
ii. red phosphorus.
Answer:
i. Properties of white phosphorus:

  • It is translucent white waxy solid.
  • It glows in the dark (chemiluminescence).
  • It is insoluble in water but dissolves in boiling NaOH solution.
  • It is poisonous.

ii. Properties of red phosphorus:

  • It is stable and less reactive.
  • It is odourless and possess iron grey lustre.
  • It does not glow in the dark.
  • It is insoluble in water.
  • It is nonpoisonous.

Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15

Question 53.
How is red phosphorus prepared?
Answer:
Red phosphorus is prepared by heating white phosphorus at 573 K in an inert atmosphere.

Question 54.
State whether the following statement is TRUE or FALSE. Correct if false.
i. Covalent molecules have irregular shape described with the help of bond lengths and bond angles.
ii. It is difficult to understand the reactivity of covalent inorganic compounds from their structures.
iii. Inorganic molecules are often represented by molecular formulae indicating their elemental composition.
Answer:
i. False
Covalent molecules have definite shape described with the help of bond lengths and bond angles.
ii. False
The reactivity of covalent inorganic compounds is better understood from their structures.
iii. True

Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15

Question 55.
Describe structure of the following molecules.
i. Boron trichloride
ii. Aluminium chloride
iii. Orthoboric acid
Answer:
i. Structure of boron trichloride (BCl3) molecule:

  • Boron trichloride (BCl3) is a covalent compound.
  • In BCl3 molecule, boron atom is sp2 hybridized having one vacant unhybridized p orbital.
  • B in BCl3 has incomplete octet.
  • BCl3 is a nonpolar trigonal planar molecule.
  • Each Cl – B – Cl bond angle is 120°.

Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15, 17

ii. Structure of aluminium chloride (AlCl3) molecule:

  • Aluminium atom in aluminium chloride is sp2 hybridized, with one vacant unhybrid p-orbital.
  • Aluminium chloride exists as the dimer (Al2Cl6) formed by overlap of vacant 3d orbital of Al with a lone pair of electrons of Cl.

Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15, 18

iii. Structure of orthoboric or boric acid (H3BO3) molecule:

  • Orthoboric acid has central boron atom bound to three -OH groups.
  • The solid orthoboric acid has layered crystal structure in which trigonal planar B(OH)3 units are joined together by hydrogen bonds.

Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15, 19

Question 56.
Which are the different crystalline forms of silica?
Answer:
Quartz, cristobalite and tridymite are the different crystalline forms of silica.
[Note: These crystalline forms are inter-convertible at a suitable temperature.]

Question 57.
Explain the structure of silicon dioxide.
Answer:

  • Silicon dioxide (SiO2), is also known as silica.
  • It is a covalent three-dimensional network solid.
  • In SiO2, each silicon atom is covalently bound in tetrahedral manner to four oxygen atoms.
  • The crystal contains eight membered rings having alternate silicon and oxygen atoms.

Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15, 20

Question 58.
Discuss the nature and structure of the following compounds.
i. Nitric acid
ii. Phosphoric acid
Answer:
i. Nitric acid:

  • Nitric acid (HNO3) is a strong, oxidizing mineral acid.
  • The central nitrogen atom is sp2 hybridized.
  • HNO3 exhibits resonance phenomenon.
  • Figure (a) represents resonating structures of HNO3 while figure (b) represents resonance hybrid of HNO3.

Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15, 21

ii. Phosphoric acid (Orthophosphoric acid):

  • Phosphorus forms number of oxyacids. Orthophosphoric acid (H3PO4) is a strong nontoxic mineral acid.
  • It contains three ionizable acidic hydrogens.
  • The central phosphorus atom is tetrahedral.

Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15, 22

Question 59.
Give the molecular formula of crystalline borax.
Answer:
The crystalline borax has formula Na2B4O7.10H2O or Na2[B4O5(OH)4].8H2O.

Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15

Question 60.
How is borax obtained from its mineral colemanite?
Answer:
Borax is obtained from its mineral colemanite by boiling it with a solution of sodium carbonate.
Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15, 23

Question 61.
Why is the aqueous solution of borax alkaline?
Answer:
On hydrolysis, borax forms a strong base (NaOH) and a weak acid (H3BO3). The presence of the strong base makes borax solution alkaline.
Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15, 24

Question 62.
What happens when borax is heated strongly?
Answer:
Borax is a white crystalline solid. On heating, borax loses water molecules and swells up. On further heating, it melts and forms a transparent liquid, which solidifies into a glass like material known as borax bead.
Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15, 25

Question 63.
Explain borax bead test.
Answer:
i. Borax bead test is used to detect coloured transition metal ions.
ii. On heating, borax loses water molecules and swells up. On further heating, it melts and forms a transparent liquid, which solidifies into a glass like material known as borax bead.
Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15, 26
iii. The borax bead consists of sodium metaborate and boric anhydride, which reacts with metals salts to form coloured bead.
e.g. When borax is heated in a Bunsen burner flame with CoO on a loop of platinum wire, a blue coloured Co(BO2)2 bead is formed.
Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15, 27

Question 64.
Write the uses of borax.
Answer:
Borax is used

  • to manufacture optical and hard borosilicate glasses.
  • as a flux for soldering and welding.
  • as a mild antiseptic in the preparation of medical soaps.
  • in qualitative analysis for borax bead test.
  • as a brightener in washing powder.

Question 65.
How are silicones prepared? Write their properties.
Answer:
i. Preparation of silicones:
a. Alkyl or aryl substituted silicon chlorides having general formula RnSiCl(4-n) (R = alkyl or aryl group) are used as the starting materials for manufacture of silicones.
b. When methyl chloride reacts with silicon in the presence of copper catalyst at a temperature 573 K, various types of methyl substituted chlorosilane of formula MeSiCl3, Me2SiCl2, Me3SiCl with small amounts of Me4Si are formed.
Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15, 28
c. Hydrolysis of dimethyldichlorosilane, (CH3)2SiCl2 followed by condensation polymerisation yields straight chain silicone polymers.
Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15, 29

d. The chain length of polymer can be controlled by adding (CH3)3SiCl at the end.
Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15, 30

ii. Properties:

  • Silicones are water repellent.
  • They have high thermal stability.
  • They are good electrical insulators.
  • They are resistant to oxidation and chemicals.

Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15

Question 66.
Explain the preparation of ammonia from nitrogeneous organic matter.
Answer:
Ammonia is formed by the decomposition of nitrogeneous organic matter such as urea. It is therefore, present naturally in small quantities in air and soil.
Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15, 31

Question 67.
Describe laboratory method for preparation of ammonia.
Answer:
Ammonia is prepared on laboratory scale by decomposition of the ammonium salts with calcium hydroxide or caustic soda.
Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15, 32

Question 68.
How is ammonia manufactured by Haber process?
Answer:

  • On the large scale, ammonia is prepared by direct combination of dinitrogen and dihydrogen by Haber process.
  • In this process, dinitrogen reacts with dihydrogen under high pressure of 200 × 105 Pa (200 atm) and temperature around 700 K to produce ammonia.
    N2(g) + 2H2(g) ⇌ 2NH3(g); ΔfH° = -46.1 kJ mol-1
  • Iron oxide with trace amounts of K2O and Al2O3 is used as catalyst in Haber process.
  • High pressure favours the formation of ammonia as equilibrium is attained rapidly under these conditions.

Question 69.
State the physical properties of ammonia.
Answer:

  • Ammonia is a colourless gas with pungent odour.
  • It has freezing point of 198.4 K and boiling point of 239.7 K.
  • It is highly soluble in water.

Question 70.
What is liquor ammonia?
Answer:
The concentrated aqueous solution of ammonia (NH3) is called liquor ammonia.

Question 71.
Give reason: Ammonia has higher melting and boiling points.
Answer:

  • In solid and liquid state, NH3 molecules get associated together through hydrogen bonding.
  • As a result, extra amount of energy is required to break such intermolecular hydrogen bonds. Hence, ammonia has higher melting and boiling points.

Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15

Question 72.
Why is ammonia basic in aqueous solution?
Answer:
i. As ammonia is highly soluble in water, it readily forms OH ions in its aqueous solution.
\(\mathrm{NH}_{3(\mathrm{~g})}+\mathrm{H}_{2} \mathrm{O}_{(l)} \rightleftharpoons \mathrm{NH}_{4(\mathrm{aq})}^{+}+\mathrm{OH}_{(a q)}^{-}\)
ii. Thus, due to the formation of OH ions, aqueous solution of ammonia is basic in nature.

Question 73.
How does the aqueous solution of ammonia react with the following salt solutions?
i. ZnSO4
ii. FeCl3
Answer:
Aqueous solution of ammonia precipitates out as hydroxides (or hydrated oxides) of metals solutions.
Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15, 33

Question 74.
Write applications of ammonia.
Answer:
Ammonia is used in

  • manufacture of fertilizers such as urea, diammonium phosphate, ammonium nitrate, ammonium sulphate etc.
  • manufacture of some inorganic compounds like nitric acid.
  • refrigerant (liq. ammonia).
  • laboratory reagent in qualitative and quantitative analysis (aq. solution of ammonia).

Question 75.
Give reactions involved in the formation of Nessler’s reagent.
Answer:
Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15, 34

Question 76.
How does ammonia react with Nessler’s reagent?
Answer:
Ammonia react with Nessler’s reagent (an alkaline solution of K2HgI4) to form a brown precipitate (Millon’s base).
Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15, 35

Question 77.
Complete and write the balanced chemical equations for:
i. Ca2B6O11 + Na2CO3
ii. CoO + B2O3
iii. AgCl + NH3
iv. ZnSO4 + 2NH4OH →
v. a. 2KI + HgCl2
b. 2KI + HgI2
Answer:
Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15, 36

Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15

Question 78.
Naina was preparing a compound in the laboratory. She added compound ‘A’ to (CaOH)2 solution. As a result of this, a compound ‘B’ was obtained which had a pungent smell. On adding Nessler’s reagent to the compound ‘B’, a brown precipitate of compound ‘C’ was obtained.
Write the chemical reactions involved and identify ‘A’, ‘B’ and ‘C’.
Answer:
i. When ammonium chloride is mixed with (CaOH)2 solution, ammonia is formed which has a pungent odour.
Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15, 37
ii. Ammonia react with Nessler’ s reagent (an alkaline solution of K2Hgl4) to form a brown precipitate (Millon’ s base).
Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15, 38

Multiple Choice Questions

1. The electronic configuration of boron family is ……………
(A) ns2 np2
(B) ns2 np5
(C) ns2 np6
(D) ns2 np1
Answer:
(D) ns2 np1

2. ………… has noble gas core plus 14 f-electrons and 10 d-electrons.
(A) Gallium
(B) Indium
(C) Thallium
(D) Boron
Answer:
(C) Thallium

3. The group 15 element having inner electronic configuration as of argon is …………..
(A) Phosphorus (Z = 15)
(B) Antimony (Z = 51)
(C) Arsenic (Z = 33)
(D) Nitrogen (Z = 7)
Answer:
(C) Arsenic (Z = 33)

Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15

4. Which of the following is NOT a metalloid?
(A) B
(B) Sn
(C) Ge
(D) Sb
Answer:
(B) Sn

5. Among the group 13 elements, melting point is highest for …………..
(A) B
(B) Al
(C) Ga
(D) In
Answer:
(A) B

6. On moving down the group 14, the ionization enthalpy
(A) increases slightly from Si to Sn and decreases slightly from Sn to Pb
(B) increases throughout uniformly from Si to Pb
(C) decreases throughout uniformly from Si to Pb
(D) decreases slightly from Si to Sn and increases slightly from Sn to Pb
Answer:
(D) decreases slightly from Si to Sn and increases slightly from Sn to Pb

7. ………… is the most electronegative element of group 14.
(A) Carbon
(B) Silicon
(C) Germanium
(D) Tin
Answer:
(A) Carbon

8. The stability of +3 oxidation state in aqueous solution is in order ……………
(A) Al > Ga > In > Tl
(B) Tl > In > Ga > Al
(C) Al > Tl > Ga > In
(D) Tl > Al > Ga > In
Answer:
(A) Al > Ga > In > Tl

9. Group oxidation state of group 15 elements is ……………
(A) +4
(B) +1
(C) +3
(D) +5
Answer:
(D) +5

Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15

10. …………. cannot expand its octet due to absence of d orbital in its valence shell.
(A) Ga
(B) C
(C) As
(D) Ge
Answer:
(B) C

11. Which one of the following statements about boron and aluminium is INCORRECT?
(A) Both exhibit oxidation state of +3.
(B) Both form oxides of the formula M2O3.
(C) Both form trihalides, MX3.
(D) Both form amphoteric oxides.
Answer:
(D) Both form amphoteric oxides.

12. Which of the following is basic oxide?
(A) Bi2O3
(B) CO2
(C) B2O3
(D) SiO2
Answer:
(A) Bi2O3

13. The reaction of Al with H2O produces ……………
(A) Al2O3
(B) AlH3
(C) Al(OH)3
(D) Al2H6
Answer:
(C) Al(OH)3

14. Which of the following is a stable halide of nitrogen?
(A) NF3
(B) NCl5
(C) NF5
(D) NBr5
Answer:
(A) NF3

Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15

15. Catenation is the ability of …………..
(A) atoms to form strong bonds with similar atoms
(B) elements to form giant molecules
(C) an element to form multiple bonds
(D) an element to form long chains of identical atoms
Answer:
(D) an element to form long chains of identical atoms

16. Among the group 13 elements, the property of allotropy is shown by ………………
(A) indium
(B) aluminium
(C) thallium
(D) boron
Answer:
(D) boron

17. Thermodynamically stable allotrope of carbon is …………..
(A) diamond
(B) graphite
(C) buckyball
(D) all of these
Answer:
(B) graphite

18. White phosphorus contains discrete …………… molecules.
(A) P5
(B) P4
(C) P6
(D) P52
Answer:
(B) P4

19. In white phosphorus, the P-P-P bond angle is ……………
(A) 60°
(B) 90°
(C) 109.5
(D) 120°
Answer:
(A) 60°

20. 3c-2e bonds are present in ………………
(A) NH3
(B) B2H6
(C) H3BO3
(D) SiCl4
Answer:
(B) B2H6

Maharashtra Board Class 11 Chemistry Important Questions Chapter 9 Elements of Group 13, 14 and 15

21. Which of the following is borax?
(A) Na2B4O7.4H2O
(B) Na2B4O7.10H2O
(C) H3BO3
(D) NaBO2
Answer:
(B) Na2B4O7.10H2O

22. In Borax bead test, the coloured ions give characteristic coloured beads due to formation of …………….
(A) metal borates
(B) metal metaborates
(C) metal phosphates
(D) metal tetraborates
Answer:
(B) metal metaborates

23. The catalyst used in Haber process contains …………..
(A) nickel
(B) palladium
(C) iron
(D) platinum
Answer:
(C) iron

24. Which of the following is used as refrigerant?
(A) Nessler’s reagent
(B) Liq. ammonia
(C) Borax
(D) Diborane
Answer:
(B) Liq. ammonia

Maharashtra Board Class 11 Hindi रचना वृत्तांत लेखन

Balbharti Maharashtra State Board Hindi Yuvakbharati 11th Digest रचना वृत्तांत लेखन Notes, Questions and Answers.

Maharashtra State Board 11th Hindi रचना वृत्तांत लेखन

वृत्तांत लेखन : किसी भी सभा, बैठक, कार्यक्रम आदि को लिखित रूप में प्रस्तुत करना ही वृत्तांत लेखन है।

  • वृत्तांत संक्षिप्त होना चाहिए और क्रमबद्धता का विशेष ध्यान रखना चाहिए।
  • वृत्तांत लेखन में उत्तम पुरुष वाचक सर्वनाम (मैं, हम) का प्रयोग नहीं करना चाहिए।
  • वृत्तांत में घटना, समय, स्थान आदि का स्पष्ट उल्लेख होना चाहिए।
  • यह सत्य घटना पर आधारित लेखन होता है।

Maharashtra Board Class 11 Hindi रचना विज्ञापन लेखन

1. हिंदी दिवस का वृत्तांत लिखिए।

एक शानदार हिंदी दिवम

15 सितंबर मुंबई : स्वामी विवेकानंद ज्युनियर कॉलेज आफॅ आर्ट्स एंड कॉमर्स के सभागार में दोपहर तीन बजे प्रधानाचार्य महोदय की अध्यक्षता में एक शानदार कार्यक्रम संपन्न हुआ। हिंदी विभागाध्यक्ष डॉ. किशोर सिंह ने राष्ट्रभाषा के रूप में हिंदी तथा बारहवीं कक्षा की छात्रा शिवानी और रुही में हिंदी की आज की स्थिति पर अपने विचार रखे। ग्यारहवी कक्षा के छात्रों ने राष्ट्रभक्ति पर गीत प्रस्तुत किए।

इस अवसर पर दोहों की प्रतियोगिता रखी गई जिसकी वजह से कार्यक्रम में जान आ गई थी। रमा राणे इस प्रतियोगिता में विजयी हुई जिसे पाँच सौ रुपए का पुरस्कार और सर्टीफिकेट प्रदान किया गया।

प्रधानाचार्य ने इस कार्यक्रम में घोषणा की, कि इस वर्ष हिंदी में सर्वाधिक अंक प्राप्त करने वाले ग्यारहवीं तथा बारहवीं के छात्र को एक हजार रूपए का नकद पुरस्कार दिया जाएगा। सभी उपस्थित लोगों को अध्यक्ष महोदय ने धन्यवाद दिया और सभा विसर्जित हुई।

(कार्यालय प्रतिनिधि द्वारा)

2. महाविद्यालय में आयोजित वृक्षारोपण समारोह का वृत्तांत लिखीए।

अध्यापक और छात्रों द्वारा वृक्षारोपण

17 जुलाई, दिल्ली: आज 16 जुलाई को विकास महाविद्यालय हरिनगर, दिल्ली के महाविद्यालय परिसर में वृक्षारोपण समारोह अत्यंत हर्षोल्लास एवं उत्साहपूर्वक मनाया गया। इस कार्यक्रम में शिक्षा निदेशक मुख्य अतिथि के रूप में उपस्थित थे। उन्हें पुष्पगुच्छ एवं एक पौधा देकर स्वागत किया गया।

इसके पश्चात छात्र-छात्राओं ने एक स्वर में, ‘नंगी धरती करे पुकार, वृक्ष लगाकर करो शृंगार’ गीत का समूहगान प्रस्तुत किया। अतिथि महोदय ने अपने भाषण में वृक्षों के महत्त्व पर प्रकाश डाला। उन्होंने स्वयं एक पौधा लगाकर कार्यक्रम का शुभारंभ किया।

फिर प्रधानाचार्य, उपप्रधानाचार्य तथा अध्यापकों ने वृक्षारोपण किया। छात्रों ने भी वृक्षारोपण करते हुए उनके देखभाल की जिम्मेदारी ली। पौधों के चारों ओर जाली लगाकर इनकी सिंचाई का प्रबंध किया गया ताकि पौधे फलें- फूलें और वृक्ष बन सकें। अंत में छात्रों में मिठाई वितरण करते हुए इस समारोह का समापन किया गया।

Maharashtra Board Class 11 Hindi रचना विज्ञापन लेखन

(कार्यालय प्रतिनिधि द्वारा)

3. समता विद्यालय में पुरस्कार वितरण समारोह संपन्न

(कार्यालय प्रतिनिधि द्वारा)

पुणे, 12 फरवरी: कल 11 फरवरी, को समता विद्यालय में वार्षिक पुरस्कार वितरण समारोह संपन्न हुआ। समारोह की अध्यक्षता मशहूर अभिनेता शेखर सेन ने की थी।

ईश – स्तवन और गणेश वंदना से कार्यक्रम आरंभ हुआ। पाँचवी कक्षा के छात्रों ने स्वागत गीत प्रस्तुत किया। तत्पश्चात विद्यालय के निरीक्षक श्री. अशोक कर्वे ने अध्यक्ष महोदय का परिचय और विद्यालय की विभिन्न गतिविधियों का विवरण प्रस्तुत किया।

अध्यक्ष महोदय के करकमलों से आदर्श विद्यार्थी, सर्वश्रेष्ठ खिलाड़ी, अभिनेता, नर्तक, गायक आदि पुरस्कार दिए गए। शालांत परीक्षा में विशेष योग्यता दिखलाने वाले मेधावी छात्रों को सम्मानित किया गया। अध्यक्ष महोदय ने अपने मार्गदर्शन पर भाषण में विद्यार्थियों को अनुशासन का पाठ पढ़ाया और उज्ज्वल भविष्य की कामना की। तत्पश्चात विद्यालय की ज्येष्ठ शिक्षिका श्रीमती चौधरी जी ने धन्यवाद यापन किया और राष्ट्रगीत के साथ समारोह का समापन हुआ।

Maharashtra Board Class 11 Hindi रचना वृत्तांत लेखन 1

नासिक, 7 फरवरी : विवेक महाविद्यालय के प्रांगण में बारहवीं कक्षा के छात्रों का बिदाई समारोह 6 फरवरी को संपन्न हुआ। अपने महाविद्यालय से विदा लेते समय विद्यार्थियों की आँखें छलक आईं। समारोह की अध्यक्षता महाविद्यालय के प्रधानाचार्य ने की।

Maharashtra Board Class 11 Hindi रचना विज्ञापन लेखन

ग्यारहवीं कक्षा के छात्रों ने बिदाई समारोह के आयोजन में अहं भूमिका निभाई। ठीक तीन बजे छात्र और शिक्षक प्रांगण में उपस्थित हुए थे। महाविद्यालय को रंगीन कागजों से सजाया गया था। एक छोटा सा वृक्ष का चित्र दीवार पर बना था और छात्र उस पर अपनी भावनाओं सरस्वती वंदना से कार्यक्रम की शुरुआत हुई। बारहवीं कक्षा के छात्रों ने अपने महाविद्यालय की यादें बताते हुए अपने शिक्षिकों का आभार प्रकट किया।

प्रधानाचार्य और वर्गशिक्षकों तथा ज्येष्ठ शिक्षकों ने विद्यार्थियों को मार्गदर्शन पर दो शब्द कहे। ग्यारहवीं कक्षा के छात्रों ने नृत्य एवं नाटिका प्रस्तुत कर सबका मनोरंजन किया। प्रधानाचार्य के करकमलों द्वारा आदर्श छात्र एवं छात्रा को पुरस्कृत किया गया। उसके बाद अल्पाहार दिया गया। छात्र अपने प्रिय शिक्षकों के साथ तस्वीरें लेते हुए देखे गए। बड़ा ही भावपूर्ण प्रसंग था यह, जो शाम सात बजे खत्म हुआ।

Maharashtra Board Class 11 Hindi रचना संभाषण लेखन

Balbharti Maharashtra State Board Hindi Yuvakbharati 11th Digest रचना संभाषण लेखन Notes, Questions and Answers.

Maharashtra State Board 11th Hindi रचना संभाषण लेखन

रोजमर्रा के जीवन में हम जो बातचीत या वार्तालाप करते हैं उसके लिखित रूप को संवाद-लेखन कहते हैं। वार्तालाप जितना चतुराई से किया गया है, उतना ही वह प्रभावशाली होता है। संवाद-लेखन करते समय ध्यान रहे –

  • संवाद की भाषा सरल एवं प्रभावशाली हो।
  • संवाद संक्षिप्त होने चाहिए।
  • संवाद विषय और पात्र के अनुकूल होने चाहिए।
  • संवाद लेखन में उचित विराम चिह्नों का प्रयोग किया जाना चाहिए।

Maharashtra Board Class 11 Hindi रचना संभाषण लेखन

प्रश्न 1.
निम्नलिखित जानकारी के आधार पर संवाद-लेखन कीजिए :
Maharashtra Board Class 11 Hindi रचना संभाषण लेखन 1
उत्तर :
अनन्या : हे रक्षिता! आज कितने दिनों बाद दिखाई दे रही हो; कहाँ थी?
रक्षिता : माँ के साथ दुर्घटना हो गई थी, इसलिए इधर आना नहीं हुआ।

अनन्या : ओह! क्या हुआ था?
रक्षिता : किसी बदमाश ने राह चलते उनके गले का मंगलसूत्र खींच लिया था। उनके गले पर जख्म हो गया था।

अनन्या : ओह! यह तो बहुत बुरा हुआ। आजकल दिनदहाड़े ऐसी वारदातें होने लगी हैं।
रक्षिता : काश! पुलिस अपनी जिम्मेदारियाँ ठीक से निभा पाते, तो बदमाशों की ऐसी हिम्मत नहीं होती।

अनन्या : (क्रोधित स्वर में) और हमारी पब्लिक तमाश-बिन की तरह केवल भीड़ इकट्ठा करती है।
रक्षिता : (सहमति प्रकट करते हुए) और नहीं तो क्या! हम लोग कब जिम्मेदार नागरिक बनेंगे?

अनन्या : क्या, बदमाश पकडा गया?
रक्षिता : नहीं तो! वह तो बाइक पर था, झपट्टा मारा और भाग गया।

अनन्या : खैर, तुम अपनी माँ का ख्याल रखो। मुझसे कोई सहायता चाहिए तो बिना हिचकिचाए बताना।
रक्षिता : अवश्य! चलती हूँ। अलविदा!

अनन्या : अलविदा !

Maharashtra Board Class 11 Hindi रचना संभाषण लेखन

प्रश्न 2.
निम्नलिखित जानकारी के आधार पर संवाद-लेखन कीजिए :
Maharashtra Board Class 11 Hindi रचना संभाषण लेखन 2
उत्तर :
पिताजी : बेटा, यहाँ आओ; बैठो। पढ़ाई के क्या हाल है? परीक्षा कब से है?
पुत्र : (घबराते हुए) परीक्षा नजदीक आई है और पढ़ाई अभी चल रही है, पूरी नहीं हुई।

पिताजी : पूरी होगी कैसे? मैं यह नहीं कहता कि दूरदर्शन मत देखो, लेकिन अपने अध्ययन के प्रति लापरवाही उचित नहीं। पूरा दिन दूरदर्शन के सामने बैठोगे तो पढ़ाई होगी ही नहीं।
पुत्र : लेकिन मैं अकेला थोड़े ही देखता हूँ, पापा? घर में सभी दूरदर्शन देखते हैं और आप मुझे ही डाँटते हो।

पिताजी : मैं तुम्हें समझा रहा हूँ। तुम्हारी दीदी को देखो, कक्षा में अव्वल आती है और तुम मात्र 50% अंक ला पाए हो।
पुत्र : (अँगूठे से जमीन कुरेदते हुए) मुझे दीदी अपने साथ पढ़ने को मना करती है।

पिताजी : मैंने दीदी को समझा दिया है, जाओ अब उसके साथ बैठकर पढ़ाई करो।
पुत्र : जी, पापा। अब मैं भी दीदी जैसे अंक लाकर दिखाऊँगा।

पिताजी : बहुत अच्छा! देर आए दुरुस्त आए। मुझे तुम पर विश्वास है। तुम जरूर अच्छे अंक ला पाओगे। भगवान तुम्हें सद्बुद्धि दें।

Maharashtra Board Class 11 Hindi रचना संभाषण लेखन

प्रश्न 3.
निम्नलिखित जानकारी के आधार पर संवाद-लेखन कीजिए :
Maharashtra Board Class 11 Hindi रचना संभाषण लेखन 3
उत्तर :
राम : नमस्ते श्याम! कैसे हो?
श्याम : नमस्ते! मैं ठीक हूँ। तुम कैसे हो? क्या कर रहे हो आजकल?

राम : मैं ठीक हूँ। आजकल मैं डाक टिकट इकट्ठा कर रहा हूँ।
श्याम : बहुत खूब! क्या तुम इन्हें अलबम में चिपकाओगे?

राम : हाँ! मैंने एक अलबम बना लिया है और टिकट चिपका भी दिए हैं।
श्याम : वाह! क्या, तुम्हारे पास सभी देशों के टिकट हैं?

राम : हाँ! ज्यादातर सभी देशों के टिकट मेरे पास हैं।
श्याम : (जिज्ञासा से) तो क्या इनमें, महँगी टिकटें भी हैं?

राम : मेरे पास बहुत सारी टिकटें है जिनमें कुछ टिकटें महँगी भी है।
श्याम : मेरे दोस्त, यह तो बता इस संग्रह से तुझे क्या लाभ होता है?

राम : यह मेरा शौक है जो मुझे बेहद सुख प्राप्त कराता है और हाँ, मुझे भूगोल पढ़ने में इनकी मदद मिलती है।
श्याम : बहुत अच्छा।

Maharashtra Board Class 11 Hindi रचना संभाषण लेखन

राम : तुम्हारा भी कोई शौक है?
श्याम : है न! मुझे जंगली फूल जमा करने का शौक है।

राम : तुम उनसे क्या करते हो?
श्याम : मैं उन्हें कागज पर चिपकाता हूँ और फिर उनका नाम लिखता हूँ।

राम : इस शौक से तुम्हें क्या लाभ होता है?
श्याम : मेरा शौक मेरा वनस्पति विज्ञान का ज्ञान बढ़ाता है।

राम : तुमसे मिलकर आज बहुत अच्छा लगा। फिर मिलेंगे, अलविदा!
श्याम : अच्छा! अलविदा!

प्रश्न 4.
निम्नलिखित जानकारी के आधार पर संवाद-लेखन कीजिए।
Maharashtra Board Class 11 Hindi रचना संभाषण लेखन 4
उत्तर :
साक्षात्कार कर्ता : जय हिंद, मेरे भाई, मेरे दोस्त!
सैनिक : जय हिंद! कहो कैसे आना हुआ?

साक्षात्कार कर्ता : आप हाल ही में सीमा पर दुश्मनों को लोहे के चने चबवाकर आए हो इसलिए मैं आपसे मिलना चाहता हूँ। आपके बारे में जानना चाहता हूँ।
सैनिक : भाई, मैंने तो सिर्फ अपना कर्तव्य निभाया है।

साक्षात्कार कर्ता : सीमा पर अपने परिवार से दूर आप कैसे रह लेते हो? क्या आपका उनके प्रति कर्तव्य नहीं है?
सैनिक : ऐसा तो नहीं। परिवार अपनी जगह है, देश अपनी जगह और देश की खातिर जो कर्तव्य है उसके आगे हमें निजी सुख-दुख बहुत छोटे नजर आते हैं।

Maharashtra Board Class 11 Hindi रचना संभाषण लेखन

साक्षात्कार कर्ता : धन्य हैं आप! सुना है आपका बेटा छह वर्ष का है!
सैनिक : सही सुना है आपने। मेरा छह वर्ष का बेटा है जो अभी से सैनिक बनने का सपना देख रहा है और हाँ मेरी तीन साल की बेटी भी, शत्रु के साथ युद्ध करने की बातें करती है।

साक्षात्कार कर्ता : बहुत अच्छा लगा सुनकर। आपका पूरापरिवार ही राष्ट्रभक्ति से ओत-प्रोत है। हम देशवासियों को और देश के बालकों को (जोर देते हुए) कुछ संदेश देना चाहेंगे?
सैनिक : जरूर! देश के बालकों आप देश का भविष्य हो। हर काम को सच्चाई, ईमानदारी और खुशी से करो। अपने सपनों को हकीकत में बदलो। ईश्वर तुम्हें सदबुद्धि दे। वंदे मातरम्! जय हिंद!

साक्षात्कार कर्ता : वंदे मातरम्! जय हिंद!

Maharashtra Board Class 11 Hindi रचना गद्य आकलन

Balbharti Maharashtra State Board Hindi Yuvakbharati 11th Digest रचना गद्य आकलन Notes, Questions and Answers.

Maharashtra State Board 11th Hindi रचना गद्य आकलन

अपठित अर्थात जो पहले से पढ़ा / पढ़ाया न गया हो ऐसा परिच्छेद परीक्षा में दिया जाता है। इसे पढ़कर इसका आशय समझना होता है। कोई शब्द परिचित न हो और अर्थ समझ में नहीं आ रहा हो तो उसके अर्थ को वाक्य के प्रसंगानुसार ग्रहण करना चाहिए और सब कुछ समझ में आ जाने पर प्रश्न बनाना आसान हो जाएगा।

Maharashtra Board Class 11 Hindi रचना गद्य आकलन

महत्त्वपूर्ण : छात्रों से अपठित गद्यांश पर आकलन हेतु मात्र प्रश्न निर्माण अपेक्षित है और प्रश्न भी ऐसे बनाने हैं जिनके उत्तर एक वाक्य में हों। हो सके उतना गद्यांश के लिए शीर्षक के बारे में प्रश्न न पूछे। आगे कुछ उदाहरण दिए हैं –

प्रश्न 1.
निम्नलिखित गद्यांश पढ़कर पाँच ऐसे प्रश्न तैयार कीजिए जिनके उत्तर एक-एक वाक्य में हों।

पंडित जवाहरलाल नेहरूजी की अंतिम इच्छा यह थी कि मैं जब मरूँ तब मैं चाहूँगा कि मेरा दाह-संस्कार हो। अगर मैं विदेश में मरूँ तो मुझे वहीं जलाया जाए पर मेरी अस्थियाँ इलाहाबाद लाई जाएँ। मुठ्ठीभर भस्म इलाहबाद की गंगा में प्रवाहित करने की मेरी इच्छा है, किंतु उसके पीछे कुछ धार्मिक भावना नहीं है, क्योंकि गंगा हमारी सदियों से पुरानी सभ्यता और संस्कृति की प्रतीक रही है।

वह मुझे हिमालय के हिमाच्छादित शिखरों और नदियों की याद दिलाती है, जिनमें मेरा लगाव और प्यार बहुत ज्यादा रहा है। गंगा मुझे शस्य-श्यामल फैले हुए मैदानों की याद दिलाती है, यहाँ मेरी जिंदगी और काम ढले हैं। गंगा में कहीं समुद्र जैसी विनाश की भी शक्ति मुझे लगती है और उसकी यह शक्ति मेरे लिए अतीत की प्रतीक व स्मृति है, जो वर्तमान में प्रवाहित है और भविष्य के महासमुद्र में आगे बढ़ते रहने की है।
उत्तर:

  1. मुठ्ठीभर भस्म का विसर्जन लेखक ने कहाँ करने के लिए कहा है?
  2. गंगा की कौन-सी शक्ति लेखक के लिए अतीत की प्रतीक व स्मृति है?
  3. लेखक की जिंदगी और काम कहाँ ढले हैं?
  4. पंडित जवाहरलाल नेहरू जी को गंगा नदी किसकी याद दिलाती है?
  5. विदेश में मरने पर पंडित जवाहरलाल नेहरू जी क्या चाहते हैं?

प्रश्न 2.
निम्नलिखित गद्यांश पढ़कर पाँच ऐसे प्रश्न तैयार कीजिए जिनके उत्तर एक-एक वाक्य में हों।
वर्तमान शासन प्रणालियों में जनतंत्र से बढ़कर उत्तम कोई प्रणाली नहीं हैं, क्योंकि उसमें जनता को स्वयं यह अधिकार प्राप्त रहता है कि वह अपने प्रतिनिधियों को चुनकर विधान सभाओं और संसद में भेजें। ऐसे प्रत्यक्ष चुनाव में प्राय: वही व्यक्ति चुना जाता है, जिसका सार्वजनिक जीवन अच्छा हो और जो जनता की सेवा करता हो। इस प्रणाली में जनता को यह अधिकार है कि यदि वह किसी दल या किसी व्यक्ति के कार्यों से संतुष्ट नहीं है तो दूसरी बार उस दल या व्यक्ति को अपना मत न दें।

निर्वाचन में विरोधी दलों के भी कुछ व्यक्ति चुने जाते हैं, जो अपनी आलोचना से शासक दल के स्वेच्छाचार पर अंकुश रखते हैं। इस प्रकार देश की शासन प्रणाली में विरोधी दलों का भी महत्त्वपूर्ण स्थान होता है।
उत्तर:

  1. जनता अपने प्रतिनिधियों को चुनकर कहाँ भेजती है?
  2. चुनाव में कैसा व्यक्ति चुना जाता है?
  3. विरोधी दल अपनी आलोचना से क्या कर सकता है?
  4. जनतंत्र में जनता को किस बात का अधिकार होता है?
  5. सबसे उत्तम शासन प्रणाली कौन-सी है?

Maharashtra Board Class 11 Hindi रचना गद्य आकलन

प्रश्न 3.
निम्नलिखित गद्यांश पढ़कर पाँच ऐसे प्रश्न तैयार कीजिए जिनके उत्तर एक-एक वाक्य में हों।
दान देने की परिपाटी प्राचीन काल से चली आ रही है। अन्नदान, गोदान, वस्त्रदान, स्वर्णदान, भूमिदान करना भारतीय अपना परम धर्म मानते हैं। धर्म से स्वर्ग की प्राप्ति होती है, ऐसा माना जाता है। प्राचीन काल में विद्यादान को सर्वश्रेष्ठ दान माना जाता था। वर्तमान काल में कुछ नए प्रकार के दान प्रचलित हुए है – नेत्रदान, रक्तदान, किडनीदान । रक्त तो हर व्यक्ति के लिए जरूरी है। पचास वर्ष तक के निरोगी स्त्री-पुरुष रक्त दान कर सकते हैं। दुर्घटनाओं से परिपूर्ण वैज्ञानिक युग में रक्तदान, सर्वश्रेष्ठ दान माना जा रहा है। नेत्रदान करने से घबराना नहीं चाहिए क्योंकि मरणोपरांत ही आँखें निकालकर, अंधों को दी जाती हैं और वे देखने लगते हैं। बीमार के प्राण बचाने के लिए हम अपनी किडनी दान दे सकते हैं।
उत्तर :

  1. प्राचीन काल से लेकर अब तक कौन-कौन से दान प्रचलित हैं?
  2. किन लोगों को रक्तदान करना चाहिए?
  3. वैज्ञानिक युग में कौन-सा दान श्रेष्ठ है?
  4. दान करना भारतीय अपना परम धर्म क्यों मानते हैं?
  5. नेत्रदान करने से घबराने की जरूरत क्यों नहीं?

प्रश्न 4.
निम्नलिखित गद्यांश पढ़कर पाँच ऐसे प्रश्न तैयार कीजिए जिनके उत्तर एक-एक वाक्य में हों।

भारत में प्राचीन काल से दहेज प्रथा चली आ रही है। कन्या के माता-पिता अपनी क्षमता के अनुसार शादी के समय दहेज देते चले आए हैं। वर एवं कन्या के परिवारवालों में आपसी प्रेम था इसलिए वरवाले कन्यावालों से किसी प्रकार की माँग करने में संकोच करते थे।

परंतु पिछले 50 वर्षों से विवाह एक व्यापार बन गया है। इससे समाज दुखी है। लड़कीवाला लड़के की योग्यता के स्थान पर धन को ही सर्वस्व मानता है और वह बड़े अमीर परिवार में अपनी लड़की को देना चाहता है। लड़का लड़कियों को देखता है।

जिस लड़की के पास धन अधिक होता है, उसे चुन लेता है। उसकी योग्यता को नहीं देखता। आज लड़की के विवाह का मूल आधार धन बन गया है। जिस दिन लड़की का जन्म होता है, उसी दिन से माता-पिता को उसके विवाह की चिंता लग जाती है।

इस बुराई को दूर करने के लिए हमें मिलकर इस प्रथा का विरोध करना चाहिए। जो दहेज लेता है, उसके लिए ऐसा कानून बनना चाहिए कि दहेज लेनेवाले को चोरी, जुआ एवं हत्या आदि अपराध करनेवालों के समान देखा जाए और सामाजिक मंच पर उसे बेइज्जत किया जाए।

इस विषय पर मात्र बोलने एवं लिखने से अब काम नहीं चलेगा। हमें एक होकर इस प्रकार के विरोध में कदम बढ़ाने होंगे।

Maharashtra Board Class 11 Hindi रचना गद्य आकलन

प्रश्न :
(1) भारत में प्राचीन काल में दहेज प्रथा का स्वरूप कैसा था?
(2) माता-पिता को लड़की के विवाह की चिंता कब से लग जाती है?
(3) आज लड़की के विवाह का मूल आधार क्या बन गया है?
(4) दहेज लेने वाले के साथ कैसा व्यवहार किया जाना चाहिए?
(5) लड़की के विवाह का मूल आधार क्या बन गया है?

प्रश्न 5.
निम्नलिखित गद्यांश पढ़कर पाँच ऐसे प्रश्न तैयार कीजिए जिनके उत्तर एक-एक वाक्य में हों।
पवन पुत्र हनुमान, भीष्म पितामह, स्वामी दयानंद सरस्वती, स्वामी विवेकानंद जैसे बाल ब्रह्मचारियों ने भारत-भूमि को पावन किया है। संसार के प्राचीन ग्रंथ वेदों में लिखा है: “ब्रह्मचारी मृत्यु को जीत लेते हैं।” ‘ब्रह्म’ शब्द के अर्थ हैं ‘परमेश्वर, विद्या और शरीर-रक्षण। ब्रह्मचर्य के पालन से शरीर स्वस्थ होता है।

जिसका शरीर स्वस्थ उसीका मन स्वस्थ, जिसका मन स्वस्थ उसकी स्मरण-शक्ति बहुत होती है। स्मरण-शक्ति से आकलन शक्ति बढ़ती है । विद्यार्थी जीवन में आकलनशक्ति का अपना विशेष महत्त्व है। ब्रह्मचर्य विद्यार्थी जीवन की कमियाँ पूरी करता है।

प्राचीन भारतीय साहित्य में ब्रह्मचर्य की महिमा लिखी है। इसका पालन करनेवाला विद्यार्थी निरोगी, बुद्धिमान, संपत्तिशाली, महान बनता है। ब्रह्मचर्य की महिमा अपार है।

प्रश्न:
(1) कौन-कौन बाल ब्रह्मचारी थे?
(2) मृत्यु को कौन जीत सकते हैं?
(3) ‘ब्रह्म’ शब्द के कितने और कौन-कौन से अर्थ हैं?
(4) विद्यार्थी जीवन में किसका विशेष महत्त्व है?
(5) ब्रह्मचर्य पालन करने वाला विद्यार्थी कैसा होता है?

Maharashtra Board Class 11 Hindi रचना गद्य आकलन

स्वाध्याय

निम्नलिखित गद्यांश पढ़कर पाँच ऐसे प्रश्न तैयार कीजिए जिनके उत्तर एक वाक्य में हों –

(1) हँसने का एक सामाजिक पक्ष भी होता है। हँसकर हम लोगों को अपने निकट ला सकते हैं और व्यंग्य उन्हें दूरस्थ बना देते हैं। जिसको भगाना हो उसकी थोड़ी देर हँसी खिल्ली उड़ाइए, वह तुरंत बोरिया-बिस्तर गोल कर पलायन करेगा। जितनी मुक्त हँसी होगी, उतना समीप व्यक्ति खींचेगा इसीलिए तो श्रोताओं की सहानुभूति अपनी ओर खींचने के लिए चतुर वक्ता अपना भाषण किसी रोचक कहानी या घटना से आरंभ करते हैं।

जनता यदि हँसी तो चंगुल में फँसी। सामाजिक मूल्यों और नियमों को मान्यता दिलाने और रूढ़ियों को निष्कासित करने में पुलिस या कानून सहायता नहीं करता, किन्तु वहाँ हास्य का चाबुक अचूक बैठता है। हास्य के कोड़े, उपहास-डंक और व्यंग्य-बाण मारकर आदमी को रास्ते पर लाया जा सकता है। इस प्रकार गुमराह बने समाज की रक्षा की जा सकती है।

(2) किसी भी देश या काल के लिए जवान तथा शिक्षक दोनों ही महत्त्वपूर्ण हैं। किसी एक के बिना समाज सुरक्षित नहीं रह सकता। दोनों ही समाज के रक्षक हैं, किन्तु कार्यों में भिन्नता दिखाई पड़ती है। एक शत्रु से रक्षा करता है तो दूसरा उसे (देश को) समृद्ध बनाता है।

फिर भी शिक्षक का उत्तरदायित्व जवान से कहीं बढ़कर है। भावी नागरिक निर्माण करने की जिम्मेदारी शिक्षक के ऊपर है। वह उसके शारीरिक, मानसिक तथा नैतिक विकास का जनक है, जिस पर व्यक्ति, समाज तथा राष्ट्र निर्भर है। शिक्षक के ही द्वारा कोई योग्य सैनिक बन सकता है।

आज शिक्षक ने सैनिक धाराओं में क्रांति पैदा कर दी है। हमारे अहिंसक आंदोलन ने दुनिया को दिखा दिया है कि शिक्षक सैनिक से श्रेष्ठ है। इसे बनावटी-शस्त्रों की जरूरत नहीं है। इसका आत्मिक बल सब शस्त्रों से बड़ा है।

Maharashtra Board Class 11 Hindi रचना गद्य आकलन

अगर आनेवाली दुनिया इसका अनुसरण करे तो शस्त्रीकरण का नामोनिशान भू-पृष्ठ से उठ जाएगा। नैतिक शक्ति का बोल-बाला होगा, सारी दुनिया में एकात्मता की लहर फैलेगी और तब ज्ञान-विज्ञान का निर्माण विकास के लिए होगा, न कि विनाश के लिए।

(3) समाजसुधार आंदोलन को निर्भीक संन्यासी स्वामी श्रद्धानंद से नई दिशा मिली। हरिजन समस्या के समाधान में कई स्थानों पर संघर्ष का भी सामना करना पड़ा। गुरुकुल काँगड़ी के छात्रवासों और भोजनालयों में बिना किसी भेदभाव के हर जाति के विद्यार्थी रहते और खाते-पीते थे।

स्वामी जी का कहना था – मनों से छुआछूत की भावना मिटाने में आवासीय शिक्षण संस्थाओं का अच्छा योगदान हो सकता है। चौबीसों घंटे एक साथ मिलकर जब रहेंगे, खेलेंगे, कूदेंगे और पढ़ेंगे, लिखेंगे तो कहाँ तक छूत-अछूत की दीवार खड़ी रह पाएगी।

आजादी के बाद भी यदि इसी रास्ते को पकड़ा गया होता तो मंजिल बहुत पहले तय हो जाती। आवासीय पद्धति पर आश्रित ऐसे गुरुकुल उन्होंने हरियाणा में झज्जर, इंद्रप्रस्थ और कुरूक्षेत्र, गुजरात में सोनगढ़ और सूपा में भी खोले। देहरादून का कन्या गुरुकुल भी उसी श्रृंखला की कड़ी है।

(4) यश और कीर्ति पैतृक संपत्ति नहीं है। जिसका सुख-भोग संतान कर सके। वास्तविक सम्मान और यश धन के द्वारा भी प्राप्त नहीं हो सकता। ये वे पदार्थ है जो घोर परिश्रम और स्वावलंबन द्वारा ही प्राप्त हो सकते हैं। ईश्वर का वरद हस्त भी उसी के शीश पर है जो स्वत: अपनी सहायता करता है।

यदि तुम अपना जीवन धन्य बनाना चाहते हो तो खड़े हो जाओ अपने पैरों पर और संसार में एक बार शक्ति से अपने कार्यो से सुख और शांति की धारा प्रवाहित कर दो। भाग्य की भाषा पढ़ने के फेरे में जो भी डूबा वह कभी भी ऊपर नहीं आ सकता। अत: यह निश्चित है कि तुम ही अपने भाग्य विधाता हो और जीवन निर्माण करने का संपूर्ण अधिकार भी तुम ही को है।

(5) संसार में कुछ भी असाध्य नहीं है। कुछ भी असम्भव नहीं है। असम्भव, असाध्य, कठिन आदि शब्द कायरों के लिए हैं।

नेपोलियन के लिए ये शब्द उसके कोष में नहीं थे। साहसी पतले बापू ने विश्व को चकित कर दिया। क्या बापू शरीर से शक्तिशाली थे? नहीं। वह तो पतली-सी एक लंगोटी पहने लकड़ी के सहारे चलते थे, परंतु विचार सशक्त थे, भावनाएँ शक्तिशाली थीं, उनके साहस को देखकर करोड़ो भारतीय उनके पीछे थे। ब्रिटिश साम्राज्य उनसे काँप गया। अहिंसा के सहारे बिना रक्त-पात के उन्होंने भारत को स्वतंत्र कराया। यह विश्व का एक अद्वितीय उदाहरण है।

Maharashtra Board Class 11 Hindi रचना गद्य आकलन

जब महात्मा गांधीजी ने अहिंसा का नारा लगाया तो लोग हँसते थे, कहते थे अहिंसा से कहीं ब्रिटिश साम्राज्य से टक्कर ली जा सकती है? परंतु वे डटे रहे, साहस नहीं छोड़ा, अंत में अहिंसा की ही विजय हुई। कहते हैं, अकेला चना क्या भाड़ फोड़ सकता है? हाँ, यदि उसमें साहस हो तो! साहसहीन के लिए सब कुछ असम्भव है। उससे अगर कहा जाए कि भाई जरा वह काम कर देना; तो वह तुरंत कहेगा, अरे! इतनी दूर!

पैदल, एक दिन में! नहीं भाई, मुझसे नहीं हो सकेगा, किसी और से करा लो। भला वह इस काम को कैसे करेगा? करने वाला दूरी और पैदल नहीं देखता! उसके मार्ग में चाहे पर्वत आकर खड़े हो जाएँ, आँधी आए या तूफान, उसको उनसे क्या वास्ता? उसको तो अपने लक्ष्य तक पहुँचना है, उसे कोई नहीं रोक सकता है। वह अपने लक्ष्य तक अवश्य पहुँच जाएगा। साहसी पुरुष दिन-रात नहीं देखा करते, आँधी तूफान, नदी-नाले, पहाड़-समुद्र नहीं देखा करते; वे तो केवल एक ही चीज देखा करते हैं कि उन्हें अपने लक्ष्य तक पहुँचना है।