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Sequences and Series Class 11 Commerce Maths 1 Chapter 4 Exercise 4.1 Answers Maharashtra Board

Balbharati Maharashtra State Board 11th Commerce Maths Solution Book Pdf Chapter 4 Sequences and Series Ex 4.1 Questions and Answers.

Std 11 Maths 1 Exercise 4.1 Solutions Commerce Maths

Question 1.
Verify whether the following sequences are G.P. If so, write tn.
(i) 2, 6, 18, 54, ……
(ii) 1, -5, 25, -125, …….
(iii) \(\sqrt{5}, \frac{1}{\sqrt{5}}, \frac{1}{5 \sqrt{5}}, \frac{1}{25 \sqrt{5}}, \ldots\)
(iv) 3, 4, 5, 6, ……
(v) 7, 14, 21, 28, …..
Solution:
(i) 2, 6, 18, 54, …….
t₁ = 2, t₂ = 6, t₃ = 18, t₄ = 54, …..
Here, \(\frac{t_{2}}{t_{1}}=\frac{t_{3}}{t_{2}}=\frac{t_{4}}{t_{3}}=3\)
Since, the ratio of any two consecutive terms is a constant, the given sequence is a geometric progression.
Here, a = 2, r = 3
t_n= ar^n-1
∴ t_n = 2(3^n-1)

(ii) 1, -5, 25, -125, ……
t₁ = 1, t₂ = -5, t₃ = 25, t₄ = -125, …..
Here, \(\frac{t_{2}}{t_{1}}=\frac{t_{3}}{t_{2}}=\frac{t_{4}}{t_{3}}=-5\)
Since, the ratio of any two consecutive terms is a constant, the given sequence is a geometric progression.
Here, a = 1, r = -5
t_n = ar^n-1
∴ t_n = (-5)^n-1

(iii) \(\sqrt{5}, \frac{1}{\sqrt{5}}, \frac{1}{5 \sqrt{5}}, \frac{1}{25 \sqrt{5}}, \ldots\)

Since, the ratio of any two consecutive terms is a constant, the given sequence is a geometric progression.

(iv) 3, 4, 5, 6,……
t₁ = 3, t₂ = 4, t₃ = 5, t₄ = 6, …..
Here, \(\frac{\mathrm{t}_{2}}{\mathrm{t}_{1}}=\frac{4}{3}, \frac{\mathrm{t}_{3}}{\mathrm{t}_{2}}=\frac{5}{4}, \frac{\mathrm{t}_{4}}{\mathrm{t}_{3}}=\frac{6}{5}\)
Since, \(\frac{t_{2}}{t_{1}} \neq \frac{t_{3}}{t_{2}} \neq \frac{t_{4}}{t_{3}}\)
∴ the given sequence is not a geometric progression.

(v) 7, 14, 21, 28, …..
t₁ = 7, t₂ = 14, t₃ = 21, t₄ = 28, …..
Here, \(\frac{t_{2}}{t_{1}}=2, \frac{t_{3}}{t_{2}}=\frac{3}{2}, \frac{t_{4}}{t_{3}}=\frac{4}{3}\)
Since, \(\frac{\mathrm{t}_{2}}{\mathrm{t}_{1}} \neq \frac{\mathrm{t}_{3}}{\mathrm{t}_{2}} \neq \frac{\mathrm{t}_{4}}{\mathrm{t}_{3}}\)
∴ the given sequence is not a geometric progression.

Question 2.
For the G.P.,
(i) if r = \(\frac{1}{3}\), a = 9, find t₇.
(ii) if a = \(\frac{7}{243}\), r = \(\frac{1}{3}\), find t₃.
(iii) if a = 7, r = -3, find t₆.
(iv) if a = \(\frac{2}{3}\), t₆ = 162, find r.
Solution:

Question 3.
Which term of the G. P. 5, 25, 125, 625, ….. is 510?
Solution:

Question 4.
For what values of x, \(\frac{4}{3}\), x, \(\frac{4}{27}\) are in G. P.?
Solution:

Question 5.
If for a sequence, \(t_{n}=\frac{5^{n-3}}{2^{n-3}}\), show that the sequence is a G. P. Find its first term and the common ratio.
Solution:
The sequence (tn) is a G.P., if \(\frac{t_{n}}{t_{n-1}}\) = constant, for all n ∈ N

∴ the sequence is a G. P. with common ratio \(\frac{5}{2}\)
First term, t₁ = \(\frac{5^{\mathrm{l}-3}}{2^{1-3}}=\frac{2^{2}}{5^{2}}=\frac{4}{25}\)

Question 6.
Find three numbers in G. P. such that their sum is 21 and sum of their squares is 189.
Solution:
Let the three numbers in G. P. be \(\frac{a}{\mathrm{r}}\), a, ar.
According to the first condition,



∴ the three numbers are 12, 6, 3 or 3, 6, 12.
Check:
First condition:
12, 6, 3 are in G.P. with r = \(\frac{1}{2}\)
12 + 6 + 3 = 21
Second condition:
12² + 6² + 3² = 144 + 36 + 9 = 189
Thus, both the conditions are satisfied.

Question 7.
Find four numbers in G. P. such that sum of the middle two numbers is \(\frac{10}{3}\) and their product is 1.
Solution:
Let the four numbers in G.P. be \(\frac{a}{r^{3}}, \frac{a}{r}, a r, a r^{3}\).
According to the second condition,
\(\frac{\mathrm{a}}{\mathrm{r}^{3}}\left(\frac{\mathrm{a}}{\mathrm{r}}\right)(\mathrm{ar})\left(\mathrm{ar}^{3}\right)=1\)
∴ a⁴ = 1
∴ a = 1
According to the first condition,

Question 8.
Find five numbers in G. P. such that their product is 1024 and the fifth term is square of the third term.
Solution:
Let the five numbers in G. P. be
\(\frac{a}{r^{2}}, \frac{a}{r}, a, a r, a r^{2}\)
According to the given conditions,

When a = 4, r = -2
\(\frac{a}{r^{2}}\) = 1, \(\frac{a}{r}\) = -2, a = 4, ar = -8, ar² = 16
∴ the five numbers in G.P. are 1, 2, 4, 8, 16 or 1, -2, 4, -8, 16.

Question 9.
The fifth term of a G. P. is x, eighth term of the G. P. is y and eleventh term of the G. P. is z. Verify whether y² = xz.
Solution:

Question 10.
If p, q, r, s are in G. P., show that p + q, q + r, r + s are also in G.P.
Solution:
p, q, r, s are in G.P.

∴ p + q, q + r, r + s are in G.P.

11th Commerce Maths Digest Pdf

• 11th Commerce Maths Exercise 4.1 Solutions
• 11th Commerce Maths Exercise 4.2 Solutions
• 11th Commerce Maths Exercise 4.3 Solutions
• 11th Commerce Maths Exercise 4.4 Solutions
• 11th Commerce Maths Exercise 4.5 Solutions
• 11th Commerce Maths Miscellaneous Exercise 4 Solutions

Functions Class 11 Commerce Maths 1 Chapter 2 Miscellaneous Exercise 2 Answers Maharashtra Board

Balbharati Maharashtra State Board 11th Commerce Maths Solution Book Pdf Chapter 2 Functions Miscellaneous Exercise 2 Questions and Answers.

Std 11 Maths 1 Miscellaneous Exercise 2 Solutions Commerce Maths

Question 1.
Which of the following relations are functions? If it is a function determine its domain and range.
(i) {(2, 1), (4, 2), (6, 3), (8, 4), (10, 5) (12, 6), (14, 7)}
(ii) {(0, 0), (1, 1), (1, -1), (4, 2), (4, -2), (9, 3), (9, -3), (16, 4), (16, -4)}
(iii) {(1, 1), (3, 1), (5, 2)}
Solution:
(i) {(2, 1), (4, 2), (6, 3), (8, 4), (10, 5) (12, 6), (14, 7)}

Every element of set A has been assigned a unique element in set B.
∴ Given relation is a function.
Domain = {2, 4, 6, 8, 10, 12, 14},
Range = {1, 2, 3, 4, 5, 6, 7}

(ii) {(0, 0), (1, 1), (1, -1), (4, 2), (4, -2), (9, 3), (9, -3), (16, 4), (16, -4)}
∴ (1, 1), (1, -1) ∈ the relation
∴ Given relation is not a function.
As element 1 of the domain has not been assigned a unique element of co-domain.

(iii) {(1, 1), (3, 1), (5, 2)}

Every element of set A has been assigned a unique element in set B.
∴ Given relation is a function.
Domain = {1, 3, 5}, Range = {1, 2}

Question 2.
A function f: R → R defined by f(x) = \(\frac{3 x}{5}\) + 2, x ∈ R. Show that f is one-one and onto. Hence, find f^-1.
Solution:
f: R → R defined by f(x) = \(\frac{3 x}{5}\) + 2
First we have to prove that f is one-one function for that we have to prove if
f(x₁) = f(x₂) then x₁ = x₂
Here f(x) = \(\frac{3 x}{5}\) + 2
Let f(x₁) = f(x₂)
∴ \(\frac{3 x_{1}}{5}+2=\frac{3 x_{2}}{5}+2\)
∴ \(\frac{3 x_{1}}{5}=\frac{3 x_{2}}{5}\)
∴ x₁ = x₂
∴ f is a one-one function.
Now, we have to prove that f is an onto function.
Let y ∈ R be such that
y = f(x)
∴ y = \(\frac{3 x}{5}\) + 2
∴ y – 2 = \(\frac{3 x}{5}\)
∴ x = \(\frac{5(y-2)}{3}\) ∈ R
∴ for any y ∈ co-domain R, there exist an element x = \(\frac{5(y-2)}{3}\) ∈ domain R such that f(x) = y
∴ f is an onto function.
∴ f is one-one onto function.
∴ f^-1 exists.
∴ \(\mathrm{f}^{-1}(y)=\frac{5(y-2)}{3}\)
∴ \(f^{-1}(x)=\frac{5(x-2)}{3}\)

Question 3.
A function f is defined as follows:
f(x) = 4x + 5, for -4 ≤ x < 0. Find the values of f(-1), f(-2), f(0), if they exist.
Solution:
f(x) = 4x + 5, -4 ≤ x < 0
f(-1) = 4(-1) + 5 = -4 + 5 = 1
f(-2) = 4(-2) + 5 = -8 + 5 = -3
x = 0 ∉ domain of f
∴ f(0) does not exist.

Question 4.
A function f is defined as follows:
f(x) = 5 – x for 0 ≤ x ≤ 4. Find the value of x such that f(x) = 3.
Solution:
f(x) = 5 – x
f(x) = 3
∴ 5 – x = 3
∴ x = 5 – 3 = 2

Question 5.
If f(x) = 3x² – 5x + 7, find f(x – 1).
Solution:
f(x) = 3x² – 5x + 7
∴ f(x – 1) = 3(x – 1)² – 5(x – 1) + 7
= 3(x² – 2x + 1) – 5(x – 1) + 7
= 3x² – 6x + 3 – 5x + 5 + 7
= 3x² – 11x + 15

Question 6.
If f(x) = 3x + a and f(1) = 7, find a and f(4).
Solution:
f(x) = 3x + a,
f(1) = 7
∴ 3(1) + a = 7
∴ a = 7 – 3 = 4
∴ f(x) = 3x + 4
∴ f(4) = 3(4) + 4
= 12 + 4
= 16

Question 7.
If f(x) = ax² + bx + 2 and f(1) = 3, f(4) = 42, find a and b.
Solution:
f(x) = ax² + bx + 2
f(1) = 3
∴ a(1)² + b(1) + 2 = 3
∴ a + b = 1 …….(i)
f(4) = 42
∴ a(4)² + b(4) + 2 = 42
∴ 16a + 4b = 40
Dividing by 4, we get
4a + b = 10 ……….(ii)
Solving (i) and (ii), we get
a = 3, b = -2

Question 8.
If f(x) = \(\frac{2 x-1}{5 x-2}, x \neq \frac{2}{5}\), verify whether (fof)(x) = x
Solution:
(fof)(x) = f(f(x))

Question 9.
If f(x) = \(\frac{x+3}{4 x-5}\), g(x) = \(\frac{3+5 x}{4 x-1}\), then verify that (fog)(x) = x.
Solution:

11th Commerce Maths Digest Pdf

• 11th Commerce Maths Exercise 2.1 Solutions
• 11th Commerce Maths Miscellaneous Exercise 2 Solutions

Determinants Class 11 Commerce Maths 1 Chapter 6 Exercise 6.2 Answers Maharashtra Board

Balbharati Maharashtra State Board 11th Commerce Maths Solution Book Pdf Chapter 6 Determinants Ex 6.2 Questions and Answers.

Std 11 Maths 1 Exercise 6.2 Solutions Commerce Maths

Question 1.
Without expanding, evaluate the following determinants.
(i) \(\left|\begin{array}{lll}
1 & a & b+c \\
1 & b & c+a \\
1 & c & a+b
\end{array}\right|\)
Solution:

(ii) \(\left|\begin{array}{ccc}
2 & 3 & 4 \\
5 & 6 & 8 \\
6 x & 9 x & 12 x
\end{array}\right|\)
Solution:

(iii) \(\left|\begin{array}{lll}
2 & 7 & 65 \\
3 & 8 & 75 \\
5 & 9 & 86
\end{array}\right|\)
Solution:

Question 2.
Using properties of determinants, show that \(\left|\begin{array}{ccc}
a+b & a & b \\
a & a+c & c \\
b & c & b+c
\end{array}\right|\) = 4abc
Solution:

Question 3.
Solve the following equation.
\(\left|\begin{array}{ccc}
x+2 & x+6 & x-1 \\
x+6 & x-1 & x+2 \\
x-1 & x+2 & x+6
\end{array}\right|=0\)
Solution:
\(\left|\begin{array}{ccc}
x+2 & x+6 & x-1 \\
x+6 & x-1 & x+2 \\
x-1 & x+2 & x+6
\end{array}\right|=0\)
Applying R₂ → R₂ – R₁ and R₃ → R₃ – R₁, we get
\(\left|\begin{array}{ccc}
x+2 & x+6 & x-1 \\
4 & -7 & 3 \\
-3 & -4 & 7
\end{array}\right|=0\)
∴ (x + 2)(-49 + 12) – (x + 6)(28 + 9) + (x – 1)(-16 – 21) = 0
∴ (x + 2) (-37) – (x + 6) (37) + (x – 1) (-37) = 0
∴ -37(x + 2 + x + 6 + x – 1) = 0
∴ 3x + 7 = 0
∴ x = \(\frac{-7}{3}\)

Question 4.
If \(\left|\begin{array}{lll}
4+x & 4-x & 4-x \\
4-x & 4+x & 4-x \\
4-x & 4-x & 4+x
\end{array}\right|=0\), then find the values of x.
Solution:

∴ (12 – x)[1(4x² – 0) – (4 – x)(0 – 0) + (4 – x)(0 – 0)] = 0
∴ (12 – x)(4x²) = 0
∴ x²(12 – x) = 0
∴ x = 0 or 12 – x = 0
∴ x = 0 or x = 12

Question 5.
Without expanding determinants, show that
\(\left|\begin{array}{ccc}
1 & 3 & 6 \\
6 & 1 & 4 \\
3 & 7 & 12
\end{array}\right|+4\left|\begin{array}{lll}
2 & 3 & 3 \\
2 & 1 & 2 \\
1 & 7 & 6
\end{array}\right|=10\left|\begin{array}{lll}
1 & 2 & 1 \\
3 & 1 & 7 \\
3 & 2 & 6
\end{array}\right|\)
Solution:

Question 6.
Without expanding determinants, find the value of
(i) \(\left|\begin{array}{lll}
10 & 57 & 107 \\
12 & 64 & 124 \\
15 & 78 & 153
\end{array}\right|\)
Solution:

(ii) \(\left|\begin{array}{lll}
2014 & 2017 & 1 \\
2020 & 2023 & 1 \\
2023 & 2026 & 1
\end{array}\right|\)
Solution:

Question 7.
Without expanding determinants, prove that
(i) \(\left|\begin{array}{lll}
a_{1} & b_{1} & c_{1} \\
a_{2} & b_{2} & c_{2} \\
a_{3} & b_{3} & c_{3}
\end{array}\right|=\left|\begin{array}{lll}
b_{1} & c_{1} & a_{1} \\
b_{2} & c_{2} & a_{2} \\
b_{3} & c_{3} & a_{3}
\end{array}\right|=\left|\begin{array}{lll}
c_{1} & a_{1} & b_{1} \\
c_{2} & a_{2} & b_{2} \\
c_{3} & a_{3} & b_{3}
\end{array}\right|\)
Solution:

(ii) \(\left|\begin{array}{lll}
1 & y z & y+z \\
1 & z x & z+x \\
1 & x y & x+y
\end{array}\right|=\left|\begin{array}{lll}
1 & x & x^{2} \\
1 & y & y^{2} \\
1 & z & z^{2}
\end{array}\right|\)
Solution:

In 1st determinant, taking (x + y + z) common from C₃ and in 2nd determinant, taking \(\frac{1}{x}, \frac{1}{y}, \frac{1}{z}\) common from R₁, R₂, R₃ respectively, we get

11th Commerce Maths Digest Pdf

• 11th Commerce Maths Exercise 6.1 Solutions
• 11th Commerce Maths Exercise 6.2 Solutions
• 11th Commerce Maths Exercise 6.3 Solutions
• 11th Commerce Maths Miscellaneous Exercise 6 Solutions

Functions Class 11 Commerce Maths 1 Chapter 2 Exercise 2.1 Answers Maharashtra Board

Balbharati Maharashtra State Board 11th Commerce Maths Solution Book Pdf Chapter 2 Functions Ex 2.1 Questions and Answers.

Std 11 Maths 1 Exercise 2.1 Solutions Commerce Maths

Question 1.
Check if the following relations are functions.


Solution:
(a) Yes
Reason: Every element of set A has been assigned a unique element in set B.

(b) No
Reason: An element of set A has been assigned more than one element from set B.

(c) No
Reason: Not every element of set A has been assigned an image from set B.

Question 2.
Which sets of ordered pairs represent functions from A = {1, 2, 3, 4} to B = {-1, 0, 1, 2, 3}? Justify.
(i) {(1, 0), (3, 3), (2, -1), (4, 1), (2, 2)}
(ii) {(1, 2), (2, -1), (3, 1), (4, 3)}
(iii) {(1, 3), (4, 1), (2, 2)}
(iv) {(1, 1), (2, 1), (3, 1), (4, 1)}
Solution:
(i) {(1, 0), (3, 3), (2, -1), (4, 1), (2, 2)} does not represent a function.
Reason: (2, -1) and (2, 2) show that element 2 ∈ A has been assigned two images -1 and 2 from set B.

(ii) {(1, 2), (2, -1), (3, 1), (4, 3)} represents a function.
Reason: Every element of set A has a unique image in set B.

(iii) {(1, 3), (4, 1), (2, 2)} does not represent a function.
Reason: 3 ∈ A does not have an image in set B.

(iv) {(1, 1), (2, 1), (3, 1), (4, 1)} represents a function
Reason: Every element of set A has been assigned a unique image in set B.

Question 3.
If f(m) = m² – 3m + 1, find
(i) f(0)
(ii) f(-3)
(iii) f(\(\frac{1}{2}\))
(iv) f(x + 1)
(v) f(-x)
Solution:
f(m) = m² – 3m + 1
(i) f(0) = 0² – 3(0) + 1 = 1

(ii) f(-3) = (-3)² – 3(-3) + 1
= 9 + 9 + 1
= 19

(iii) \(f\left(\frac{1}{2}\right)=\left(\frac{1}{2}\right)^{2}-3\left(\frac{1}{2}\right)+1\)
= \(\frac{1}{4}-\frac{3}{2}+1\)
= \(\frac{1-6+4}{4}\)
= \(-\frac{1}{4}\)

(iv) f(x + 1) = (x + 1)² – 3(x + 1) + 1
= x² + 2x + 1 – 3x – 3 + 1
= x² – x – 1

(v) f(-x) = (-x)² – 3(-x) + 1 = x² + 3x + 1

Question 4.
Find x, if g(x) = 0 where
(i) g(x) = \(\frac{5 x-6}{7}\)
(ii) g(x) = \(\frac{18-2 x^{2}}{7}\)
(iii) g(x) = 6x² + x – 2
Solution:
(i) g(x) = \(\frac{5 x-6}{7}\)
g(x) = 0
∴ \(\frac{5 x-6}{7}\) = 0
∴ 5x – 6 = 0
∴ x = \(\frac{6}{5}\)

(ii) g(x) = \(\frac{18-2 x^{2}}{7}\)
g(x) = 0
∴ \(\frac{18-2 x^{2}}{7}\) = 0
∴ 18 – 2x² = 0
∴ x² = 9
∴ x = ±3

(iii) g(x) = 6x² + x – 2
g(x) = 0
∴ 6x² + x – 2 = 0
∴ 6x² + 4x – 3x – 2 = 0
∴ 2x(3x + 2) – 1(3x + 2) = 0
∴ (2x – 1)(3x + 2) = 0
∴ 2x – 1 = 0 or 3x + 2 = 0
∴ x = \(\frac{1}{2}\) or x = \(\frac{-2}{3}\)

Question 5.
Find x, if f(x) = g(x) where f(x) = x⁴ + 2x², g(x) = 11x².
Solution:
f(x) = x⁴ + 2x², g(x) = 11x²
f(x) = g(x)
∴ x⁴ + 2x² = 11x²
∴ x⁴ – 9x² = 0
∴ x²(x² – 9) = 0
∴ x² = 0 or x² – 9 = 0
∴ x = 0 or x² = 9
∴ x = 0 or x = ±3

Question 6.
If f(x) = \(\begin{cases}x^{2}+3, & x \leq 2 \\ 5 x+7, & x>2\end{cases}\), then find
(i) f(3)
(ii) f(2)
(iii) f(0)
Solution:
f(x) = x² + 3, x ≤ 2
= 5x + 7, x > 2
(i) f(3) = 5(3) + 7 = 15 + 7 = 22
(ii) f(2) = 2² + 3 = 4 + 3 = 7
(iii) f(0) = 0² + 3 = 3

Question 7.
If f(x) = \(\left\{\begin{array}{cl}
4 x-2, & x \leq-3 \\
5, & -3<x<3 \\
x^{2}, & x \geq 3
\end{array}\right.\), then fmd
(i) f(-4)
(ii) f(-3)
(iii) f(1)
(iv) f(5)
Solution:
f(x) = 4x – 2, x ≤ -3
= 5, -3 < x < 3
= x², x ≥ 3
(i) f(-4) = 4(-4) – 2 = -16 – 2 = -18
(ii) f(-3) = 4(-3) – 2 = -12 – 2 = -14
(iii) f(1) = 5
(iv) f(5) = 5² = 25

Question 8.
If f(x) = 3x + 5, g(x) = 6x – 1, then find
(i) (f + g)(x)
(ii) (f – g)(2)
(iii) (fg)(3)
(iv) \(\left(\frac{\mathbf{f}}{\mathbf{g}}\right)(x)\) and its domain
Solution:
f(x) = 3x + 5, g(x) = 6x – 1
(i) (f + g)(x) = f(x) + g(x)
= 3x + 5 + 6x – 1
= 9x + 4

(ii) (f – g) (2) = f(2) – g(2)
= [3(2) + 5] – [6(2) – 1]
= 6 + 5 – 12 + 1
= 0

(iii) (fg)(3) = f(3) g(3)
= [3(3) + 5] [6(3) – 1]
= (14) (17)
= 238

(iv) \(\left(\frac{\mathrm{f}}{\mathrm{g}}\right) x=\frac{\mathrm{f}(x)}{\mathrm{g}(x)}=\frac{3 x+5}{6 x-1}, x \neq \frac{1}{6}\)
Domain = R – {\(\frac{1}{6}\)}

Question 9.
If f(x) = 2x² + 3, g(x) = 5x – 2, then find
(i) fog
(ii) gof
(iii) fof
(iv) gog
Solution:
f(x) = 2x² + 3, g(x) = 5x – 2
(i) (fog)(x) = f(g(x))
= f(5x – 2)
= 2(5x – 2)² + 3
= 2(25x² – 20x + 4) + 3
= 50x² – 40x + 8 + 3
= 50x² – 40x + 11

(ii) (gof)(x) = g(f(x))
= g(2x² + 3)
= 5(2x² + 3) – 2
= 10x² + 15 – 2
= 10x² + 13

(iii) (fof)(x) = f(f(x))
= f(2x² + 3)
= 2(2x² + 3)² + 3
= 2(4x⁴ + 12x² + 9) + 3
= 8x⁴ + 24x² + 18 + 3
= 8x⁴ + 24x² + 21

(iv) (gog)(x) = g(g(x))
= g(5x – 2)
= 5(5x – 2) – 2
= 25x – 10 – 2
= 25x – 12

11th Commerce Maths Digest Pdf

• 11th Commerce Maths Exercise 2.1 Solutions
• 11th Commerce Maths Miscellaneous Exercise 2 Solutions

Limits Class 11 Commerce Maths 1 Chapter 7 Exercise 7.4 Answers Maharashtra Board

Balbharati Maharashtra State Board 11th Commerce Maths Solution Book Pdf Chapter 7 Limits Ex 7.4 Questions and Answers.

Std 11 Maths 1 Exercise 7.4 Solutions Commerce Maths

I. Evaluate the following:

Question 1.
\(\lim _{x \rightarrow 0}\left[\frac{9^{x}-5^{x}}{4^{x}-1}\right]\)
Solution:

Question 2.
\(\lim _{x \rightarrow 0}\left[\frac{5^{x}+3^{x}-2^{x}-1}{x}\right]\)
Solution:

Question 3.
\(\lim _{x \rightarrow 0}\left[\frac{\log (2+x)-\log (2-x)}{x}\right]\)
Solution:

II. Evaluate the following:

Question 1.
\(\lim _{x \rightarrow 0}\left[\frac{3^{x}+3^{-x}-2}{x^{2}}\right]\)
Solution:

Question 2.
\(\lim _{x \rightarrow 0}\left[\frac{3+x}{3-x}\right]^{\frac{1}{x}}\)
Solution:

Question 3.
\(\lim _{x \rightarrow 0}\left[\frac{\log (3-x)-\log (3+x)}{x}\right]\)
Solution:

III. Evaluate the following:

Question 1.
\(\lim _{x \rightarrow 0}\left[\frac{a^{3 x}-b^{2 x}}{\log (1+4 x)}\right]\)
Solution:

Question 2.
\(\lim _{x \rightarrow 0}\left[\frac{\left(2^{x}-1\right)^{2}}{\left(3^{x}-1\right) \cdot \log (1+x)}\right]\)
Solution:

Question 3.
\(\lim _{x \rightarrow 0}\left[\frac{15^{x}-5^{x}-3^{x}+1}{x^{2}}\right]\)
Solution:

Question 4.
\(\lim _{x \rightarrow 2}\left[\frac{3^{\frac{x}{2}}-3}{3^{x}-9}\right]\)
Solution:

IV. Evaluate the following:

Question 1.
\(\lim _{x \rightarrow 0}\left[\frac{(25)^{x}-2(5)^{x}+1}{x^{2}}\right]\)
Solution:

Question 2.
\(\lim _{x \rightarrow 0}\left[\frac{(49)^{x}-2(35)^{x}+(25)^{x}}{x^{2}}\right]\)
Solution:

11th Commerce Maths Digest Pdf

• 11th Commerce Maths Exercise 7.1 Solutions
• 11th Commerce Maths Exercise 7.2 Solutions
• 11th Commerce Maths Exercise 7.3 Solutions
• 11th Commerce Maths Exercise 7.4 Solutions
• 11th Commerce Maths Miscellaneous Exercise 7 Solutions

Sets and Relations Class 11 Commerce Maths 1 Chapter 1 Miscellaneous Exercise 1 Answers Maharashtra Board

Balbharati Maharashtra State Board 11th Commerce Maths Solution Book Pdf Chapter 1 Sets and Relations Miscellaneous Exercise 1 Questions and Answers.

Std 11 Maths 1 Miscellaneous Exercise 1 Solutions Commerce Maths

Question 1.
Write the following sets in set builder form:
(i) {10, 20, 30, 40, 50}
(ii) {a, e, i, o, u}
(iii) {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}
Solution:
(i) Let A = {10, 20, 30, 40, 50}
∴ A = {x / x = 10n, n ∈ N and n ≤ 5}

(ii) Let B = {a, e, i, o, u}
∴ B = {x / x is a vowel of English alphabets}

(iii) Let C = {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}
∴ C = {x / x represents days of a week}

Question 2.
If U = {x / x ∈ N, 1 ≤ x ≤ 12}, A = {1, 4, 7, 10}, B = {2, 4, 6, 7, 11}, C = {3, 5, 8, 9, 12}.
Write the sets
(i) A ∪ B
(ii) B ∩ C
(iii) A – B
(iv) B – C
(v) A ∪ B ∪ C
(vi) A ∩ (B ∪ C)
Solution:
U = {x / x ∈ N, 1 ≤ x ≤ 12} = {1, 2, 3, …., 12}
A = {1, 4, 7, 10}, B = {2, 4, 6, 7, 11}, C = {3, 5, 8, 9, 12}
(i) A ∪ B = {1, 2, 4, 6, 7, 10, 11}

(ii) B ∩ C = { }

(iii) A – B = {1, 10}

(iv) B – C = {2, 4, 6, 7, 11}

(v) A ∪ B ∪ C = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

(vi) B ∪ C = {2, 3, 4, 5, 6, 7, 8, 9, 11, 12}
∴ A ∩ (B ∪ C) = {4, 7}

Question 3.
In a survey of 425 students in a school, it was found that 115 drink apple juice, 160 drink orange juice, and 80 drink both apple as well as orange juice. How many drinks neither apple juice nor orange juice?
Solution:
Let A = set of students who drink apple juice
B = set of students who drink orange juice
X = set of all students
∴ n(X) = 425, n(A) = 115, n(B) = 160, n(A ∩ B) = 80

No. of students who neither drink apple juice nor orange juice
n(A’ ∩ B’) = n(A ∪ B)’
= n(X) – n(A ∪ B)
= 425 – [n(A) + n(B) – n(A ∩ B)]
= 425 – (115 + 160 – 80)
= 230

Question 4.
In a school, there are 20 teachers who teach Mathematics or Physics. of these, 12 teach Mathematics and 4 teach both Physics and Mathematics. How many teachers teach Physics?
Solution:
Let A = set of teachers who teach Mathematics
B = set of teachers who teach Physics
n(A ∪ B) = 20, n(A) = 12, n(A ∩ B) = 4

Since, n(A ∪ B) = n(A) + n(B) – n(A ∩ B)
∴ 20 = 12 + n(B) – 4
∴ n(B) = 12
∴ Number of teachers who teach physics = 12

Question 5.
(i) If A = {1, 2, 3} and B = {2, 4}, state the elements of A × A, A × B, B × A, B × B, (A × B) ∩ (B × A).
(ii) If A = {-1, 1}, find A × A × A.
Solution:
(i) A = {1, 2, 3} and B = {2, 4}
A × A = {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}
A × B = {(1, 2), (1, 4), (2, 2), (2, 4), (3, 2), (3, 4)}
B × A = {(2, 1), (2, 2), (2, 3), (4, 1), (4, 2), (4, 3)}
B × B = {(2, 2), (2, 4), (4, 2), (4, 4)}
(A × B) ∩ (B × A) = {(2, 2)}

(ii) A = {-1, 1}
∴ A × A × A = {(-1, -1, -1), (-1, -1, 1), (-1, 1, -1), (-1, 1, 1), (1, -1, -1), (1, -1, 1), (1, 1, -1),(1, 1, 1)}

Question 6.
If A = {1, 2, 3}, B = {4, 5, 6}, which of the following are relations from A to B.
(i) R₁ = {(1, 4), (1, 5), (1, 6)}
(ii) R₂ = {(1, 5), (2, 4), (3, 6)}
(iii) R₃ = {(1, 4), (1, 5), (3, 6), (2, 6), (3, 4)}
(iv) R₄ = {(4, 2), (2, 6), (5, 1), (2, 4)}
Solution:
A = {1, 2, 3}, B = {4, 5, 6}
∴ A × B = {(1, 4), (1, 5), (1, 6), (2, 4), (2, 5), (2, 6), (3, 4), (3, 5), (3, 6)}
(i) R₁ = {(1, 4), (1, 5), (1, 6)}
Since, R₁ ⊆ A × B
∴ R₁ is a relation from A to B.

(ii) R₂ = {(1, 5), (2, 4), (3, 6)}
Since, R₂ ⊆ A × B
∴ R₂ is a relation from A to B.

(iii) R₃ = {(1, 4), (1, 5), (3, 6), (2, 6), (3, 4)}
Since, R₃ ⊆ A × B
∴ R₃ is a relation from A to B.

(iv) R₄ = {(4,2), (2, 6), (5,1), (2, 4)}
Since, (4, 2) ∈ R₄, but (4, 2) ∉ A × B
∴ R₄ ⊄ A × B
∴ R₄ is not a relation from A to B.

Question 7.
Determine the domain and range of the following relation.
R = {(a, b) / a ∈ N, a < 5, b = 4}
Solution:
R = {(a, b) / a ∈ N, a < 5, b = 4}
∴ Domain (R) = {a / a ∈ N, a < 5} = {1, 2, 3, 4}
Range (R) = {b / b = 4} = {4}

11th Commerce Maths Digest Pdf

• 11th Commerce Maths Exercise 1.1 Solutions
• 11th Commerce Maths Exercise 1.2 Solutions
• 11th Commerce Maths Miscellaneous Exercise 1 Solutions

Limits Class 11 Commerce Maths 1 Chapter 7 Exercise 7.3 Answers Maharashtra Board

Balbharati Maharashtra State Board 11th Commerce Maths Solution Book Pdf Chapter 7 Limits Ex 7.3 Questions and Answers.

Std 11 Maths 1 Exercise 7.3 Solutions Commerce Maths

I. Evaluate the following limits:

Question 1.
\(\lim _{x \rightarrow 0}\left[\frac{\sqrt{6+x+x^{2}}-\sqrt{6}}{x}\right]\)
Solution:

Question 2.
\(\lim _{y \rightarrow 0}\left[\frac{\sqrt{1-y^{2}}-\sqrt{1+y^{2}}}{y^{2}}\right]\)
Solution:

Question 3.
\(\lim _{x \rightarrow 2}\left[\frac{\sqrt{2+x}-\sqrt{6-x}}{\sqrt{x}-\sqrt{2}}\right]\)
Solution:

II. Evaluate the following limits:

Question 1.
\(\lim _{x \rightarrow a}\left[\frac{\sqrt{a+2 x}-\sqrt{3 x}}{\sqrt{3 a+x}-2 \sqrt{x}}\right]\)
Solution:

Question 2.
\(\lim _{x \rightarrow 2}\left[\frac{x^{2}-4}{\sqrt{x+2}-\sqrt{3 x-2}}\right]\)
Solution:

III. Evaluate the following limits:

Question 1.
\(\lim _{x \rightarrow 1}\left[\frac{x^{2}+x \sqrt{x}-2}{x-1}\right]\)
Solution:

Question 2.
\(\lim _{x \rightarrow 0}\left[\frac{\sqrt{1+x^{2}}-\sqrt{1+x}}{\sqrt{1+x^{3}}-\sqrt{1+x}}\right]\)
Solution:

Question 3.
\(\lim _{x \rightarrow 4}\left[\frac{x^{2}+x-20}{\sqrt{3 x+4}-4}\right]\)
Solution:

Question 4.
\(\lim _{x \rightarrow 2}\left[\frac{x^{3}-8}{\sqrt{x+2}-\sqrt{3 x-2}}\right]\)
Solution:

IV. Evaluate the following limits:

Question 1.
\(\lim _{y \rightarrow 2}\left[\frac{2-y}{\sqrt{3-y}-1}\right]\)
Solution:

Question 2.
\(\lim _{z \rightarrow 4}\left[\frac{3-\sqrt{5+z}}{1-\sqrt{5-z}}\right]\)
Solution:

11th Commerce Maths Digest Pdf

• 11th Commerce Maths Exercise 7.1 Solutions
• 11th Commerce Maths Exercise 7.2 Solutions
• 11th Commerce Maths Exercise 7.3 Solutions
• 11th Commerce Maths Exercise 7.4 Solutions
• 11th Commerce Maths Miscellaneous Exercise 7 Solutions

Sets and Relations Class 11 Commerce Maths 1 Chapter 1 Exercise 1.2 Answers Maharashtra Board

Balbharati Maharashtra State Board 11th Commerce Maths Solution Book Pdf Chapter 1 Sets and Relations Ex 1.2 Questions and Answers.

Std 11 Maths 1 Exercise 1.2 Solutions Commerce Maths

Question 1.
If (x – 1, y + 4) = (1, 2), find the values of x and y.
Solution:
(x – 1, y + 4) = (1, 2)
By the definition of equality of ordered pairs, we have
x – 1 = 1 and y + 4 = 2
∴ x = 2 and y = -2

Question 2.
If \(\left(x+\frac{1}{3}, \frac{y}{3}-1\right)=\left(\frac{1}{3}, \frac{3}{2}\right)\), find x and y.
Solution:
\(\left(x+\frac{1}{3}, \frac{y}{3}-1\right)=\left(\frac{1}{3}, \frac{3}{2}\right)\)
By the definition of equality of ordered pairs, we have
\(x+\frac{1}{3}=\frac{1}{3}\) and \(\frac{y}{3}-1=\frac{3}{2}\)
\(x=\frac{1}{3}-\frac{1}{3}\) and \(\frac{y}{3}=\frac{3}{2}+1=\frac{5}{2}\)
x = 0 and y = \(\frac{15}{2}\)

Question 3.
If A = {a, b, c}, B = {x, y}, find A × B, B × A, A × A, B × B.
Solution:
A = {a, b, c}, B = {x, y}
A × B = {(a, x), (a, y), (b, x), (b, y), (c, x), (c, y)}
B × A = {(x, a), (x, b), (x, c), (y, a), (y, b), (y, c)}
A × A = {(a, a), (a, b), (a, c), (b, a), (b, b), (b, c), (c, a), (c, b), (c, c)}
B × B = {(x, x), (x, y), (y, x), (y, y)}

Question 4.
If P = {1, 2, 3} and Q = {6, 4}, find the sets P × Q and Q × P.
Solution:
P = {1, 2, 3}, Q = {6, 4}
P × Q = {(1, 6), (1, 4), (2, 6), (2, 4), (3, 6), (3, 4)}
Q × P = {(6, 1), (6, 2), (6, 3), (4, 1), (4, 2), (4, 3)}

Question 5.
Let A = {1, 2, 3, 4}, B = {4, 5, 6}, C = {5, 6}. Find
(i) A × (B ∩ C)
(ii) (A × B) ∩ (A × C)
(iii) A × (B ∪ C)
(iv) (A × B) ∪ (A × C)
Solution:
A= {1, 2, 3, 4}, B = {4, 5, 6}, C = {5, 6}
(i) B ∩ C = {5, 6}
∴ A × (B ∩ C) = {(1, 5), (1, 6), (2, 5), (2, 6), (3, 5), (3, 6), (4, 5), (4, 6)}

(ii) A × B = {(1, 4), (1, 5), (1, 6), (2, 4), (2, 5), (2, 6), (3, 4), (3, 5), (3, 6), (4, 4), (4, 5), (4, 6)}
A × C = {(1, 5), (1, 6), (2, 5), (2, 6), (3, 5), (3, 6), (4, 5), (4, 6)}
∴ (A × B) ∩ (A × C) = {(1, 5), (1, 6), (2, 5), (2, 6), (3, 5), (3, 6), (4, 5), (4, 6)}

(iii) B ∪ C = {4, 5, 6}
∴ A × (B ∪ C) = {(1, 4), (1, 5), (1, 6), (2, 4), (2, 5), (2, 6), (3, 4), (3, 5), (3, 6), (4, 4), (4, 5), (4, 6)}

(iv) A × B = {(1, 4), (1, 5), (1, 6), (2, 4), (2, 5), (2, 6), (3, 4), (3, 5), (3, 6), (4, 4), (4, 5), (4, 6)}
A × C = {(1, 5), (1, 6), (2, 5), (2, 6), (3, 5), (3, 6), (4, 5), (4, 6)}
∴ (A × B) ∪ (A × C) = {(1, 4), (1, 5), (1, 6), (2, 4), (2, 5), (2, 6), (3, 4), (3, 5), (3, 6), (4, 4), (4, 5), (4, 6)}

Question 6.
Express {(x, y) / x² + y² = 100, where x, y ∈ W} as a set of ordered pairs.
Solution:
{(x, y) / x² + y² = 100, where x, y ∈ W}
We have, x² + y² = 100
When x = 0 and y = 10,
x² + y² = 0² + 10² = 100
When x = 6 andy = 8,
x² + y² = 6² + 8² = 100
When x = 8 and y = 6,
x² + y² = 8² + 6² = 100
When x = 10 and y = 0,
x² + y² = 10² + 0² = 100
∴ Set of ordered pairs = {(0, 10), (6, 8), (8, 6), (10, 0)}

Question 7.
Write the domain and range of the following relations.
(i) {(a, b) / a ∈ N, a < 6 and b = 4}
(ii) {(a, b) / a, b ∈ N, a + b = 12}
(iii) {(2, 4), (2, 5), (2, 6), (2, 7)}
Solution:
(i) Let R₁ = {(a, b)/ a ∈ N, a < 6 and b = 4}
Set of values of ‘a’ are domain and set of values of ‘b’ are range.
a ∈ N and a < 6
∴ a = 1, 2, 3, 4, 5 and b = 4
Domain (R₁) = {1, 2, 3, 4, 5}
Range (R₁) = {4}

(ii) Let R₂ = {(a, b)/a, b ∈ N and a + b = 12}
Now, a, b ∈ N and a + b = 12
When a = 1, b = 11
When a = 2, b = 10
When a = 3, b = 9
When a = 4, b = 8
When a = 5, b = 7
When a = 6, b = 6
When a = 7, b = 5
When a = 8, b = 4
When a = 9, b = 3
When a = 10, b = 2
When a = 11, b = 1
∴ Domain (R₂) = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}
Range (R₂) = {11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1}

(iii) Let R₃ = {(2, 4), (2, 5), (2, 6), (2, 7)}
Domain (R₃) = {2}
Range (R₃) = {4, 5, 6, 7}

Question 8.
Let A = {6, 8} and B = {1, 3, 5}.
Let R = {(a, b) / a ∈ A, b ∈ B, a – b is an even number}.
Show that R is an empty relation from A to B.
Solution:
A= {6, 8}, B = {1, 3, 5}
R = {(a, b)/ a ∈ A, b ∈ B, a – b is an even number}
a ∈ A
∴ a = 6, 8
b ∈ B
∴ b = 1, 3, 5
When a = 6 and b = 1, a – b = 5 which is odd
When a = 6 and b = 3, a – b = 3 which is odd
When a = 6 and b = 5, a – b = 1 which is odd
When a = 8 and b = 1, a – b = 7 which is odd
When a = 8 and b = 3, a – b = 5 which is odd
When a = 8 and b = 5, a – b = 3 which is odd
Thus, no set of values of a and b gives a – b even.
∴ R is an empty relation from A to B.

Question 9.
Write the relation in the Roster form and hence find its domain and range.
(i) R₁ = {(a, a²) / a is a prime number less than 15}
(ii) R₂ = {(a, \(\frac{1}{a}\)) / 0 < a ≤ 5, a ∈ N}
Solution:
(i) R₁ = {(a, a²) / a is a prime number less than 15}
∴ a = 2, 3, 5, 7, 11, 13
∴ a2 = 4, 9, 25, 49, 121, 169
∴ R₁ = {(2, 4), (3, 9), (5, 25), (7, 49), (11, 121), (13, 169)}
∴ Domain (R₁) = {a/a is a prime number less than 15} = {2, 3, 5, 7, 11, 13}
Range (R₁) = {a²/a is a prime number less than 15} = {4, 9, 25, 49, 121, 169}

Question 10.
R = {(a, b) / b = a + 1, a ∈ Z, 0 < a < 5}. Find the range of R.
Solution:
R = {(a, b) / b = a + 1, a ∈ Z, 0 < a < 5}
∴ a = 1, 2, 3, 4
∴ b = 2, 3, 4, 5
∴ Range (R) = {2, 3, 4, 5}

Question 11.
Find the following relations as sets of ordered pairs.
(i) {(x, y) / y = 3x, x ∈ {1, 2, 3}, y ∈ {3, 6, 9, 12}}
(ii) {(x,y) / y > x + 1, x ∈ {1, 2} and y ∈ {2, 4, 6}}
(iii) {(x, y) / x + y = 3, x, y ∈ {0, 1, 2, 3}}
Solution:
(i) {(x, y) / y = 3x, x ∈ {1, 2, 3}, y ∈ {3, 6, 9, 12}}
Here y = 3x
When x = 1, y = 3(1) = 3
When x = 2, y = 3(2) = 6
When x = 3, y = 3(3) = 9
∴ Ordered pairs are {(1, 3), (2, 6), (3, 9)}

(ii) {(x, y) / y > x + 1, x ∈ {1, 2} and y ∈ {2, 4, 6}}
Here, y > x + 1
When x = 1 and y = 2, 2 ≯ 1 + 1
When x = 1 and y = 4, 4 > 1 + 1
When x = 1 and y = 6, 6 > 1 + 1
When x = 2 and y = 2, 2 ≯ 2 + 1
When x = 2 and y = 4, 4 > 2 + 1
When x = 2 and y = 6, 6 > 2 + 1
∴ Ordered pairs are {(1, 4), (1, 6), (2, 4), (2, 6)}

(iii) {(x, y) / x + y = 3, x, y ∈ {0, 1, 2, 3}}
Here, x + y = 3
When x = 0, y = 3
When x = 1, y = 2
When x = 2, y = 1
When x = 3, y = 0
∴ Ordered pairs are {(0, 3), (1, 2), (2, 1), (3, 0)}

11th Commerce Maths Digest Pdf

• 11th Commerce Maths Exercise 1.1 Solutions
• 11th Commerce Maths Exercise 1.2 Solutions
• 11th Commerce Maths Miscellaneous Exercise 1 Solutions

Differentiation Class 11 Commerce Maths 1 Chapter 9 Exercise 9.2 Answers Maharashtra Board

Balbharati Maharashtra State Board 11th Commerce Maths Solution Book Pdf Chapter 9 Differentiation Ex 9.2 Questions and Answers.

Std 11 Maths 1 Exercise 9.2 Solutions Commerce Maths

I. Differentiate the following functions w.r.t. x.

Question 1.
\(\frac{x}{x+1}\)
Solution:

Question 2.
\(\frac{x^{2}+1}{x}\)
Solution:

Question 3.
\(\frac{1}{e^{x}+1}\)
Solution:

Question 4.
\(\frac{e^{x}}{e^{x}+1}\)
Solution:

Question 5.
\(\frac{x}{\log x}\)
Solution:

Question 6.
\(\frac{2^{x}}{\log x}\)
Solution:

Question 7.
\(\frac{\left(2 e^{x}-1\right)}{\left(2 e^{x}+1\right)}\)
Solution:

Question 8.
\(\frac{(x+1)(x-1)}{\left(e^{x}+1\right)}\)
Solution:

II. Solve the following examples:

Question 1.
The demand D for a price P is given as D = \(\frac{27}{P}\), find the rate of change of demand when the price is 3.
Solution:
Demand, D = \(\frac{27}{P}\)
Rate of change of demand = \(\frac{dD}{dP}\)

When price P = 3,
Rate of change of demand,
\(\left(\frac{\mathrm{dD}}{\mathrm{dP}}\right)_{\mathrm{P}=3}=\frac{-27}{(3)^{2}}=-3\)
∴ When price is 3, Rate of change of demand is -3.

Question 2.
If for a commodity; the price-demand relation is given as D = \(\frac{P+5}{P-1}\). Find the marginal demand when the price is 2.
Solution:

Question 3.
The demand function of a commodity is given as P = 20 + D – D². Find the rate at which price is changing when demand is 3.
Solution:
Given, P = 20 + D – D²
Rate of change of price = \(\frac{dP}{dD}\)
= \(\frac{d}{dD}\)(20 + D – D²)
= 0 + 1 – 2D
= 1 – 2D
Rate of change of price at D = 3 is
\(\left(\frac{\mathrm{dP}}{\mathrm{dD}}\right)_{\mathrm{D}=3}\) = 1 – 2(3) = -5
∴ Price is changing at a rate of -5, when demand is 3.

Question 4.
If the total cost function is given by; C = 5x³ + 7x² + 7; find the average cost and the marginal cost when x = 4.
Solution:
Total cost function, C = 5x³ + 7x² + 7
Average cost = \(\frac{C}{x}\)

When x = 4, Marginal cost = \(\left(\frac{\mathrm{dC}}{\mathrm{d} x}\right)_{x=4}\)
= 15(4)² + 4(4)
= 240 + 16
= 256
∴ the average cost and marginal cost at x = 4 are \(\frac{359}{4}\) and 256 respectively.

Question 5.
The total cost function of producing n notebooks is given by
C = 1500 – 75n + 2n² + \(\frac{n^{3}}{5}\)
Find the marginal cost at n = 10.
Solution:
The total cost function,


∴ Marginal cost at n = 10 is 25.

Question 6.
The total cost of ‘t’ toy cars is given by C = 5(2t) + 17. Find the marginal cost and average cost at t = 3.
Solution:
Total cost of ‘t’ toy cars, C = 5(2t) + 17

∴ at t = 3, the Marginal cost is 40 log 2 and the Average cost is 19.

Question 7.
If for a commodity; the demand function is given by, D = \(\sqrt{75-3 P}\). Find the marginal demand function when P = 5.
Solution:
Demand function, D = \(\sqrt{75-3 P}\)
Now, Marginal demand = \(\frac{dD}{dP}\)

Question 8.
The total cost of producing x units is given by C = 10e^2x, find its marginal cost and average cost when x = 2.
Solution:

Question 9.
The demand function is given as P = 175 + 9D + 25D². Find the revenue, average revenue, and marginal revenue when demand is 10.
Solution:
Given, P = 175 + 9D + 25D²
Total revenue, R = P.D
= (175 + 9D + 25D²)D
= 175D + 9D² + 25D³
Average revenue = P = 175 + 9D + 25D²
Marginal revenue = \(\frac{dR}{dD}\)
= \(\frac{d}{dD}\) (175D + 9D² + 25D³)
= 175 \(\frac{d}{dD}\) (D) + 9 \(\frac{d}{dD}\) (D²) + 25 \(\frac{d}{dD}\) (D³)
= 175(1) + 9(2D) + 25(3D²)
= 175 + 18D + 75D²
When D = 10,
Total revenue = 175(10) + 9(10)² + 25(10)³
= 1750 + 900 + 25000
= 27650
Average revenue = 175 + 9(10) + 25(10)²
= 175 + 90 + 2500
= 2765
Marginal revenue = 175 + 18(10) + 75(10)²
= 175 + 180 + 7500
= 7855
∴ When Demand = 10,
Total revenue = 27650, Average revenue = 2765, Marginal revenue = 7855.

Question 10.
The supply S for a commodity at price P is given by S = P² + 9P – 2. Find the marginal supply when the price is 7.
Solution:
Given, S = P² + 9P – 2

∴ The marginal supply is 23, at P = 7.

Question 11.
The cost of producing x articles is given by C = x² + 15x + 81. Find the average cost and marginal cost functions. Find marginal cost when x = 10. Find x for which the marginal cost equals the average cost.
Solution:
Given, cost C = x² + 15x + 81

If marginal cost = average cost, then
2x + 15 = x + 15 + \(\frac{81}{x}\)
∴ x = \(\frac{81}{x}\)
∴ x² = 81
∴ x = 9 …..[∵ x > 0]

11th Commerce Maths Digest Pdf

• 11th Commerce Maths Exercise 9.1 Solutions
• 11th Commerce Maths Exercise 9.2 Solutions
• 11th Commerce Maths Miscellaneous Exercise 9 Solutions

Company Meetings – II 11th Secretarial Practice Chapter 8 Solutions Maharashtra Board

Balbharti Maharashtra State Board Class 11 Secretarial Practice Solutions Chapter 8 Company Meetings – II Textbook Exercise Questions and Answers.

Class 11 Secretarial Practice Chapter 8 Exercise Solutions

1A. Select the correct answer from the options given below and rewrite the statements.

Question 1.
The secretary has to prepare ____________ before the meeting.
(a) resolution
(b) notice
(c) minutes
Answer:
(b) notice

Question 2.
General Meeting must have a notice of at least ____________ clear days.
(a) 7
(b) 21
(c) 14
Answer:
(b) 21

Question 3.
The gap between two annual general meetings should not be more than months.
(a) 15
(b) 18
(c) 24
Answer:
(a) 15

Question 4.
Auditor is appointed in ____________
(a) Annual General Meeting
(b) Creditors Meeting
(c) Extra ordinary General Meeting
Answer:
(a) Annual General Meeting

Question 5.
The business transacted in extra ordinary general meeting is ____________ business.
(a) Ordinary
(b) Routine
(c) Special
Answer:
(c) Special

Question 6.
An extraordinary general meeting is held ____________
(a) once in a year
(b) once in the life time
(c) under special circumstances
Answer:
(c) under special circumstances

Question 7.
Writing the minutes of meeting is the duty of the ____________
(a) Chairman
(b) Director
(c) Secretary
Answer:
(c) Secretary

Question 8.
New Directors in place of retiring Directors are appointed in the ____________
(a) Class Meeting
(b) Annual General Meeting
(c) Creditors Meeting
Answer:
(b) Annual General Meeting

Question 9.
Before the Annual General Meeting is held the Secretary has to compile ____________
(a) Auditor’s Report
(b) Committee Report
(c) Annual Report
Answer:
(c) Annual Report

Question 10.
The Secretary has to draft the minutes of the meeting within ____________ days of the meeting.
(a) Fifteen
(b) Sixty
(c) Forty
Answer:
(a) Fifteen

1B. Match the pairs.

Question 1.

Group ‘A’	Group ‘B’
(a) Board Meeting	(1) Once in the year
(b) Auditor	(2) Minimum 4 meetings in a year
(c) Annual General Meeting	(3) Prepare at any time
(d) Extraordinary General Meeting	(4) Auditor’s Report
(e) Minutes of Meeting	(5) Prepared before the meeting
	(6) Prepared after the meeting
	(7) Under special circumstance
	(8) Prepared during the meeting
	(9) Once in Six months
	(10) Meeting of creditors

Answer:

Group ‘A’	Group ‘B’
(a) Board Meeting	(2) Minimum 4 meetings in a year
(b) Auditor	(4) Auditor’s Report
(c) Annual General Meeting	(1) Once in the year
(d) Extraordinary General Meeting	(7) Under special circumstance
(e) Minutes of Meeting	(6) Prepared after the meeting

1C. Write a word or a term or a phrase that can substitute each of the following statements.

Question 1.
A meeting of shareholders which is held once a year.
Answer:
Annual General Meeting

Question 2.
A meeting of shareholders is held between two annual general meetings.
Answer:
Extra Ordinary General Meeting

Question 3.
A Report which is prepared before Annual General Meeting.
Answer:
Annual Report

Question 4.
A meeting of the shareholders is held under special circumstances.
Answer:
Extra Ordinary General Meeting

Question 5.
A meeting of the preference shareholders of a company.
Answer:
Class Meeting

1D. State whether the following statements are True or False.

Question 1.
Only special business is transacted at Annual General Meeting.
Answer:
False

Question 2.
Annual Report is prepared by members.
Answer:
False

Question 3.
Extraordinary General Meeting is called for a special and urgent purpose.
Answer:
True

Question 4.
A member has a right to attend a Board meeting.
Answer:
False

Question 5.
A class meeting is the meeting of a particular class of shareholders.
Answer:
True

Question 6.
Director cannot appoint a proxy to attend and vote at the board meetings.
Answer:
True

Question 7.
21 clear days notice should be given in case of Board Meeting.
Answer:
False

Question 8.
Secretary has to arrange to take down the notes of the proceedings of the meeting.
Answer:
True

1E. Find the odd one.

Question 1.
Annual General Meeting, Extra Ordinary General Meeting, Board Meeting.
Answer:
Board Meeting

Question 2.
Annual General Meeting, Committee Meeting, Class Meeting.
Answer:
Committee Meeting

1F. Complete the sentences.

Question 1.
The meeting which is held once in every financial year is called as ____________
Answer:
Annual General Meeting

Question 2.
The meeting held to discuss and decide any special or urgent matters is called as ____________
Answer:
Extra-ordinary General Meeting

Question 3.
Meeting of all Directors is called as ____________
Answer:
Board Meeting

Question 4.
The Authority to convene the Annual General Meeting is with the ____________
Answer:
Board of Directors

1G. Select the correct option from the bracket.

Question 1.

Group ‘A’	Group ‘B’
(1) Interval between two Annual General Meetings	…………………….
(2) First Board Meeting	…………………….
(3) ……………………….	21 clear days
(4) ……………………….	Alteration in Memorandum of Association

(Notice of General Meeting, Extraordinary General Meeting, Not more than 15 months, within 30 days of company’s Incorporation)
Answer:

Group ‘A’	Group ‘B’
(1) Interval between two Annual General Meeting	Not more than 15 months
(2) First Board Meeting	within 30 days of companies Incorporations
(3) Notice of General Meeting	21 clear days
(4) Extraordinary General meeting	Alteration in Memorandum of Association

1H. Answer in one sentence.

Question 1.
Who can attend Board Meetings?
Answer:
Board of Directors attends Board Meetings.

Question 2.
When can an Adjourned Annual General meeting be held?
Answer:
Adjourned Annual General Meeting will be held on the same day, same place and same time in the next week.

Question 3.
Why is Extra-Ordinary General Meeting held?
Answer:
Extra Ordinary General Meeting is held under special circumstances to discuss urgent and important matters.

Question 4.
When should the first Annual General Meeting be held?
Answer:
The first Annual General Meeting must be held within nine (9) months of the closing of the first financial year of the company.

Question 5.
What should be the interval between two Annual General Meetings?
Answer:
The time interval between two Annual General Meetings should not be more than 15 months.

Question 6.
What should be the gap between two Board Meetings?
Answer:
The gap between the two Board Meetings should not be more than 120 days.

1I. Correct the underlined word and rewrite the following sentences.

Question 1.
Minimum four Committee Meetings must be held in a year.
Answer:
Minimum four Directors Meetings must be held in a year.

Question 2.
Creditors Meeting must be held once a year.
Answer:
Annual General Meeting must be held once a year.

Question 3.
Extra Ordinary Meeting must be held every year.
Answer:
Annual General Meeting must be held every year.

Question 4.
The Meeting of all Directors is called Annual General Meeting.
Answer:
The Meeting of all Directors is Called Director’s Meeting.

2. Explain the following terms/concepts.

Question 1.
Committee Meeting
Answer:
The Board of Directors may from the committee and delegate some of its powers to them. This committee should consist of only directors. The delegation of such powers to committees is to be authorized by Articles of Association and should be subject to the provisions of the Companies Act. A meeting of Allotment Committee, Transfer Committee, etc are examples of Committee Meeting.

Question 2.
Virtual Meeting
Answer:
If there is any urgency to call Board Meeting and practically not possible for Directors to attend the meeting in a short notice, so with the help of technology, a virtual meeting can be held. The meetings are recorded and active participation of Directors is also possible through video conferencing. Certain special and confidential matters cannot be dealt with through video conferencing as per the directions of the Central Government.

Question 3.
Creditor’s Meeting
Answer:
Creditors of the company include Debenture holders, Depositors, Banks, and others. The meeting of creditors is called when the company proposes to make a scheme for an arrangement with its creditors.
E.g. In the case of winding up of the company, a meeting of creditors and of contributors is held to ascertain the total amount due by the company and also to appoint a liquidator to wind up the affairs of the company.

3. Study the following case/situation and express your opinion.

1. Platinum Limited Company was incorporated on 1st January 2018. Advice the Board of Directors on the following matters.

Question (a).
Within what period should the company hold its 1st Annual General Meeting?
Answer:
First Annual General Meeting must be held within Nine (9) months of the closing of its financial year.

Question (b).
At which place should the Annual General Meeting be held?
Answer:
Annual General Meeting should be held at the registered office or at some other place within the city or village which the registered office of the company is situated.

Question (c).
How many days in advance should the notice and agenda be sent to members?
Answer:
Notice and Agenda should be sent 21 clear days before the Annual General Meeting.

2. XYZ Ltd held its Annual General Meeting on 11th May 2018. On 1st June Mr. X, a Director of the company was arrested for a financial scam. Hence the shareholders of the company want to remove him.

Question (a).
Can the company call for the next Annual General Meeting to be held immediately to remove the Director?
Answer:
No, the company cannot call the next Annual General Meeting immediately to remove the director.

Question (b).
Which meeting of members should the company hold to discuss the removal of the Director?
Answer:
Extra Ordinary General Meeting should be called to discuss removal of the director.

Question (c).
What type of resolution will be passed in this meeting for the removal of the Director?
Answer:
A resolution requiring special notice (section 115) will be passed in the meeting for the removal of the director.

4. Distinguish between the following.

Question 1.
Shareholders’ Meeting and Board Meeting.
Answer:

Basis	Shareholder’s Meeting (Annual General Meeting)	Board Meeting (Director’s Meeting)
1. Meaning	It is the meeting of all the members or shareholders of the company.	It is the meeting of all the Directors of the company.
2. Number of times	Annual General Meeting is held once every year. Extra Ordinary General Meeting is held to discuss urgent and important matters. It is called in between two Annual General Meetings.	Minimum 4 meetings to be held in a year and the gap between two meetings should not be more than 120 days.
3. Quorum	Quorum in case of Public Company is as follows: No. of Shareholders Quorum Up to 1000 5 members 1000-5000 15 members More than 5000 30 members In the case of the private company, 2 members present in person shall be the quorum.	The quorum for the Board Meeting is a minimum of 2 Directors or 1/3rd of its total Directors whichever is higher.
4. Proxy	Members can appoint a proxy to attend and vote at the meeting on their behalf.	Directors cannot appoint a proxy to attend and vote at the meeting.
5. Notice	Notice of shareholder’s meeting must be sent to all the members at least 21 clear days before the meeting.	Notice of Board Meeting must be sent to every Director at least 7 days before the meeting.
6. Purpose	The purpose of this meeting is to take approvals from shareholders on the Annual Report, Appointment of Auditor, etc.	The purpose of this meeting is to discuss policy matters, future planning, etc.

Question 2.
Annual General Meeting and Extra-ordinary General Meeting
Answer:

Points	Annual General Meeting	Extraordinary General Meeting
1. Meaning	It is a meeting of shareholders held once in a year as per the provisions of Section 96 of the Companies Act, 2013.	It is a meeting of shareholders held under special conditions to discuss urgent and important matters as per the provisions of Section 100 of the Companies Act, 2013.
2. Purpose	The purpose is to give information about the progress made by the company during the year.	The purpose is to discuss urgent and important matters that cannot be postponed till next Annual General Meeting.
3. Number of times	The first Annual General Meeting is held within 9 months of the completion of the financial year and the subsequent Annual General Meeting is held within 6 months of the completion of the financial year.	Extraordinary General Meeting is held anytime between two Annual General Meetings.
4. Authority	Annual General Meeting is normally called by the Board of Directors and can be called by the Central Government under special circumstances.	Extra-ordinary General Meeting can be called by: Board of Directors Board of Directors on the requisition of members The National Company Law Tribunal
5. Business transacted	To approve annual accounts To approve Auditors and Director’s Report To appoint Auditors, Directors, etc. To declare a dividend.	To alter Articles of Association. To alter Memorandum of Association. Removal of Director. Voluntary winding up of a company.
6. Penalty	In case of default in holding Annual General Meeting, every officer concerned is punishable with a fine upto ₹ 1,00,000 and fine of ₹ 5,000 per day till default continues.	If the directors fail to hold this meeting on requisition, the requisionist themselves can hold it and recover the expenses from defaulting directors.

5. Answer in brief.

Question 1.
List the duties of a Secretary after the Annual General Meeting.
Answer:
Duties of a Secretary after Annual General Meeting:

• Drafting Minutes: The secretary drafts the minutes of the meeting on the basis of the notes taken down during the proceedings of the meeting.
• Implementation of decisions: The secretary instructs the concerned department for implementing the decisions taken at the meeting.
• Dispatch Dividend Warrants: The secretary has to make an arrangement to dispatch dividend warrants to the shareholders.
• Filing Resolutions: The secretary has to file an Annual Report with the Registrar. In case of a special resolution, the secretary has to file the special resolution with the Registrar within 30 days of passing that resolution.
• Information to Absent Members: The secretary sends the information of the proceedings of the meeting to the absent members.

Question 2.
State the objectives of the Annual General Meeting.
Answer:
Purpose or objectives of Annual General Meeting: The main objective of holding an Annual General Meeting is to provide an opportunity to members to discuss the functioning of the company.

• To review the progress and performance of the company.
• To approve annual accounts, director’s report, and auditor’s report.
• To appoint auditors and fix their remuneration.
• To declare a dividend.
• To elect new directors in place of retiring Directors To transact special business, if any like-
  • business of the company taken over by the Government.
  • the company did not function well through the year, etc.

Question 3.
State any four legal provisions regarding Board Meeting.
Answer:
Legal Provisions relating to Board Meeting:
(i) Authority to Convene:
The chairman of the Board is the proper authority to call the Board Meeting.

(ii) Number of Board Meetings:

• First Board Meeting shall be held within 30 days of the date of its incorporation.
• Subsequently, there should be four meetings in a year.
• The gap between the two meetings should not be more than 120 days.
• At least one Board meeting should be held in 6 months in case of one person company, small company and dormant company.
• The gap between two meetings should not be more than 90 days in the case of one person company, a small company, and a dormant company.

(iii) Notice [Section 173(3)]:

• The notice of every Board Meeting must be given to every director at least 7 days before the meeting at his registered address along with an agenda of the meeting.
• It can also be sent by post or by hand delivery or by E-mail.
• If a company fails to send notice within a given time, then every officer in default shall be liable to a penalty of Rs. 25,000.

(iv) Quorum (Section – 174):

• The quorum for the Board Meeting is 1/3rd of the total number of directors or two directors whichever is higher.
• Any fraction should be rounded to one.
• Interested directors (personal interest in any matter) are not included in the quorum.

Question 4.
What are Class Meetings?
Answer:
Class Meetings (Section 48):

• When the meeting of a particular class of shareholders takes place such as a preference shareholders meeting, it is known as a class meeting.
• Such a meeting may be called for an alteration in the right and privileges of that particular class of shareholders.
• It can also be called for the purpose of conversion of one class of shares into another.
• The class meeting is not held regularly.

Question 5.
What is the purpose of conducting an Extra-ordinary General Meeting?
Answer:
Purpose or Objectives of Extra-Ordinary General Meeting:
The main purpose behind calling this meeting is to make shareholders aware of the happenings in the company or to seek their approval on some urgent and important matters of the company.

• Alteration in the Memorandum of Association.
• Alteration in Articles of Association.
• Reduction of Share Capital.
• Removal of a Director before the expiry of his term.
• Removal of an Auditor before the expiry of his term.
• Voluntary winding up of the company.

6. Justify the following statements.

Question 1.
Annual General Meeting is adjourned in absence of a quorum.
Answer:

• Quorum is a minimum number of members to be present for a valid Meeting.
• As per the Companies Act, 2013, the quorum for the Annual General Meeting of a public company is as follows:

No. of Shareholders	Quorum
Up to 1000	5 Members
1000 – 5000	15 Members
More than 5000	30 Members

• For a private company minimum of 2 members should be present
• If the meeting is conducted without a quorum, it is considered invalid.
• In the absence of a quorum, Annual General Meeting is adjourned.
• Adjourned Meeting will be convened on the same day, same time, and the same place in the next week.

Question 2.
A notice of the meeting must be sent to all members in case of the General Meeting.
Answer:

• The Board of Directors is the authority to convene general meetings.
• The notice must be given 21 clear days before the meeting to all those who are entitled to receive it.
• The notice must be sent to all the members at their registered address in India.
• Notice can be sent by registered post, if requested by the member, charges for the same is to be paid in advance.
• Notice is also given in all leading newspapers.
• Notice is accompanied by agenda and also Annual report, Auditor report, Annual accounts, etc. in case of Annual General Meeting.

Question 3.
Extra Ordinary General Meeting is called under special circumstances.
Answer:

1. Extraordinary General Meeting is held in between two Annual General Meetings.
2. It is held when the matter is of utmost importance and it cannot be postponed till the next Annual General Meeting.
3. It can be called to make shareholders aware of the happenings in the company.
4. It can also be called to seek their approval on important and urgent matters.
5. Important and urgent matters can be:
   • Alteration in Memorandum of Association
   • Alteration in Articles of Association
   • Reduction in share capital
   • Removal of a director or auditor
   • Voluntary winding up of a company, etc.

Question 4.
Board of Directors can exercise their powers collectively through the Board Meeting.
Answer:

1. Directors collectively are called as Board of Directors.
2. The Board of Directors has to meet frequently with reference to the day-to-day working of the company.
3. All important decisions can be taken by the Board of Directors in the Board Meeting collectively.
4. Powers which can be exercised at Board Meeting are:
   • The power to fill casual vacancies in the Board.
   • To recommend the rate of dividend.
   • The power to appoint managing Director, Manager, etc.
   • To sanction a contract.
   • To give loans or invest in shares of other companies.

Question 5.
The Quorum for the Annual General Meeting of a public company depends upon the number of shareholders.
Answer:

• Quorum is a minimum number of members to be present for a valid meeting.
• As per the Companies Act, 2013, the quorum for the Annual General Meeting of a public company are as follows:

No. of Shareholders	Quorum
Up to 1000	5 Members
1000 – 5000	15 Members
More than 5000	30 Members

• • If the number of shareholders is up to 1000, then the required quorum is 5 members.
  • If the number of shareholders is more than 1000 and up to 5000, then the required quorum is 15 members.
  • If the number of shareholders exceeds 5000 then the quorum is 30 members for a public company.
• In the absence of a quorum, if the meeting is conducted or a resolution is passed, it is considered illegal and invalid.
• In the absence of a quorum, the meeting is adjourned to the same day, same place, and same time in the next week.

Question 6.
When a company is in financial difficulty it may call a meeting of creditors.
Answer:

• Creditors of the company include Debenture holders, Depositors, Banks, financial institutions, etc.
• When a company is in a financial crisis, it seeks the help of creditors.
• The company can request creditors to alter the rate of interest, to alter the terms of security, etc.
• Meeting of creditors can be called to compromise in a dispute with them.
• Creditors and contributors can provide financial help to the company to safeguard them and also to protect them to avoid winding up of a company.

Question 7.
As per the Companies Act, every company has to hold the Annual General Meeting.
Answer:

• Every company other than one person company must hold an Annual General Meeting once a year.
• Annual General Meeting is held to review the working of the company during the previous financial year and take an effective decision for the next year.
• It is mainly called to approve Annual Report, Auditor Report, Annual Accounts, to declare a dividend, to elect Directors, Auditors and to transact special business, if any.
• First AGM shall be held within 9 months of the closing of the first financial year of the company.
• The time interval between two Annual General Meetings should not be more than 15 months.
• If the default is made in holding AGM then the company and every officer of the company in default are punishable with a fine which may extend up to ₹ 1,00,000 and up to ₹ 5,000 per day till default continues.

7. Answer the following questions.

Question 1.
Explain the functions of a secretary related to the Annual General Meeting.
Answer:
Functions (duties) of secretary related with Annual General Meeting:
Secretary has to perform many functions related to all types of meetings like Board Meeting, Annual General Meeting, etc. He has to perform all duties in compliance with the provisions of the Act. Secretary has to perform functions before, during, and after the meeting.

Functions of a secretary related to Annual General Meeting are as follows:
Functions Before the Annual General Meeting:
(i) Preparation of Accounts and Reports:
The secretary has to ensure that the Annual Accounts, Annual Report, and Auditor Report are prepared for consideration and approval.

(ii) Board Meeting:
Board Meeting is called to transact the following business:

• To adopt Annual Accounts duly audited.
• To adopt Director’s Report
• To consider the rate of dividend
• To fix the day, date, time, and place of the Annual General Meeting.

(iii) Notice:
The secretary has to send 21 clear days’ notice of a meeting to members and auditors before the meeting along with the auditor report, annual report, annual accounts, agenda, etc. The secretary has to arrange for publishing the notice of the meeting in leading newspapers.

(iv) Chairman Speech:
The secretary has to assist the chairman in drafting his speech to be delivered at the meeting.

(v) Preliminary Arrangement:
The secretary has to make arrangements such as booking of hall, refreshments, Register of Members, Members Attendance Book, etc.

Functions During the Annual General Meeting:
(i) Attendance:
The secretary has to get signatures of members present at the meeting in ‘Members Attendance Book.’

(ii) Quorum:
The secretary should see that the required quorum is present at the meeting and informs the chairman of the meeting.

(iii) Reading of Reports:
The secretary has to read the notice, Director’s Report, and Auditor’s Report as directed by the Chairman.

(iv) Notice of Proceedings:
The secretary should take down the notes of the proceedings of the meeting. He has to draft the minutes on the basis of these notes.

(v) Assist Chairman:
The secretary has to assist the chairman during the meeting. He should provide information, documents, etc as and when needed.

(vi) Conduct of Poll:
Secretary makes arrangements for conducting the poll if it is demanded by the members.

Functions After the Annual General Meeting:
(i) Drafting Minutes:
The secretary drafts the minutes of the meeting on the basis of the notes taken down during the proceedings of the meeting.

(ii) Implementation of decisions:
The secretary instructs the concerned department for implementing the decisions taken at the meeting.

(iii) Dispatch Dividend Warrants:
The secretary has to make an arrangement to dispatch dividend warrants to the shareholders.

(iv) Filing Resolutions:
The secretary has to file the Annual Report with the Registrar. In case of a special resolution, the secretary has to file the special resolution with the Registrar within 30 days of passing that resolution.

(v) Information to Absent Members :
The secretary sends the information of the proceedings of the meeting to the absent members.

Question 2.
Explain the different types of General Meetings of a company.
Answer:
General Meetings are meetings of Equity shareholders of the company. There are two types of General Meetings:

• Annual General Meeting
• Extraordinary General Meeting.

1. Annual General Meeting (Section 96):
Every Company other than One Person Company must hold Annual General Meeting once in a year.
Purpose or Objective of Annual General Meeting:

• To review the progress and performance of the company.
• To approve annual accounts, director’s report, and auditor’s report.
• To appoint auditors and fix their remuneration.
• To declare a dividend.
• To elect new directors in place of retiring directors.
• To transact special business, if any.

Legal Provisions relating to Annual General Meeting:
(i) Time Limit:

• The time interval between two Annual General Meetings should not be more than 15 months.
• Extension of time for holding the Annual General Meeting may be given by the Registrar to the company for a period not more than three months, except for the first Annual General Meeting.
• One person company is exempted from the provision of Annual General Meeting.

(ii) Authority to Convene Annual General Meeting:
The Board of Directors is the proper authority to convene the Annual General Meeting.

(iii) Notice:
The notice must be given to all those who are entitled to receive it, at least 21 clear days in advance of the meeting. It is sent to the members at their registered address by post or through electronic mode like Email, etc.

(iv) Time, Day, and Place of Annual General Meeting:
Annual General Meeting shall be called during business hours between 9 am to 6 pm on any day other than National holidays. It can be held at the registered office or at some other place within the city or village in which the registered office of the company is situated.

(v) Quorum:
According to the Companies Act, 2013, the quorum for the Annual General Meeting of a public company is as follows:

No. of Shareholders	Quorum
Upto 1000	5 Members
1000 – 5000	15 Members
More than 5000	30 Members

For a private company minimum of two members should be present in person.

(vi) Adjournment:
In the absence of a quorum, Annual General Meeting stands adjourned. It will be convened on the same day, same time and the same place in the next week.

(vii) Default:
If the default is made in holding the Annual General Meeting as per the provisions of the Act or as per the instructions of the Central Government, the company and every officer of the company who is in default are punishable with a fine which may extend up to ₹ 1,00,000/-. If the default continues fine may extend up to ₹ 5,000 per day till the default continues.

2. Extra-Ordinary General Meeting (Section – 100):
Meaning:
It is a meeting which is called in between two Annual General Meetings under special circumstances, which cannot be postponed till the next Annual General Meeting. The matters discussed under this meeting is of utmost importance and urgent by nature.

Purpose or Objectives of Extra-Ordinary General Meeting:
The main purpose behind calling this meeting is to make shareholders aware of the happenings in the company or to seek their approval on some urgent and important matters of the company.

• Alteration in the Memorandum of Association.
• Alteration in Articles of Association.
• Reduction of Share Capital of the company.
• Removal of a Director before the expiry of his term.
• Removal of an Auditor before the expiry of his term.
• Voluntary winding up of the company.

Legal Provisions relating to Extra Ordinary General Meeting:
(i) Time for holding a meeting:
Extra Ordinary General Meeting is held between two Annual General Meetings under special circumstances. It can be held at any time as per the requirements of the company.

(ii) Authority to Convene:

• The Board of Directors has the right to call an Extra-Ordinary General Meeting by sending a proper notice to the shareholders.
• Extra-Ordinary General Meeting can be called by the members holding at least 1/10th of the paid-up capital or 1/10th of voting power in the company. Board must call Extra-Ordinary General Meeting within 45 days of receiving the requisition from the members.
• If the Board fails to call such a meeting then the requisitionists themselves call this meeting within 3 months from the date of deposit of the requisition. The company shall pay all the expenses incurred for holding such meetings by the requisitionist.
• National Company Law Tribunal (NCLT) can order such meetings on its own or at the request of a director or any members having voting rights.

(iii) Notice:
The notice must be given to all those who are entitled to receive it, at least 21 clear days in advance of the meeting. It is sent to the members at their registered address by post or through electronic mode like Email, etc.

(iv) Quorum:
According to the Companies Act, 2013, the quorum for the Annual General Meeting of a public company is as follows:

No. of Shareholders	Quorum
Up to 1000	5 Members
1000 – 5000	15 Members
More than 5000	30 Members

For a private company minimum of two members should be present in person.

Question 3.
Explain the different types of Director’s Meetings.
Answer:
Director’s meeting:
Directors collectively are called as Board of Directors. They meet regularly to discuss and decide upon company matters, implementation of plans and review the progress made by the company.

(i) Board Meeting (Section 173):
It is a meeting of the Board of Directors. The Board of Directors has to meet frequently with reference to day to day working of the company. The Board of Directors can exercise their powers collectively through the Board Meeting.

Legal Provisions relating to Board Meeting:
(a) Authority to Convene:
The chairman of the Board is the proper authority to call the Board Meeting.

(b) Number of Board Meetings:

• First Board Meeting shall be held within 30 days of the date of its incorporation.
• Subsequently, there should be four meetings in a year.
• The gap between the two meetings should not be more than 120 days.
• At least one Board meeting should be held in 6 months in case of one person company, small company, and dormant company.
• The gap between two meetings should not be more than 90 days in the case of one person company.

(c) Virtual Meeting/Video Conferencing:
If there is any urgency to call Board Meeting and practically not possible for Directors to attend the meeting in a short notice, so with the help of technology, a virtual meeting can be held. The meetings are recorded and active participation of Directors is also possible through video conferencing. Certain special and confidential matters cannot be dealt with through video conferencing as per the directions of the Central Government.

(d) Notice [Section 173(3)]:
The notice of every Board Meeting must be given to every director at least 7 days before the meeting at his registered address along with an agenda of the meeting.
It can also be sent by post or by hand delivery or by E-mail.
If a company fails to send notice within a given time, then every officer in default shall be liable to a penalty of Rs. 25,000.

(e) Quorum (Section – 174)

• The quorum for the Board Meeting is 1/3rd of the total number of directors or two directors whichever is higher.
  Any fraction should be rounded to one.
• Interested directors (personal interest in any matter) are not included in the quorum.

(f) Adjournment:
The meeting of the Board of directors can be adjourned in the absence of a quorum.
The adjourned meeting can be held on the same day, same time and the same place in the next week.

(ii) Committee Meeting:

• The Board of Directors may from the committee and delegate some of its powers to them.
• This committee should consist of only directors. The delegation of such powers to committees is to be authorized by Articles of Association and should be subject to the provisions of the Companies Act.
• A meeting of Allotment Committee, Transfer Committee, etc are examples of Committee Meeting.

Activity 1 (Textbook Page No. 123)

Visit the website of any public company and study its Annual Report/Director’s Report.
Answer:
[Note: Students should do this activity by themselves with the help of the internet.]

Activity 2 (Textbook Page No. 127)

Enlist the functions of a Secretary related to an Extra-Ordinary Meeting.
Answer:
Functions of Secretary Relating to Extra Ordinary General Meeting:
(i) Before the Meeting:

• Time of Holding: To decide the date, time, place, and agenda for holding the meeting in consultation with the chairman of the company and to convene the Board Meeting for finalization of these things by passing a resolution to that effect.
• Drafting of Resolution: To draft the proposed resolution and explanatory statements and to get them approved by the board.
• Notice: To draft the notice of the meeting and arrange to get the same printed together with the explanatory statements, proxy forms, and admission cards. To send the notice with agenda to all the members, at least 21 days before the meeting.
• Public Notice: The notice is also published in the leading newspapers for the information of the general public.
• Notice to Stock Authorities: If the shares are listed on the stock exchange, a notice of the meeting is sent to stock exchange authorities too.
• Proxy: To verify proxy forms received and enter them in the proxy register.
• Preparing Documents: The important documents, records, Books, etc. are kept ready for reference during the meeting.

(ii) During the Meeting:

• Checking: To check the admission card of the members.
• Attendance: To take attendance of members.
• Ascertaining Quorum: To ascertain the quorum.
• Reading Notice: To read the notice convening meeting.
• Assist Chairman: To assist the chairman, by providing necessary information, records documents.
• Notes of Proceedings: To make notes of the proceedings of the meeting.

(iii) After the Meeting:

• Drafting Minutes: To draft the minutes of the meeting and enter the same in the Minute Book.
• Signed by the Chairman: To get Minutes signed by the chairman of the meeting.
• Filing of Resolution: To file the certified copy of the resolution passed at the meeting, to the Registrar of Companies.
• Sending Intimation: To send intimation to the concerned persons.
• Implementations: To carry out decisions taken at the meeting.

Maharashtra State Board 11th Std Secretarial Practice Textbook Solutions Digest

• 11th Secretarial Practice Chapter 1 Practical Problems
• 11th Secretarial Practice Chapter 2 Practical Problems
• 11th Secretarial Practice Chapter 3 Practical Problems
• 11th Secretarial Practice Chapter 4 Practical Problems
• 11th Secretarial Practice Chapter 5 Practical Problems
• 11th Secretarial Practice Chapter 6 Practical Problems
• 11th Secretarial Practice Chapter 7 Practical Problems
• 11th Secretarial Practice Chapter 8 Practical Problems
• 11th Secretarial Practice Chapter 9 Practical Problems
• 11th Secretarial Practice Chapter 10 Practical Problems
• 11th Secretarial Practice Chapter 11 Practical Problems
• 11th Secretarial Practice Chapter 12 Practical Problems