# Maharashtra Board 10th Class Maths Part 1 Practice Set 2.5 Solutions Chapter 2 Quadratic Equations

Balbharti Maharashtra State Board Class 10 Maths Solutions covers the Practice Set 2.5 Algebra 10th Class Maths Part 1 Answers Solutions Chapter 2 Quadratic Equations.

## Practice Set 2.5 Algebra 10th Std Maths Part 1 Answers Chapter 2 Quadratic Equations

Question 1.
Fill in the gaps and complete.

Question 2.
Find the value of discriminant.
i. x2 + 7x – 1 = 0
ii. 2y2 – 5y + 10 = 0
iii. √2 x2 + 4x + 2√2 = 0
Solution:
i. x2 +7 x – 1 = 0
Comparing the above equation with
ax2 + bx + c = 0, we get
a = 1, b = 7, c = -1
∴ b2– 4ac = (7)2 – 4 × 1 × (-1)
= 49 + 4
∴ b2 – 4ac = 53

ii. 2y2 – 5y + 10 = 0
Comparing the above equation with
ay2 + by + c = 0, we get
a = 2, b = -5, c = 10
∴ b2 – 4ac = (-5)2 -4 × 2 × 10
= 25 – 80
∴ b2 – 4ac = -55

iii. √2 x2 + 4x + 2√2 = 0
Comparing the above equation with
ax + bx + c = 0, we get
a = √2,b = 4, c = 2√2
∴ b2 – 4ac = (4)2 – 4 × √2 × 2√2
= 16 – 16
∴ b2 – 4ac =0

Question 3.
Determine the nature of roots of the following quadratic equations.
i. x2 – 4x + 4 = 0
ii. 2y2 – 7y + 2 = 0
iii. m2 + 2m + 9 = 0
Solution:
i. x2 – 4x + 4= 0
Comparing the above equation with
ax2 + bx + c = 0, we get
a = 1,b = -4, c = 4
∴ ∆ = b2 – 4ac
= (- 4)2 – 4 × 1 × 4
= 16 – 16
∴ ∆ = 0
∴ Roots of the given quadratic equation are real and equal.

ii. 2y2 – 7y + 2 = 0
Comparing the above equation with
ay2 + by + c = 0, we get
a = 2, b = -7, c = 2
∴ ∆ = b2 – 4ac
= (- 7)2 – 4 × 2 × 2
= 49 – 16
∴ ∆ = 33
∴ ∆ > 0
∴ Roots of the given quadratic equation are real and unequal.

iii. m2 + 2m + 9 = 0
Comparing the above equation with
am2 + bm + c = 0, we get
a = 1,b = 2, c = 9
∴ ∆ = b2 – 4ac
= (2)2 – 4 × 1 × 9
= 4 – 36
∴ ∆ = -32
∴ ∆ < 0
∴ Roots of the given quadratic equation are not real.

Question 4.
Form the quadratic equation from the roots given below.
i. 0 and 4
ii. 3 and -10
iii. $$\frac { 1 }{ 2 }$$ , $$\frac { 1 }{ 2 }$$
iv. 2 – √5, 2 + √5
Solution:
i. Let a = 0 and β = 4
∴ α + β = 0 + 4 = 4
and α × β = 0 × 4 = 0
∴ The required quadratic equation is
x2 – (α + β) x + αβ = 0
∴ x2 – 4x + 0 = 0
∴ x2 – 4x = 0

ii. Let α = 3 and β = -10
∴ α + β = 3 – 10 = -7
and α × β = 3 × -10 = -30
∴ The required quadratic equation is
x2 – (α + β)x + αβ = 0
∴ x2 – (-7) x + (-30) = 0

∴ The required quadratic equation is
x2 – (α + β)x + αβ = 0
∴ x2 – 4x – 1 = 0

Question 5.
Sum of the roots of a quadratic equation is double their product. Find k if equation is x2 – 4kx + k + 3 = 0.
Solution:
x2 – 4kx + k + 3 = 0
Comparing the above equation with
ax2 + bx + c = 0, we get
a = 1, b = – 4k, c = k + 3
Let α and β be the roots of the given quadratic equation.
Then, α + β  = $$\frac { -b }{ a }$$ and αβ = $$\frac { c }{ a }$$
According to the given condition,

Question 6.
α, β are roots of y2 – 2y – 7 = 0 find,
i. α2 + β2
ii. α3 + β3
Solution:
y2 – 2y – 7 = 0
Comparing the above equation with
ay2 + by + c = 0, we get
a = 1, b = -2, c = -7

Question 7.
The roots of each of the following quadratic equations are real and equal, find k.
i. 3y2 + ky + 12 = 0
ii. kx (x-2) + 6 = 0
Solution:
i. 3y2 + kg + 12 = 0
Comparing the above equation with
ay2 + by + c = 0, we get
a = 3, b = k, c = 12
∴ ∆ = b2 – 4ac
= (k)2 – 4 × 3 × 12
= k2 – 144 = k2 – (12)2
∴ ∆ = (k + 12) (k – 12) …[∵ a2 – b2 = (a + b) (a – b)]
Since, the roots are real and equal.
∴ ∆ = 0
∴ (k + 12) (k – 12) = 0
∴ k + 12 = 0 or k – 12 = 0
∴ k = -12 or k = 12

ii. kx (x – 2) + 6 = 0
∴ kx2 – 2kx + 6 = 0
Comparing the above equation with
ax2 + bx + c = 0, we get
a = k, b = -2k, c = 6
∴ ∆ = b2 – 4ac
= (-2k)2 – 4 × k × 6
= 4k2 – 24k
∴ ∆ = 4k (k – 6)
Since, the roots are real and equal.
∴ ∆ = 0
∴ 4k (k – 6) = 0
∴ k(k – 6) = 0
∴ k = 0 or k – 6 = 0
But, if k = 0 then quadratic coefficient becomes zero.
∴ k ≠ 0
∴ k = 6

Question 1.
Fill in the blanks. (Textbook pg. no. 44)
Solution:

Question 2.
Determine nature of roots of the quadratic equation: x2 + 2x – 9 = 0 (Textbook pg. no. 45)
Solution:

∴ The roots of the given equation are real and unequal.

Question 3.
Fill in the empty boxes properly. (Textbook pg. no. 46)
Solution:
10x2 + 10x + 1 = 0
Comparing the above equation with
ax2 + bx + c = 0, we get
a = 10, b = 10, c = 1

Question 4.
Write the quadratic equation if addition of the roots is 10 and product of the roots is 9. (Textbook pg, no. 48)

Question 5.
What will be the quadratic equation if α = 2, β = 5. (Textbook pg. no, 48)
Solution: