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NCERT Exemplar · Class 10 · Maths

Chapter 4: Quadratic Equations

Practice each question, then open the answer or worked solution.

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Exercise 4.1

5 questions
Q1Written answer

Which of the following is a quadratic equation?

(A) x2 + 2x + 1 = (4 – x)2 + 3 (B) –2x2 = (5 – x)(2x-(2/5))

(C) (k + 1) x2 + (3/2) x = 7, where k = –1 (D) x3 – x2 = (x – 1)3

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Worked solution

(D) x3 – x2 = (x – 1)3

Explanation:

The standard form of a quadratic equation is given by,

ax2 + bx + c = 0, a ≠ 0

(A) Given, x2 + 2x + 1 = (4 – x)2 + 3

x2 + 2x + 1 = 16 – 8x + x2 + 3

10x – 18 = 0

which is not a quadratic equation.

(B) Given, -2x2 = (5 – x) (2x – 2/5)

-2x2 = 10x – 2x2 – 2 +2/5x

52x – 10 = 0

which is not a quadratic equation.

(C) Given, (k + 1) x2 + 3/2 x  = 7, where k = -1

(-1 + 1) x2 + 3/2 x = 7

3x – 14 = 0

which is not a quadratic equation.

(D) Given, x3 – x2 = (x – 1)3

x3 – x2 = x3 – 3x2 + 3x – 1

2x2 – 3x + 1 = 0

which represents a quadratic equation.

Q2Written answer

Which of the following is not a quadratic equation?

(A) 2(x – 1)2 = 4x2 – 2x + 1 (B) 2x – x2 = x2 + 5

(C) (√2x + √3)2 + x2 = 3x2 − 5x (D) (x2 + 2x)2 = x4 + 3 + 4x3

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Worked solution

(D) (x2 + 2x)2 = x4 + 3 + 4x3

A quadratic equation is represented by the form,

ax2 + bx + c = 0, a ≠ 0

(A) Given, 2(x – 1)2 = 4x2 – 2x + 1

2(x2 – 2x + 1) = 4x2 – 2x + 1

2x2 + 2x – 1 = 0

which is a quadratic equation.

(B) Given, 2x – x2 = x2 + 5

2x2 – 2x + 5 = 0

which is a quadratic equation.

(C) Given, (√2x + √3)2  = 3x2 – 5x

2x2 + 2√6x + 3  = 3x2 – 5x

x2 – (5 + 2√6)x – 3 = 0

which is a quadratic equation.

(D) Given, (x2 + 2x)2 = x4 + 3 + 4x2

x4 + 4x3 + 4x2 = x4 + 3 + 4x2

4x3 – 3 = 0

which is a cubic equation and not a quadratic equation.

Q3Written answer

Which of the following equations has 2 as a root?

(A) x2 – 4x + 5 = 0 (B) x2 + 3x – 12 = 0

(C) 2x2 – 7x + 6 = 0 (D) 3x2 – 6x – 2 = 0

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Worked solution

(C) 2x2 – 7x + 6 = 0

If 2 is a root then substituting the value 2 in place of x should satisfy the equation.

(A) Given,

x2 – 4x + 5 = 0

(2)2 – 4(2) + 5 = 1 ≠ 0

So, x = 2 is not a root of x2 – 4x + 5 = 0

(B) Given, x2 + 3x – 12 = 0

(2)2 + 3(2) – 12 = -2 ≠ 0

So, x = 2 is not a root of x2 + 3x – 12 = 0

(C) Given, 2x2 – 7x + 6 = 0

2(2)2 – 7(2) + 6 = 0

Here, x = 2 is a root of 2x2 – 7x + 6 = 0

(D) Given, 3x2 – 6x – 2 = 0

3(2)2 – 6(2) – 2 = -2 ≠ 0

So, x = 2 is not a root of 3x2 – 6x – 2 = 0

Q4Written answer

If ½ is a root of the equation x2 + kx – 5/4 = 0, then the value of k is

(A) 2 (B) – 2

(C) ¼ (D) ½

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Worked solution

(A) 2

If ½ is a root of the equation

x2 + kx – 5/4 = 0 then, substituting the value of ½ in place of x should give us the value of k.

Given, x2 + kx – 5/4 = 0 where, x = ½

(½)2 + k (½) – (5/4) = 0

(k/2) = (5/4) – ¼

k = 2

Q5Written answer

Which of the following equations has the sum of its roots as 3?

(A) 2x2 – 3x + 6 = 0 (B) –x2 + 3x – 3 = 0

(C) √2x2 – 3/√2x+1=0 (D) 3x2 – 3x + 3 = 0

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Worked solution

(B) –x2 + 3x – 3 = 0

The sum of the roots of a quadratic equation ax2 + bx + c = 0, a ≠ 0 is given by,

Coefficient of x / coefficient of x2 = – (b/a)

(A) Given, 2x2 – 3x + 6 = 0

Sum of the roots = – b/a = -(-3/2) = 3/2

(B) Given, -x2 + 3x – 3 = 0

Sum of the roots = – b/a = -(3/-1) = 3

(C) Given, √2x2 – 3/√2x+1=0

2x2 – 3x + √2 = 0

Sum of the roots = – b/a = -(-3/2) = 3/2

(D) Given, 3x2 – 3x + 3 = 0

Sum of the roots = – b/a = -(-3/3) = 1