Find below the NCERT Exemplar problems and their solutions for Class 10 Mathematics Chapter, Quadratic Equations:

Exercise 4.1

Multiple Choice Questions (MCQs)

Question1. Which of the following is a quadratic equation?

Solution: (d)

Explanation:

An equation which is of the form ax² + bx + c = 0, a ≠ 0 is called a quadratic equation.

⟹ x³ - x² = x³ -3x² + 3x - 1

⟹-x² + 3x² - 3x + 1 = 0

⟹ 2x² - 3x + 1 = 0

This equation is not of the form ax²+ bx + c, a ≠ 0. Thus, the equation is a quadratic equation.

Question2. Which of the following is not a quadratic equation?

This equation is not of the form ax² + bx + c, a ≠ 0.

Thus, the equation is not quadratic.

Question3. 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

Solution: (c)

Explanation:

If k is one of the root of any quadratic equation then x = k

i.e., k satisfies the equation ak² + bk + c = 0.

i.e., f(k) = ak² + bk + c = 0,

(a) Putting x = 2 in x² - 4x + 5, we get

(2)² - 4(2) + 5 = 4 - 8 + 5 = 1≠0.

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

(b) Putting x = 2 in x² + 3x - 12, we get

(2)² + 3(2) - 12 = 4 + 6 – 12 = –2 ≠ 0

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

(c) Putting x = 2 in 2x² – 7x + 6, we get

2(2)² – 7(2) + 6 = 2 (4) – 14+ 6 = 8 – 14 + 6 = 14 – 14 = 0

So, x = 2 is root of the equation 2x² – 7x + 6 = 0.

(d) Putting x = 2 in 3x² –6x –2, we get

3(2)² –6(2) –2 = 12 –12 –2 = –2 ≠ 0

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

⟹ k = 2

Question5. Which of the following equations has the sum of its roots as 3?

So, option (b) is correct.

Question6. Value(s) of k for which the quadratic equation 2x² –kx + k = 0 has equal roots is/are

(a) 0  

(b) 4  

(c) 8  

(d) 0, 8

Solution: (d)

Explanation:

For a quadratic equation of the form, ax² + bx + c = 0, a ≠ 0 to have two equal real roots, its discriminant must be equal to zero, i.e., D = b² –4ac = 0

Given equation is 2x² – kx + k = 0

On comparing with ax² + bx + c = 0, we get

a = 2, b = – k and c = k

For equal roots, the discriminant must be zero.

i.e., D = b² – 4ac = 0

⟹ (– k)² – 4 (2) k = 0

⟹ k² – 8k = 0

⟹ k (k – 8) = 0

Thus, k = 0, 8

Hence, the required values of k are 0 and 8.

Question10. Which of the following equations has no real roots?

Hence, the equation has two distinct real roots.

Question11. (x² + l)² – x² = 0 has

(a) four real roots       

(b) two real roots        

(c) no real roots          

(d) one real root

Solution: (c)