CBSE Class 10 Mathematics Quadratic Equations MCQs Set 14

Question. The value(s) of k for which the quadratic equation \( 2x^2 + kx + 2 = 0 \) has equal roots, is:
(a) 4
(b) \( \pm 4 \)
(c) – 4
(d) 0
Answer: (b) \( \pm 4 \)

Question. The quadratic equations \( x^2 – 4x + k = 0 \) has distinct real roots if
(a) \( k = 4 \)
(b) \( k > 4 \)
(c) \( k = 16 \)
(d) \( k < 4 \)
Answer: (d) \( k < 4 \)
 

Question. The equations \( x^2 – 8x + k = 0 \) has real and distinct roots if:
(a) \( k = 16 \)
(b) \( k > 16 \)
(c) \( k = 8 \)
(d) \( k < 16 \)
Answer: (d) \( k < 16 \)
 

Question. Which of the following is a quadratic equation?
(a) \( x^2 + 2x + 1 = (4 – x)^2 + 3 \)
(b) \( –2x^2 = (5 – x) \left( 2x - \frac{2}{5} \right) \)
(c) \( (k + 1)x^2 + \frac{3}{2}x = 7 \), where \( k = –1 \)
(d) \( x^3 – x^2 = (x – 1)^3 \)
Answer: (d) \( x^3 – x^2 = (x – 1)^3 \)

Question. Which of the following is not a quadratic equation?
(a) \( 2(x – 1)^2 = 4x^2 – 2x + 1 \)
(b) \( 2x – x^2 = x^2 + 5 \)
(c) \( (\sqrt{2}x + \sqrt{3})^2 + x^2 = 3x^2 – 5x \)
(d) \( (x^2 + 2x)^2 = x^4 + 3 + 4x^3 \)
Answer: (c) \( (\sqrt{2}x + \sqrt{3})^2 + x^2 = 3x^2 – 5x \)

Definition

An equation which is of the form \( ax^2 + bx + c = 0, a \neq 0 \) is called a quadratic equation.

Question. For what value of k, \( kx^2 + 8x + 2 = 0 \) has real roots
(a) \( k < 8 \)
(b) \( k > 8 \)
(c) \( k = 8 \)
(d) None of the options
Answer: (a) \( k < 8 \)

Question. Which of the following equations has 2 as a root?
(a) \( x^2 – 4x + 5 = 0 \)
(b) \( x^2 + 3x – 12 = 0 \)
(c) \( 2x^2 – 7x + 6 = 0 \)
(d) \( 3x^2 – 6x – 2 = 0 \)
Answer: (c) \( 2x^2 – 7x + 6 = 0 \)

Trick Applied

A real number \( \alpha \) is said to be the root of the quadratic equation \( ax^2 + bx + c = 0 \) if \( a\alpha^2 + b\alpha + c = 0 \).

Question. The positive root of \( \sqrt{3x^2 + 6} = 9 \) is
(a) 2
(b) 1
(c) 4
(d) 3
Answer: (b) 1

Question. Which of the following equations has the sum of its roots as 3?
(a) \( 2x^2 – 3x + 6 = 0 \)
(b) \( -x^2 + 3x – 3 = 0 \)
(c) \( \sqrt{2}x^2 – \frac{3}{\sqrt{2}}x + 1 = 0 \)
(d) \( 3x^2 – 3x + 3 = 0 \)
Answer: (b) \( -x^2 + 3x – 3 = 0 \)
 

Question. Is \( x^3 – 4x^2 – x + 1 = (x – 2)^3 \) a quadratic equation?
(a) yes
(b) No
(c) Can’t say
(d) This is a cubic equation
Answer: (a) yes
 

Question. For equal root, \( kx(x – 2) + 6 = 0 \), the value of k is
(a) \( k = 0, 6 \)
(b) \( k = 6, – 6 \)
(c) \( k = 2, 3 \)
(d) \( k = 0, 3 \)
Answer: (a) \( k = 0, 6 \)
 

Question. Roots of \( -x^2 + \frac{1}{2}x + \frac{1}{2} = 0 \) are
(a) \( -\frac{1}{2}, 1 \)
(b) \( \frac{1}{2}, 1 \)
(c) \( -\frac{1}{2}, -1 \)
(d) \( \frac{1}{2}, \frac{1}{2} \)
Answer: (a) \( -\frac{1}{2}, 1 \)
 

Question. The quadratic equation \( 2x^2 – \sqrt{5}x + 1 = 0 \) has:
(a) two distinct real roots
(b) two equal real roots
(c) no real roots
(d) more than 2 real roots
Answer: (c) no real roots
 

Question. Which of the following equations has two distinct real roots?
(a) \( 2x^2 – 3\sqrt{2}x + \frac{9}{4} = 0 \)
(b) \( x^2 + x – 5 = 0 \)
(c) \( x^2 + 3x + 2\sqrt{2} = 0 \)
(d) \( 5x^2 – 3x + 1 = 0 \)
Answer: (b) \( x^2 + x – 5 = 0 \)
 

Trick Applied

A quadratic equation \( ax^2 + bx + c = 0 \) has:
(i) two distinct real roots if \( b^2 – 4ac > 0 \).
(ii) two equal real roots if \( b^2 – 4ac = 0 \).
(iii) no real roots if \( b^2 – 4ac < 0 \).

Question. \( (x^2 + 1)^2 – x^2 = 0 \) has:
(a) four real roots
(b) two real roots
(c) no real root
(d) one real root
Answer: (c) no real root
 

Fill in the Blanks

Question. The quadratic equation \( 2x^2 + px + 3 = 0 \) has two equal roots if p = .................... .
Answer: \( 2x^2 + px + 3 = 0 \) will have equal roots, when \( p^2 – 4(2)(3) = 0 \) i.e. when \( p^2 – 24 = 0 \), or \( p = \sqrt{24} \) or \( p = \pm 2\sqrt{6} \)

Question. Equation \( ax^2 + bx + c = 0 \) represents a quadratic equation if and only if .................... .
Answer: \( a \neq 0 \). [In case \( a = 0 \), the equation reduces to \( bx + c = 0 \), which is a linear equation]

Question. Sum of roots of quadratic equation \( x^2 – 4x + 2 = 0 \) is ................... of product of roots.
Answer: Twice
Sum of roots \( = -\text{coefficient of } x / \text{coefficient of } x^2 = -(-4) = 4 \)
Product of roots \( = \frac{\text{constant term}}{\text{Coefficient of } x^2} = \frac{2}{1} = 2 \)

Question. The quadratic equation \( 2x^2 + x + 4 \) has .................... real roots.
Answer: No. Here, \( b^2 – 4ac = 1 – 32 \), which is less than 0.

Question. The roots of \( x + \frac{1}{x} = 2 \) are ........................ .
Answer: [1, 1]. The equation is \( x^2 – 2x + 1 = 0 \), or \( (x – 1)^2 = 0 \). So, roots are 1, 1.

Question. The sum of the roots of the quadratic equation \( 2x^2 + 14x + 24 = 0 \) is .................... .
Answer: [–7]. The sum roots of the equation \( 2x^2 + 14x + 24 = 0 \) is \( -\frac{14}{2} \), i.e., –7