CBSE Class 10 Mathematics Quadratic Equations MCQs Set 24

Mathematics Objective Questions and Answers: Chapter 04 Quadratic Equations

Access targeted multiple-choice questions for Chapter 04 Quadratic Equations designed to align with the latest CBSE academic syllabus for Class 10 Mathematics. These objective practice sets help students evaluate their conceptual understanding and improve exam readiness.

Download Chapter 04 Quadratic Equations MCQs with Answers

Navigate directly to the 50 objective questions for Chapter 04 Quadratic Equations using the digital viewer below. Each practice set includes verified answer keys, allowing students to instantly cross-check their work and identify areas requiring further revision.

Multiple Choice Questions

Question 1: The values of \( p \) for which \( 2x^2 + px + 8 = 0 \) has real roots are
(a) \( -8 \le p \le 8 \)
(b) \( p \ge 8 \) only
(c) \( p \ge 8 \) or \( p \le -8 \)
(d) \( p \le -8 \) only
Show Answer & Explanation

Answer: (c) \( p \ge 8 \) or \( p \le -8 \)

Explanation:
1. \( D = p^2 - 64 \ge 0 \), so \( p \ge 8 \) or \( p \le -8 \).

Teacher's Note:
1. p² ≥ 64 gives two intervals.
2. Test p = 0: D = -64 < 0, so values between -8 and 8 fail.

Question 2: The values of \( p \) for which \( 2x^2 + px + 3 = 0 \) has real roots are
(a) \( p^2 \le 24 \)
(b) \( p \ge 24 \)
(c) \( p^2 \ge 24 \)
(d) \( p^2 \ge 6 \)
Show Answer & Explanation

Answer: (c) \( p^2 \ge 24 \)

Explanation:
1. \( D = p^2 - 24 \ge 0 \).

Teacher's Note:
1. Real roots: D ≥ 0.
2. p ≥ 2√6 or p ≤ -2√6.

Question 3: The value(s) of \( m \) for which the roots of \( (4 - m)x^2 + (2m + 4)x + (8m + 1) = 0 \) are equal are
(a) \( m = 0 \) or \( m = 3 \)
(b) \( m = 3 \) only
(c) \( m = 0 \) only
(d) \( m = 4 \)
Show Answer & Explanation

Answer: (a) \( m = 0 \) or \( m = 3 \)

Explanation:
1. \( D = (2m + 4)^2 - 4(4 - m)(8m + 1) = 36m^2 - 108m = 36m(m - 3) \).
2. So \( m = 0 \) or \( m = 3 \) (both keep \( 4 - m \ne 0 \)).

Teacher's Note:
1. Expand 4(4 - m)(8m + 1) = 124m + 16 - 32m².
2. Factor out 36m.

Question 4: The discriminant of \( x^2 - 3x + 1 = 0 \) is
(a) 13
(b) 5
(c) -5
(d) 3
Show Answer & Explanation

Answer: (b) 5

Explanation:
1. \( D = 9 - 4 = 5 \).

Teacher's Note:
1. D = b² - 4ac.
2. Here a = 1, b = -3, c = 1.

Question 5: The discriminant of \( 4x^2 + 3x - 2 = 0 \) is
(a) -23
(b) 41
(c) 9
(d) 25
Show Answer & Explanation

Answer: (b) 41

Explanation:
1. \( D = 9 + 32 = 41 \).

Teacher's Note:
1. The product -4ac = -4(4)(-2) = +32.
2. Signs matter when c is negative.

Question 6: The discriminant of \( \sqrt{2}x^2 + 5\sqrt{3}x - 2\sqrt{2} = 0 \) is
(a) 91
(b) 59
(c) 75
(d) -91
Show Answer & Explanation

Answer: (a) 91

Explanation:
1. \( D = (5\sqrt{3})^2 - 4(\sqrt{2})(-2\sqrt{2}) = 75 + 16 = 91 \).

Teacher's Note:
1. (√2)(2√2) = 4, so -4ac = +16.
2. (5√3)² = 75.

Question 7: The discriminant of \( 3\sqrt{5}x^2 - 8x + 2\sqrt{5} = 0 \) is
(a) \( 56 \)
(b) \( 64 \)
(c) \( -56 \)
(d) \( -64 \)
Show Answer & Explanation

Answer: (c) \( -56 \)

Explanation:
1. \( D = 64 - 4(3\sqrt{5})(2\sqrt{5}) = 64 - 120 = -56 \).

Teacher's Note:
1. (3√5)(2√5) = 30, so 4ac = 120.
2. D < 0 means no real roots.

Question 8: The discriminant of \( ax^2 - 3x - 5 = 0 \) is
(a) \( 9 - 20a \)
(b) \( 9 + 20a \)
(c) \( -9 + 20a \)
(d) \( 3 + 20a \)
Show Answer & Explanation

Answer: (b) \( 9 + 20a \)

Explanation:
1. \( D = 9 - 4a(-5) = 9 + 20a \).

Teacher's Note:
1. c = -5, so -4ac = +20a.
2. Do not drop the minus sign of c.

Question 9: The set of values of \( p \) for which \( 3x^2 - 2px - 5 = 0 \) has real roots is
(a) all real \( p \)
(b) \( p \ge 0 \) only
(c) \( |p| \ge 3 \)
(d) \( p \ge \sqrt{15} \)
Show Answer & Explanation

Answer: (a) all real \( p \)

Explanation:
1. \( D = 4p^2 + 60 > 0 \) for every real \( p \).
2. So the roots are always real.

Teacher's Note:
1. When a and c have opposite signs, D is always positive.
2. Here a = 3 and c = -5.

Question 10: The equation \( x^2 + 6x + 6 = 0 \) has roots
(a) \( -3 \pm \sqrt{3} \)
(b) \( 3 \pm \sqrt{3} \)
(c) \( -3 \pm \sqrt{15} \)
(d) \( -6 \pm \sqrt{3} \)
Show Answer & Explanation

Answer: (a) \( -3 \pm \sqrt{3} \)

Explanation:
1. \( D = 36 - 24 = 12 \), \( \sqrt{D} = 2\sqrt{3} \).
2. \( x = \dfrac{-6 \pm 2\sqrt{3}}{2} = -3 \pm \sqrt{3} \).

Teacher's Note:
1. Check that D > 0 first.
2. Sum of roots -6, product 6.

Question 11: The equation \( 3x^2 - 3x - 1 = 0 \) has roots
(a) \( \dfrac{3 \pm \sqrt{21}}{3} \)
(b) \( \dfrac{-3 \pm \sqrt{21}}{6} \)
(c) \( \dfrac{3 \pm \sqrt{9}}{6} \)
(d) \( \dfrac{3 \pm \sqrt{21}}{6} \)
Show Answer & Explanation

Answer: (d) \( \dfrac{3 \pm \sqrt{21}}{6} \)

Explanation:
1. \( D = 9 + 12 = 21 \).
2. \( x = \dfrac{3 \pm \sqrt{21}}{6} \).

Teacher's Note:
1. The denominator is 2a = 6.
2. The sign of -b is +3.

Question 12: The sum and the product of the roots of \( x^2 + 3x - 6 = 0 \) are respectively
(a) \( 3 \) and \( -6 \)
(b) \( -3 \) and \( -6 \)
(c) \( -3 \) and \( 6 \)
(d) \( -\dfrac{3}{2} \) and \( -6 \)
Show Answer & Explanation

Answer: (b) \( -3 \) and \( -6 \)

Explanation:
1. Here \( a = 1, b = 3, c = -6 \).
2. Sum \( = -\dfrac{b}{a} = -3 \) and product \( = \dfrac{c}{a} = -6 \).

Teacher's Note:
1. Sum of roots = -b/a and product of roots = c/a.
2. Always bring the equation to the form ax² + bx + c = 0 first.

Question 13: The sum and the product of the roots of \( 2x^2 - 7x + 4 = 0 \) are respectively
(a) \( -\dfrac{7}{2} \) and \( 2 \)
(b) \( \dfrac{7}{2} \) and \( 4 \)
(c) \( \dfrac{7}{2} \) and \( 2 \)
(d) \( 7 \) and \( 2 \)
Show Answer & Explanation

Answer: (c) \( \dfrac{7}{2} \) and \( 2 \)

Explanation:
1. Sum \( = \dfrac{7}{2} \) and product \( = \dfrac{4}{2} = 2 \).

Teacher's Note:
1. Sum of roots = -b/a and product of roots = c/a.
2. Always bring the equation to the form ax² + bx + c = 0 first.

Question 14: The sum and the product of the roots of \( 2x^2 + 5\sqrt{3}x - 3 = 0 \) are respectively
(a) \( \dfrac{5\sqrt{3}}{2} \) and \( -\dfrac{3}{2} \)
(b) \( -\dfrac{5\sqrt{3}}{2} \) and \( \dfrac{3}{2} \)
(c) \( -5\sqrt{3} \) and \( -3 \)
(d) \( -\dfrac{5\sqrt{3}}{2} \) and \( -\dfrac{3}{2} \)
Show Answer & Explanation

Answer: (d) \( -\dfrac{5\sqrt{3}}{2} \) and \( -\dfrac{3}{2} \)

Explanation:
1. Sum \( = -\dfrac{5\sqrt{3}}{2} \) and product \( = -\dfrac{3}{2} \).

Teacher's Note:
1. Sum of roots = -b/a and product of roots = c/a.
2. Always bring the equation to the form ax² + bx + c = 0 first.

Question 15: The sum and the product of the roots of \( 5x^2 - 10x + 3\sqrt{5} = 0 \) are respectively
(a) \( -2 \) and \( \dfrac{3\sqrt{5}}{5} \)
(b) \( 2 \) and \( \dfrac{3\sqrt{5}}{5} \)
(c) \( 2 \) and \( 3\sqrt{5} \)
(d) \( 10 \) and \( \dfrac{3\sqrt{5}}{5} \)
Show Answer & Explanation

Answer: (b) \( 2 \) and \( \dfrac{3\sqrt{5}}{5} \)

Explanation:
1. Sum \( = \dfrac{10}{5} = 2 \) and product \( = \dfrac{3\sqrt{5}}{5} \).

Teacher's Note:
1. Sum of roots = -b/a and product of roots = c/a.
2. Always bring the equation to the form ax² + bx + c = 0 first.

Question 16: The quadratic equation whose roots are \( 3 + \sqrt{3} \) and \( 3 - \sqrt{3} \) is
(a) \( x^2 + 6x + 6 = 0 \)
(b) \( x^2 - 6x + 12 = 0 \)
(c) \( x^2 - 3x + 6 = 0 \)
(d) \( x^2 - 6x + 6 = 0 \)
Show Answer & Explanation

Answer: (d) \( x^2 - 6x + 6 = 0 \)

Explanation:
1. Sum \( = 6 \), product \( = 9 - 3 = 6 \).
2. The equation is \( x^2 - 6x + 6 = 0 \).

Teacher's Note:
1. Sum of roots = -b/a and product of roots = c/a.
2. Always bring the equation to the form ax² + bx + c = 0 first.

Question 17: The quadratic equation whose roots are \( \dfrac{2 + \sqrt{5}}{2} \) and \( \dfrac{2 - \sqrt{5}}{2} \) is
(a) \( 4x^2 - 8x - 1 = 0 \)
(b) \( 4x^2 - 8x + 1 = 0 \)
(c) \( x^2 - 2x + \dfrac{1}{4} = 0 \)
(d) \( 4x^2 + 8x - 1 = 0 \)
Show Answer & Explanation

Answer: (a) \( 4x^2 - 8x - 1 = 0 \)

Explanation:
1. Sum \( = 2 \), product \( = \dfrac{4 - 5}{4} = -\dfrac{1}{4} \).
2. \( x^2 - 2x - \dfrac{1}{4} = 0 \), i.e. \( 4x^2 - 8x - 1 = 0 \).

Teacher's Note:
1. Sum of roots = -b/a and product of roots = c/a.
2. Always bring the equation to the form ax² + bx + c = 0 first.

Question 18: The quadratic equation whose roots are \( -8 \) and \( 3 \) is
(a) \( x^2 - 5x - 24 = 0 \)
(b) \( x^2 + 5x + 24 = 0 \)
(c) \( x^2 + 11x - 24 = 0 \)
(d) \( x^2 + 5x - 24 = 0 \)
Show Answer & Explanation

Answer: (d) \( x^2 + 5x - 24 = 0 \)

Explanation:
1. Sum \( = -5 \), product \( = -24 \).
2. The equation is \( x^2 - (-5)x - 24 = x^2 + 5x - 24 = 0 \).

Teacher's Note:
1. Sum of roots = -b/a and product of roots = c/a.
2. Always bring the equation to the form ax² + bx + c = 0 first.

Question 19: The quadratic equation whose roots are \( \sqrt{3} \) and \( 5\sqrt{3} \) is
(a) \( x^2 + 6\sqrt{3}x + 15 = 0 \)
(b) \( x^2 - 6\sqrt{3}x + 15 = 0 \)
(c) \( x^2 - 6\sqrt{3}x + 5\sqrt{3} = 0 \)
(d) \( x^2 - 4\sqrt{3}x + 15 = 0 \)
Show Answer & Explanation

Answer: (b) \( x^2 - 6\sqrt{3}x + 15 = 0 \)

Explanation:
1. Sum \( = 6\sqrt{3} \), product \( = 15 \).
2. The equation is \( x^2 - 6\sqrt{3}x + 15 = 0 \).

Teacher's Note:
1. Sum of roots = -b/a and product of roots = c/a.
2. Always bring the equation to the form ax² + bx + c = 0 first.

Question 20: The quadratic equation whose roots are \( -3\sqrt{5} \) and \( -4\sqrt{5} \) is
(a) \( x^2 - 7\sqrt{5}x + 60 = 0 \)
(b) \( x^2 + 7\sqrt{5}x - 60 = 0 \)
(c) \( x^2 + 7\sqrt{5}x + 60 = 0 \)
(d) \( x^2 + 7\sqrt{5}x + 12 = 0 \)
Show Answer & Explanation

Answer: (c) \( x^2 + 7\sqrt{5}x + 60 = 0 \)

Explanation:
1. Sum \( = -7\sqrt{5} \), product \( = 12 \times 5 = 60 \).

Teacher's Note:
1. Sum of roots = -b/a and product of roots = c/a.
2. Always bring the equation to the form ax² + bx + c = 0 first.

Question 21: The quadratic equation whose roots are \( 1 + \sqrt{2} \) and \( 1 - \sqrt{2} \) is
(a) \( x^2 - 2x - 1 = 0 \)
(b) \( x^2 - 2x + 1 = 0 \)
(c) \( x^2 + 2x - 1 = 0 \)
(d) \( x^2 - 2x + 3 = 0 \)
Show Answer & Explanation

Answer: (a) \( x^2 - 2x - 1 = 0 \)

Explanation:
1. Sum \( = 2 \), product \( = 1 - 2 = -1 \).

Teacher's Note:
1. Sum of roots = -b/a and product of roots = c/a.
2. Always bring the equation to the form ax² + bx + c = 0 first.

Question 22: The quadratic equation whose roots are \( \dfrac{3 + \sqrt{2}}{3} \) and \( \dfrac{3 - \sqrt{2}}{3} \) is
(a) \( 9x^2 - 18x + 11 = 0 \)
(b) \( 3x^2 - 6x + 7 = 0 \)
(c) \( 9x^2 - 18x + 7 = 0 \)
(d) \( 9x^2 + 18x + 7 = 0 \)
Show Answer & Explanation

Answer: (c) \( 9x^2 - 18x + 7 = 0 \)

Explanation:
1. Sum \( = 2 \), product \( = \dfrac{9 - 2}{9} = \dfrac{7}{9} \).
2. \( x^2 - 2x + \dfrac{7}{9} = 0 \), i.e. \( 9x^2 - 18x + 7 = 0 \).

Teacher's Note:
1. Sum of roots = -b/a and product of roots = c/a.
2. Always bring the equation to the form ax² + bx + c = 0 first.

Question 23: A quadratic equation with rational coefficients has \( 2 + \sqrt{3} \) as one root. The equation is
(a) \( x^2 - 4x - 1 = 0 \)
(b) \( x^2 - 4x + 1 = 0 \)
(c) \( x^2 + 4x + 1 = 0 \)
(d) \( x^2 - 2x + 1 = 0 \)
Show Answer & Explanation

Answer: (b) \( x^2 - 4x + 1 = 0 \)

Explanation:
1. The other root is the conjugate \( 2 - \sqrt{3} \).
2. Sum \( = 4 \) and product \( = 4 - 3 = 1 \), so the equation is \( x^2 - 4x + 1 = 0 \).

Teacher's Note:
1. Sum of roots = -b/a and product of roots = c/a.
2. Always bring the equation to the form ax² + bx + c = 0 first.

Question 24: For \( ax^2 - 5x + c = 0 \) the sum of the roots is 10 and the product of the roots is 10. Then \( (a, c) \) is
(a) \( \left(\dfrac{1}{2}, 5\right) \)
(b) \( \left(2, 20\right) \)
(c) \( \left(\dfrac{1}{2}, 10\right) \)
(d) \( \left(5, 5\right) \)
Show Answer & Explanation

Answer: (a) \( \left(\dfrac{1}{2}, 5\right) \)

Explanation:
1. Sum \( = \dfrac{5}{a} = 10 \), so \( a = \dfrac{1}{2} \).
2. Product \( = \dfrac{c}{a} = 10 \), so \( c = 5 \).

Teacher's Note:
1. Sum of roots = -b/a and product of roots = c/a.
2. Always bring the equation to the form ax² + bx + c = 0 first.

Question 25: For \( ax^2 + 7x + c = 0 \) the sum of the roots is \( -1 \) and the product of the roots is \( 1 \). Then \( (a, c) \) is
(a) \( (-7, 7) \)
(b) \( (7, -7) \)
(c) \( (1, 7) \)
(d) \( (7, 7) \)
Show Answer & Explanation

Answer: (d) \( (7, 7) \)

Explanation:
1. Sum \( = -\dfrac{7}{a} = -1 \), so \( a = 7 \).
2. Product \( = \dfrac{c}{a} = 1 \), so \( c = 7 \).

Teacher's Note:
1. Sum of roots = -b/a and product of roots = c/a.
2. Always bring the equation to the form ax² + bx + c = 0 first.

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Multiple Choice Questions (MCQs) for Class 10 Mathematics Chapter 04 Quadratic Equations

Download Multiple Choice Questions: Chapter 04 Quadratic Equations (Class 10 Mathematics)

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