CBSE Class 10 Mathematics Quadratic Equations MCQs Set 23

Download CBSE MCQs for Class 10 Mathematics: 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.

Chapter-wise Objective Questions: Chapter 04 Quadratic Equations

Access the complete set of multiple-choice questions for Chapter 04 Quadratic Equations below. This focused format allows students to isolate specific topics for thorough review and uninterrupted practice alongside official CBSE textbooks.

Multiple Choice Questions

Question 1: The roots of \( 10x^2 - 7x - 12 = 0 \) are
(a) \( -\dfrac{3}{2} \) and \( \dfrac{4}{5} \)
(b) \( \dfrac{3}{2} \) and \( -\dfrac{4}{5} \)
(c) \( \dfrac{3}{2} \) and \( \dfrac{4}{5} \)
(d) \( 2 \) and \( -\dfrac{3}{5} \)
Show Answer & Explanation

Answer: (b) \( \dfrac{3}{2} \) and \( -\dfrac{4}{5} \)

Explanation:
1. \( 10x^2 - 15x + 8x - 12 = 5x(2x - 3) + 4(2x - 3) = (2x - 3)(5x + 4) = 0 \).
2. So \( x = \dfrac{3}{2} \) or \( x = -\dfrac{4}{5} \).

Teacher's Note:
1. Factorise or use the quadratic formula, and then check using the sum and product of roots.
2. Take care with signs of the coefficients.

Question 2: The roots of \( 8x^2 - 2x - 3 = 0 \) are
(a) \( \dfrac{3}{4} \) and \( -\dfrac{1}{2} \)
(b) \( -\dfrac{3}{4} \) and \( \dfrac{1}{2} \)
(c) \( \dfrac{3}{4} \) and \( \dfrac{1}{2} \)
(d) \( \dfrac{3}{2} \) and \( -\dfrac{1}{4} \)
Show Answer & Explanation

Answer: (a) \( \dfrac{3}{4} \) and \( -\dfrac{1}{2} \)

Explanation:
1. \( 8x^2 - 6x + 4x - 3 = 2x(4x - 3) + 1(4x - 3) = (4x - 3)(2x + 1) = 0 \).
2. So \( x = \dfrac{3}{4} \) or \( x = -\dfrac{1}{2} \).

Teacher's Note:
1. Factorise or use the quadratic formula, and then check using the sum and product of roots.
2. Take care with signs of the coefficients.

Question 3: The roots of \( x(4x - 7) = 0 \) are
(a) \( 0 \) and \( \dfrac{7}{4} \)
(b) \( 0 \) and \( -\dfrac{7}{4} \)
(c) \( \dfrac{7}{4} \) only
(d) \( 4 \) and \( -7 \)
Show Answer & Explanation

Answer: (a) \( 0 \) and \( \dfrac{7}{4} \)

Explanation:
1. \( x = 0 \) or \( 4x - 7 = 0 \), so \( x = \dfrac{7}{4} \).

Teacher's Note:
1. Factorise or use the quadratic formula, and then check using the sum and product of roots.
2. Take care with signs of the coefficients.

Question 4: The roots of \( x(x + 1) + (x + 2)(x + 3) = 26 \) are
(a) \( -2 \) and \( 5 \)
(b) \( 3 \) and \( -6 \)
(c) \( 1 \) and \( -4 \)
(d) \( 2 \) and \( -5 \)
Show Answer & Explanation

Answer: (d) \( 2 \) and \( -5 \)

Explanation:
1. \( 2x^2 + 6x + 6 = 26 \) gives \( x^2 + 3x - 10 = (x + 5)(x - 2) = 0 \).
2. So \( x = 2 \) or \( x = -5 \).

Teacher's Note:
1. Factorise or use the quadratic formula, and then check using the sum and product of roots.
2. Take care with signs of the coefficients.

Question 5: The roots of \( 4x^2 = 25 \) are
(a) \( \dfrac{5}{2} \) only
(b) \( \pm\dfrac{5}{2} \)
(c) \( \pm\dfrac{25}{4} \)
(d) \( \pm\dfrac{2}{5} \)
Show Answer & Explanation

Answer: (b) \( \pm\dfrac{5}{2} \)

Explanation:
1. \( x^2 = \dfrac{25}{4} \), so \( x = \pm\dfrac{5}{2} \).

Teacher's Note:
1. Factorise or use the quadratic formula, and then check using the sum and product of roots.
2. Take care with signs of the coefficients.

Question 6: The roots of \( x^2 - 2\sqrt{2}x - 6 = 0 \) are
(a) \( -3\sqrt{2} \) and \( \sqrt{2} \)
(b) \( \sqrt{2} \) and \( 3\sqrt{2} \)
(c) \( 3\sqrt{2} \) and \( -\sqrt{2} \)
(d) \( 2\sqrt{2} \) and \( -3 \)
Show Answer & Explanation

Answer: (c) \( 3\sqrt{2} \) and \( -\sqrt{2} \)

Explanation:
1. \( D = 8 + 24 = 32 \), \( \sqrt{D} = 4\sqrt{2} \).
2. \( x = \dfrac{2\sqrt{2} \pm 4\sqrt{2}}{2} = 3\sqrt{2} \) or \( -\sqrt{2} \).

Teacher's Note:
1. Factorise or use the quadratic formula, and then check using the sum and product of roots.
2. Take care with signs of the coefficients.

Question 7: The roots of \( 16x^2 - 1 = 0 \) are
(a) \( \dfrac{1}{4} \) only
(b) \( \pm 4 \)
(c) \( \pm\dfrac{1}{16} \)
(d) \( \pm\dfrac{1}{4} \)
Show Answer & Explanation

Answer: (d) \( \pm\dfrac{1}{4} \)

Explanation:
1. \( x^2 = \dfrac{1}{16} \), so \( x = \pm\dfrac{1}{4} \).

Teacher's Note:
1. Factorise or use the quadratic formula, and then check using the sum and product of roots.
2. Take care with signs of the coefficients.

Question 8: The roots of \( 5x^2 - x - 4 = 0 \) are
(a) \( -1 \) and \( \dfrac{4}{5} \)
(b) \( 1 \) and \( -\dfrac{4}{5} \)
(c) \( 1 \) and \( \dfrac{4}{5} \)
(d) \( 4 \) and \( -\dfrac{1}{5} \)
Show Answer & Explanation

Answer: (b) \( 1 \) and \( -\dfrac{4}{5} \)

Explanation:
1. \( D = 1 + 80 = 81 \), so \( x = \dfrac{1 \pm 9}{10} = 1 \) or \( -\dfrac{4}{5} \).

Teacher's Note:
1. Factorise or use the quadratic formula, and then check using the sum and product of roots.
2. Take care with signs of the coefficients.

Question 9: The roots of \( 3x^2 - 5x = 0 \) are
(a) \( 0 \) and \( -\dfrac{5}{3} \)
(b) \( \dfrac{5}{3} \) only
(c) \( 3 \) and \( -5 \)
(d) \( 0 \) and \( \dfrac{5}{3} \)
Show Answer & Explanation

Answer: (d) \( 0 \) and \( \dfrac{5}{3} \)

Explanation:
1. \( x(3x - 5) = 0 \), so \( x = 0 \) or \( x = \dfrac{5}{3} \).

Teacher's Note:
1. Factorise or use the quadratic formula, and then check using the sum and product of roots.
2. Take care with signs of the coefficients.

Question 10: The roots of \( \sqrt{6}x^2 - 4x - 2\sqrt{6} = 0 \) are
(a) \( \sqrt{6} \) and \( -\dfrac{2}{\sqrt{6}} \)
(b) \( -\sqrt{6} \) and \( \dfrac{2}{\sqrt{6}} \)
(c) \( \sqrt{6} \) and \( \dfrac{2}{\sqrt{6}} \)
(d) \( 2 \) and \( -\sqrt{6} \)
Show Answer & Explanation

Answer: (a) \( \sqrt{6} \) and \( -\dfrac{2}{\sqrt{6}} \)

Explanation:
1. \( D = 16 + 48 = 64 \), so \( x = \dfrac{4 \pm 8}{2\sqrt{6}} \).
2. \( x = \dfrac{12}{2\sqrt{6}} = \sqrt{6} \) or \( x = -\dfrac{4}{2\sqrt{6}} = -\dfrac{2}{\sqrt{6}} \).

Teacher's Note:
1. Factorise or use the quadratic formula, and then check using the sum and product of roots.
2. Take care with signs of the coefficients.

Question 11: The roots of \( 4x^2 + x - 5 = 0 \) are
(a) \( -1 \) and \( \dfrac{5}{4} \)
(b) \( 1 \) and \( \dfrac{5}{4} \)
(c) \( 5 \) and \( -\dfrac{1}{4} \)
(d) \( 1 \) and \( -\dfrac{5}{4} \)
Show Answer & Explanation

Answer: (d) \( 1 \) and \( -\dfrac{5}{4} \)

Explanation:
1. The coefficients add up to 0 (\( 4 + 1 - 5 = 0 \)), so \( x = 1 \) is a root; the product of roots is \( -\dfrac{5}{4} \), so the other root is \( -\dfrac{5}{4} \).

Teacher's Note:
1. Factorise or use the quadratic formula, and then check using the sum and product of roots.
2. Take care with signs of the coefficients.

Question 12: The roots of \( 5x^2 - 6x - 2 = 0 \) are
(a) \( \dfrac{3 \pm \sqrt{11}}{5} \)
(b) \( \dfrac{3 \pm \sqrt{19}}{5} \)
(c) \( \dfrac{6 \pm \sqrt{19}}{5} \)
(d) \( \dfrac{3 \pm \sqrt{19}}{10} \)
Show Answer & Explanation

Answer: (b) \( \dfrac{3 \pm \sqrt{19}}{5} \)

Explanation:
1. \( D = 36 + 40 = 76 \), \( \sqrt{D} = 2\sqrt{19} \).
2. \( x = \dfrac{6 \pm 2\sqrt{19}}{10} = \dfrac{3 \pm \sqrt{19}}{5} \).

Teacher's Note:
1. Factorise or use the quadratic formula, and then check using the sum and product of roots.
2. Take care with signs of the coefficients.

Question 13: The nature of the roots of \( 2x^2 - 7x + 3 = 0 \) is
(a) real and equal
(b) real, irrational and unequal
(c) real, rational and unequal
(d) imaginary
Show Answer & Explanation

Answer: (c) real, rational and unequal

Explanation:
1. \( D = 49 - 24 = 25 > 0 \) and 25 is a perfect square.
2. So the roots are rational and unequal.

Teacher's Note:
1. Compute D = b² - 4ac first; its sign and whether it is a perfect square decide the nature.
2. D > 0: two distinct real roots; D = 0: equal real roots; D < 0: no real roots.

Question 14: The nature of the roots of \( x^2 - 5x - 2 = 0 \) is
(a) real, irrational and unequal
(b) real, rational and unequal
(c) real and equal
(d) imaginary
Show Answer & Explanation

Answer: (a) real, irrational and unequal

Explanation:
1. \( D = 25 + 8 = 33 > 0 \) and 33 is not a perfect square.
2. So the roots are irrational and unequal.

Teacher's Note:
1. Compute D = b² - 4ac first; its sign and whether it is a perfect square decide the nature.
2. D > 0: two distinct real roots; D = 0: equal real roots; D < 0: no real roots.

Question 15: The nature of the roots of \( 7x^2 - 9x + 2 = 0 \) is
(a) real and equal
(b) real, irrational and unequal
(c) real, rational and unequal
(d) imaginary
Show Answer & Explanation

Answer: (c) real, rational and unequal

Explanation:
1. \( D = 81 - 56 = 25 > 0 \), a perfect square.
2. So the roots are rational and unequal.

Teacher's Note:
1. Compute D = b² - 4ac first; its sign and whether it is a perfect square decide the nature.
2. D > 0: two distinct real roots; D = 0: equal real roots; D < 0: no real roots.

Question 16: The nature of the roots of \( 2x^2 - 7x + 4 = 0 \) is
(a) real, rational and unequal
(b) real, irrational and unequal
(c) real and equal
(d) imaginary
Show Answer & Explanation

Answer: (b) real, irrational and unequal

Explanation:
1. \( D = 49 - 32 = 17 > 0 \), not a perfect square.
2. So the roots are irrational and unequal.

Teacher's Note:
1. Compute D = b² - 4ac first; its sign and whether it is a perfect square decide the nature.
2. D > 0: two distinct real roots; D = 0: equal real roots; D < 0: no real roots.

Question 17: The roots of \( x^2 - 2\sqrt{2}x + 1 = 0 \) are
(a) real and distinct, but not rational
(b) real and rational
(c) real and equal
(d) imaginary
Show Answer & Explanation

Answer: (a) real and distinct, but not rational

Explanation:
1. \( D = 8 - 4 = 4 > 0 \), so the roots are real and distinct.
2. The roots are \( \sqrt{2} \pm 1 \), which are irrational because the coefficients are not rational.

Teacher's Note:
1. Even if D is a perfect square, the roots need not be rational when coefficients are irrational.
2. Solve to confirm: x = √2 ± 1.

Question 18: If \( a, b, c \) are rational numbers, the roots of \( (b + c)x^2 - (a + b + c)x + a = 0 \) are
(a) imaginary
(b) real and irrational
(c) real and equal for every \( a, b, c \)
(d) real and rational
Show Answer & Explanation

Answer: (d) real and rational

Explanation:
1. \( D = (a + b + c)^2 - 4a(b + c) = (a - b - c)^2 \), a perfect square of a rational number.
2. So the roots are real and rational.

Teacher's Note:
1. Show that the discriminant is a perfect square.
2. The roots are 1 and a/(b+c).

Question 19: If the roots of \( x^2 + 2x + P = 0 \) are real, then
(a) \( P \ge 1 \)
(b) \( P \le 1 \)
(c) \( P < 1 \)
(d) \( P > 1 \)
Show Answer & Explanation

Answer: (b) \( P \le 1 \)

Explanation:
1. \( D = 4 - 4P \ge 0 \), so \( P \le 1 \).

Teacher's Note:
1. Real roots mean D ≥ 0 (equal roots are allowed).
2. Keep the inequality sign when dividing by the positive number 4.

Question 20: If \( r \) and \( s \) are positive numbers, the roots of \( x^2 - rx - s = 0 \) are
(a) real and both positive
(b) real and both negative
(c) non-real
(d) real and of opposite signs
Show Answer & Explanation

Answer: (d) real and of opposite signs

Explanation:
1. \( D = r^2 + 4s > 0 \), so the roots are real and distinct.
2. Product of roots \( = -s < 0 \), so one root is positive and one is negative.

Teacher's Note:
1. The product of roots decides whether the signs are the same or opposite.
2. Positive D plus negative product gives opposite signs.

Question 21: Which of the following quadratic equations has real and equal roots?
(a) \( x^2 - 3x + 4 = 0 \)
(b) \( 3x^2 - 3x + 1 = 0 \)
(c) \( 4x^2 - 12x + 9 = 0 \)
(d) \( 3x^2 - 3x - 1 = 0 \)
Show Answer & Explanation

Answer: (c) \( 4x^2 - 12x + 9 = 0 \)

Explanation:
1. \( 4x^2 - 12x + 9 \): \( D = 144 - 144 = 0 \).
2. The others have \( D = -7 \), \( -3 \) and \( 21 \) respectively.

Teacher's Note:
1. Check the discriminant of each option.
2. 4x² - 12x + 9 = (2x - 3)².

Question 22: The values of \( k \) for which \( 2x^2 + kx + 3 = 0 \) has equal roots are
(a) \( k = \pm 6 \)
(b) \( k = \pm 2\sqrt{3} \)
(c) \( k = \pm 24 \)
(d) \( k = \pm 2\sqrt{6} \)
Show Answer & Explanation

Answer: (d) \( k = \pm 2\sqrt{6} \)

Explanation:
1. \( D = k^2 - 24 = 0 \), so \( k = \pm 2\sqrt{6} \).

Teacher's Note:
1. Equal roots: D = 0.
2. Both signs are valid.

Question 23: The value of \( k \) for which \( (k - 4)x^2 + 2(k - 4)x + 4 = 0 \) has equal roots is
(a) \( k = 8 \)
(b) \( k = 4 \)
(c) \( k = 4 \) or \( k = 8 \)
(d) \( k = 2 \)
Show Answer & Explanation

Answer: (a) \( k = 8 \)

Explanation:
1. \( D = 4(k - 4)^2 - 16(k - 4) = 4(k - 4)(k - 8) = 0 \).
2. \( k = 4 \) makes the coefficient of \( x^2 \) zero (no longer quadratic), so \( k = 8 \).

Teacher's Note:
1. Always reject the value that makes the leading coefficient zero.
2. Check k = 8: 4x² + 8x + 4 = 4(x + 1)².

Question 24: The value of \( k \) for which \( kx(x - 2) + 6 = 0 \) has equal roots is
(a) \( k = 0 \)
(b) \( k = -6 \)
(c) \( k = 6 \)
(d) \( k = 3 \)
Show Answer & Explanation

Answer: (c) \( k = 6 \)

Explanation:
1. \( kx^2 - 2kx + 6 = 0 \): \( D = 4k^2 - 24k = 4k(k - 6) = 0 \).
2. \( k = 0 \) is rejected because the equation would not be quadratic; so \( k = 6 \).

Teacher's Note:
1. Expand the bracket before computing D.
2. Reject k = 0.

Question 25: The values of \( p \) for which \( 4x^2 + 8x - p = 0 \) has real roots are
(a) \( p \le -4 \)
(b) \( p \ge 4 \)
(c) \( p \le 4 \)
(d) \( p \ge -4 \)
Show Answer & Explanation

Answer: (d) \( p \ge -4 \)

Explanation:
1. \( D = 64 + 16p \ge 0 \), so \( p \ge -4 \).

Teacher's Note:
1. D ≥ 0 gives 64 + 16p ≥ 0.
2. Divide by 16.

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

About Chapter 04 Quadratic Equations MCQs for Class 10 Mathematics

Review structured objective questions for Class 10 Mathematics Chapter 04 Quadratic Equations. Built according to official CBSE guidelines, these MCQ sets support daily revision and core concept reinforcement.

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