CBSE Class 10 Mathematics Quadratic Equations MCQs Set 15

Download CBSE MCQs for Class 10 Mathematics: Chapter 04 Quadratic Equations

Explore reliable objective questions for Chapter 04 Quadratic Equations tailored for Class 10 learners. Utilizing these Mathematics multiple-choice formats ensures thorough preparation and strengthens problem-solving speed for upcoming school assessments.

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: If the roots of the equation \(ax^2 + bx + c = 0\), \(a \ne 0\) are real and equal, then which of the following relations is true?
(a) \(a = b^2/c\)
(b) \(b^2 = ac\)
(c) \(ac = b^2/4\)
(d) \(c = b^2/a\)
Show Answer & Explanation

Answer: (c) \(ac = b^2/4\)

Explanation:
1. For equal roots, \(D = b^2 - 4ac = 0\), so \(b^2 = 4ac\), i.e. \(ac = b^2/4\).

Question: If the quadratic equation \(ax^2 + bx + c = 0\) has two real and equal roots, then \(c\) is equal to
(a) \(-\frac{b}{2a}\)
(b) \(\frac{b}{2a}\)
(c) \(-\frac{b^2}{4a}\)
(d) \(\frac{b^2}{4a}\)
Show Answer & Explanation

Answer: (d) \(\frac{b^2}{4a}\)

Explanation:
1. \(D = b^2 - 4ac = 0\), so \(c = \frac{b^2}{4a}\).

Question: The sum and product of the zeroes of a quadratic polynomial are 3 and \(-10\) respectively. The quadratic polynomial is
(a) \(x^2 - 3x + 10\)
(b) \(x^2 + 3x - 10\)
(c) \(x^2 - 3x - 10\)
(d) \(x^2 + 3x + 10\)
Show Answer & Explanation

Answer: (c) \(x^2 - 3x - 10\)

Explanation:
1. \(p(x) = x^2 - (\text{sum})x + \text{product} = x^2 - 3x - 10\).

Question: If the sum of the zeroes of the quadratic polynomial \(kx^2 + 2x + 3k\) is equal to their product, then \(k\) equals
(a) \(\frac{1}{3}\)
(b) \(-\frac{1}{3}\)
(c) \(\frac{2}{3}\)
(d) \(-\frac{2}{3}\)
Show Answer & Explanation

Answer: (d) \(-\frac{2}{3}\)

Explanation:
1. Sum \(= -\frac{2}{k}\) and product \(= \frac{3k}{k} = 3\).
2. \(-\frac{2}{k} = 3\), so \(k = -\frac{2}{3}\).

Question: If \(\alpha\) and \(\beta\) are the zeroes of the polynomial \(x^2 + 2x + 1\), then \(\frac{1}{\alpha} + \frac{1}{\beta}\) is equal to
(a) \(-2\)
(b) \(2\)
(c) \(0\)
(d) \(3\)
Show Answer & Explanation

Answer: (a) \(-2\)

Explanation:
1. Sum \(= -2\), product \(= 1\).
2. \(\frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha\beta} = -2\).

Question: The linear factors of the quadratic equation \(x^2 + kx + 1 = 0\) are
(a) \(k \ge 2\)
(b) \(k \le 2\)
(c) \(k \ge -2\)
(d) \(k \ge 2\) or \(k \le -2\)
Show Answer & Explanation

Answer: (d) \(k \ge 2\) or \(k \le -2\)

Explanation:
1. For real linear factors \(b^2 - 4ac \ge 0\), so \(k^2 - 4 \ge 0\).
2. Hence \(k \ge 2\) or \(k \le -2\).

Teacher's Note:
1. Option (d) is printed in the source as \(2 \le k \le -2\). That is impossible, so it is corrected here to 'or'. The answer letter is unchanged.

Question: If one root of the quadratic equation \(ax^2 + bx + c = 0\) is the reciprocal of the other, then
(a) \(b = c\)
(b) \(a = b\)
(c) \(ac = 1\)
(d) \(a = c\)
Show Answer & Explanation

Answer: (d) \(a = c\)

Explanation:
1. If one root is \(\alpha\), the other is \(\frac{1}{\alpha}\).
2. Product of roots \(= \frac{c}{a} = 1\), so \(a = c\).

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
Show Answer & Explanation

Answer: (c) no real roots

Explanation:
1. \(D = (-\sqrt{5})^2 - 4(2)(1) = 5 - 8 = -3 < 0\).

Question: The real roots of the equation \(x^{2/3} + x^{1/3} - 2 = 0\) are
(a) \(1, 8\)
(b) \(-1, -8\)
(c) \(-1, 8\)
(d) \(1, -8\)
Show Answer & Explanation

Answer: (d) \(1, -8\)

Explanation:
1. Put \(y = x^{1/3}\): \(y^2 + y - 2 = 0\), so \((y - 1)(y + 2) = 0\).
2. \(y = 1\) gives \(x = 1\). \(y = -2\) gives \(x = -8\).

Question: \(\left(x^2 + 1\right)^2 - x^2 = 0\) has
(a) four real roots
(b) two real roots
(c) no real roots
(d) one real root
Show Answer & Explanation

Answer: (c) no real roots

Explanation:
1. Expanding: \(x^4 + x^2 + 1 = 0\). Put \(y = x^2\): \(y^2 + y + 1 = 0\).
2. \(D = 1 - 4 = -3 < 0\), so there are no real roots.

Question: The equation \(2x^2 + 2(p + 1)x + p = 0\), where \(p\) is real, always has roots that are
(a) Equal
(b) Equal in magnitude but opposite in sign
(c) Irrational
(d) Real
Show Answer & Explanation

Answer: (d) Real

Explanation:
1. \(D = 4(p + 1)^2 - 8p = 4(p^2 + 1) > 0\).
2. So the roots are real (and distinct).

Question: The quadratic equation \(5x^2 - 3x + 1 = 0\) has
(a) two distinct real roots
(b) two equal real roots
(c) no real roots
(d) more than 2 real roots
Show Answer & Explanation

Answer: (c) no real roots

Explanation:
1. \(D = (-3)^2 - 4(5)(1) = 9 - 20 = -11 < 0\).

Question: The quadratic equation \(x^2 - 4x + 3\sqrt{2} = 0\) has
(a) two distinct real roots
(b) two equal real roots
(c) no real roots
(d) more than 2 real roots
Show Answer & Explanation

Answer: (c) no real roots

Explanation:
1. \(D = 16 - 12\sqrt{2} \approx 16 - 16.97 = -0.97 < 0\).

Teacher's Note:
1. The source shows the value as \(-0.92\) (from \(16.92\)). The correct value is \(-0.97\). The conclusion is unchanged.

Question: The quadratic equation \(x^2 + 4x - 3\sqrt{2} = 0\) has
(a) two distinct real roots
(b) two equal real roots
(c) no real roots
(d) more than 2 real roots
Show Answer & Explanation

Answer: (a) two distinct real roots

Explanation:
1. \(D = 16 + 12\sqrt{2} > 0\), so there are two distinct real roots.

Question: The quadratic equation \(x^2 - 4x - 3\sqrt{2} = 0\) has
(a) two distinct real roots
(b) two equal real roots
(c) no real roots
(d) more than 2 real roots
Show Answer & Explanation

Answer: (a) two distinct real roots

Explanation:
1. \(D = 16 + 12\sqrt{2} > 0\), so there are two distinct real roots.

Question: The quadratic equation \(3x^2 + 4\sqrt{3}\,x + 4 = 0\) has
(a) two distinct real roots
(b) two equal real roots
(c) no real roots
(d) more than 2 real roots
Show Answer & Explanation

Answer: (b) two equal real roots

Explanation:
1. \(D = 48 - 48 = 0\), so the roots are real and equal.

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) \(3x^2 - 6x - 2 = 0\)
Show Answer & Explanation

Answer: (c) \(2x^2 - 7x + 6 = 0\)

Explanation:
1. Substitute \(x = 2\) in each equation.
2. In (c): \(2(4) - 7(2) + 6 = 8 - 14 + 6 = 0\). The other options do not give 0.

Question:
Assertion (A): The roots of the quadratic equation \(x^2 + 2x + 2 = 0\) are imaginary.
Reason (R): If discriminant \(D = b^2 - 4ac < 0\), then the roots of \(ax^2 + bx + c = 0\) are imaginary.
(a) Both A and R are true, and R is the correct explanation of A.
(b) Both A and R are true, but R is not the correct explanation of A.
(c) A is true, but R is false.
(d) A is false, but R is true.
Show Answer & Explanation

Answer: (a) Both A and R are true, and R is the correct explanation of A.

Explanation:
1. \(D = 4 - 8 = -4 < 0\), so the roots are imaginary.
2. The reason is true and explains the assertion.

Question:
Assertion (A): The equation \(x^2 + 3x + 1 = (x - 2)^2\) is a quadratic equation.
Reason (R): Any equation of the form \(ax^2 + bx + c = 0\), where \(a \ne 0\), is called a quadratic equation.
(a) Both A and R are true, and R is the correct explanation of A.
(b) Both A and R are true, but R is not the correct explanation of A.
(c) A is true, but R is false.
(d) A is false, but R is true.
Show Answer & Explanation

Answer: (d) A is false, but R is true.

Explanation:
1. \(x^2 + 3x + 1 = x^2 - 4x + 4\), which simplifies to \(7x - 3 = 0\).
2. This is not of the form \(ax^2 + bx + c = 0\), so the assertion is false. The reason is true.

Question:
Assertion (A): The equation \(8x^2 + 3kx + 2 = 0\) has equal roots, then the value of \(k\) is \(\pm\frac{8}{3}\).
Reason (R): The equation \(ax^2 + bx + c = 0\) has equal roots if \(D = b^2 - 4ac = 0\).
(a) Both A and R are true, and R is the correct explanation of A.
(b) Both A and R are true, but R is not the correct explanation of A.
(c) A is true, but R is false.
(d) A is false, but R is true.
Show Answer & Explanation

Answer: (a) Both A and R are true, and R is the correct explanation of A.

Explanation:
1. \(D = 9k^2 - 64 = 0\), so \(k = \pm\frac{8}{3}\).
2. The reason is the condition for equal roots and explains the assertion.

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Practice MCQs for Class 10 Mathematics Chapter 04 Quadratic Equations

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

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FAQs

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