Gyanyaan हिंदी

Chapter 4: Quadratic Equations

Class 10 Mathematics · MCQs with answers

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  1. Which of these is a quadratic equation?

    1. (A) x(x + 1) + 8 = (x + 2)(x − 2)
    2. (B) (x + 1)² = 2(x − 3)
    3. (C) x² + 3x = x² − 5
    4. (D) 2x + 3 = 7
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    Correct answer: (B) (x + 1)² = 2(x − 3)

    Explanation: (x + 1)² = 2(x − 3) simplifies to x² + 7 = 0, which still has x², so it is quadratic. In B and C the x² terms cancel and they become linear; D is linear already.

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  2. ax² + bx + c = 0 is a quadratic equation only when:

    1. (A) b ≠ 0
    2. (B) c ≠ 0
    3. (C) a = 0
    4. (D) a ≠ 0
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    Correct answer: (D) a ≠ 0

    Explanation: If a = 0, the x² term disappears and the equation becomes linear. So a ≠ 0 is required; b and c can be zero.

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  3. What are the roots of 2x² − 5x + 3 = 0?

    1. (A) −1 and −3/2
    2. (B) 1 and −3/2
    3. (C) 1 and 3/2
    4. (D) 3 and 1/2
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    Correct answer: (C) 1 and 3/2

    Explanation: 2x² − 5x + 3 = 2x² − 2x − 3x + 3 = (x − 1)(2x − 3). So x = 1 or x = 3/2.

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  4. What are the roots of x² − 3x − 10 = 0?

    1. (A) 5 and −2
    2. (B) −5 and 2
    3. (C) 10 and −1
    4. (D) 5 and 2
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    Correct answer: (A) 5 and −2

    Explanation: x² − 3x − 10 = (x − 5)(x + 2). So x = 5 or x = −2.

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  5. What are the roots of 2x² + x − 6 = 0?

    1. (A) −3/2 and 2
    2. (B) 2 and −3
    3. (C) 3 and −2
    4. (D) 3/2 and −2
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    Correct answer: (D) 3/2 and −2

    Explanation: 2x² + x − 6 = 2x² + 4x − 3x − 6 = (2x − 3)(x + 2). So x = 3/2 or x = −2.

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  6. What is the discriminant of 2x² − 4x + 3 = 0?

    1. (A) 40
    2. (B) 8
    3. (C) −8
    4. (D) −40
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    Correct answer: (C) −8

    Explanation: D = b² − 4ac = (−4)² − 4 × 2 × 3 = 16 − 24 = −8. Since D < 0, it has no real roots.

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  7. What is the nature of the roots of x² − 4x + 4 = 0?

    1. (A) Two equal real roots
    2. (B) Two distinct real roots
    3. (C) No real roots
    4. (D) Infinitely many roots
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    Correct answer: (A) Two equal real roots

    Explanation: D = 16 − 16 = 0. When D = 0 there are two equal real roots; here (x − 2)² = 0, so x = 2, 2.

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  8. What is the nature of the roots of 2x² − 6x + 3 = 0?

    1. (A) Two equal real roots
    2. (B) No real roots
    3. (C) Two distinct real roots
    4. (D) Only one root
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    Correct answer: (C) Two distinct real roots

    Explanation: D = 36 − 24 = 12 > 0. When D is positive there are two distinct real roots.

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  9. What is the discriminant of the quadratic equation ax² + bx + c = 0?

    1. (A) b² + 4ac
    2. (B) b² − 4ac
    3. (C) √(b² − 4ac)
    4. (D) −b/2a
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    Correct answer: (B) b² − 4ac

    Explanation: The discriminant D = b² − 4ac tells the nature of the roots: two distinct if D > 0, two equal if D = 0, and no real roots if D < 0.

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  10. For which values of k does 2x² + kx + 3 = 0 have equal roots?

    1. (A) ±√6
    2. (B) ±6
    3. (C) ±24
    4. (D) ±2√6
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    Correct answer: (D) ±2√6

    Explanation: For equal roots, D = 0: k² − 4 × 2 × 3 = 0, so k² = 24 and k = ±√24 = ±2√6.

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  11. For which non-zero value of k does kx(x − 2) + 6 = 0 have equal roots?

    1. (A) −6
    2. (B) 0
    3. (C) 3
    4. (D) 6
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    Correct answer: (D) 6

    Explanation: kx² − 2kx + 6 = 0. For equal roots, D = 4k² − 24k = 4k(k − 6) = 0. Since k ≠ 0, k = 6.

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  12. What is the quadratic formula for the roots of ax² + bx + c = 0?

    1. (A) x = [−b ± √(b² − 4ac)] / 2a
    2. (B) x = [b ± √(b² − 4ac)] / 2a
    3. (C) x = [−b ± √(b² + 4ac)] / 2a
    4. (D) x = [−b ± (b² − 4ac)] / 2a
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    Correct answer: (A) x = [−b ± √(b² − 4ac)] / 2a

    Explanation: The quadratic formula is x = [−b ± √(b² − 4ac)] / 2a. It gives real roots only when b² − 4ac ≥ 0.

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  13. How many real roots does x² + 4x + 5 = 0 have?

    1. (A) 1
    2. (B) 2
    3. (C) Infinitely many
    4. (D) 0
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    Correct answer: (D) 0

    Explanation: D = 16 − 20 = −4 < 0, so there are no real roots.

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  14. What are the roots of x² − 2x − 1 = 0?

    1. (A) −1 ± √2
    2. (B) 1 ± √2
    3. (C) 2 ± √2
    4. (D) 1 ± √3
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    Correct answer: (B) 1 ± √2

    Explanation: x = [2 ± √(4 + 4)] / 2 = [2 ± 2√2] / 2 = 1 ± √2.

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  15. The product of two consecutive positive integers is 306. What are the integers?

    1. (A) 17 and 18
    2. (B) 16 and 17
    3. (C) 18 and 19
    4. (D) 15 and 16
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    Correct answer: (A) 17 and 18

    Explanation: x(x + 1) = 306, so x² + x − 306 = 0 = (x − 17)(x + 18). The positive solution is x = 17, so the integers are 17 and 18.

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  16. The sum of a number and its reciprocal is 10/3. What is the number?

    1. (A) 2 or 1/2
    2. (B) 4 or 1/4
    3. (C) 3 or 1/3
    4. (D) 5 or 1/5
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    Correct answer: (C) 3 or 1/3

    Explanation: x + 1/x = 10/3 gives 3x² − 10x + 3 = 0 = (3x − 1)(x − 3), so x = 3 or x = 1/3.

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  17. A rectangular plot has an area of 528 m². Its length is 1 m more than twice its breadth. What are its length and breadth?

    1. (A) Breadth 12 m, length 44 m
    2. (B) Breadth 16 m, length 33 m
    3. (C) Breadth 22 m, length 24 m
    4. (D) Breadth 11 m, length 48 m
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    Correct answer: (B) Breadth 16 m, length 33 m

    Explanation: x(2x + 1) = 528, so 2x² + x − 528 = 0 = (x − 16)(2x + 33). So the breadth is 16 m and the length is 2 × 16 + 1 = 33 m.

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  18. The altitude of a right triangle is 7 cm less than its base, and its hypotenuse is 13 cm. What are the base and the altitude?

    1. (A) Base 13 cm, altitude 6 cm
    2. (B) Base 10 cm, altitude 3 cm
    3. (C) Base 8 cm, altitude 1 cm
    4. (D) Base 12 cm, altitude 5 cm
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    Correct answer: (D) Base 12 cm, altitude 5 cm

    Explanation: x² + (x − 7)² = 13², so x² − 7x − 60 = 0 = (x − 12)(x + 5). So the base is 12 cm and the altitude 5 cm (check: 144 + 25 = 169).

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  19. If −5 is a root of 2x² + px − 15 = 0, what is the value of p?

    1. (A) −7
    2. (B) 3
    3. (C) 5
    4. (D) 7
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    Correct answer: (D) 7

    Explanation: Putting x = −5 gives 2 × 25 − 5p − 15 = 0, so 35 − 5p = 0 and p = 7.

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  20. Which quadratic equation has the roots 2 and −3?

    1. (A) x² + x − 6 = 0
    2. (B) x² − x − 6 = 0
    3. (C) x² + x + 6 = 0
    4. (D) x² − 5x + 6 = 0
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    Correct answer: (A) x² + x − 6 = 0

    Explanation: (x − 2)(x + 3) = x² + x − 6. So the equation is x² + x − 6 = 0.

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  21. Which of these equations has real and equal roots?

    1. (A) x² + 9 = 0
    2. (B) x² − 9 = 0
    3. (C) x² − 6x + 8 = 0
    4. (D) x² − 6x + 9 = 0
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    Correct answer: (D) x² − 6x + 9 = 0

    Explanation: For x² − 6x + 9 = (x − 3)², D = 36 − 36 = 0, so there are two equal roots (3, 3). B has no real roots; C and D have two distinct roots.

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  22. At most how many roots can a quadratic equation have?

    1. (A) 4
    2. (B) 3
    3. (C) 2
    4. (D) 1
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    Correct answer: (C) 2

    Explanation: A quadratic equation has degree 2, so it can have at most 2 roots.

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  23. Using completing the square, how can x² + 6x + 5 = 0 be written?

    1. (A) (x + 3)² = 4
    2. (B) (x + 3)² = 14
    3. (C) (x + 6)² = 31
    4. (D) (x − 3)² = 4
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    Correct answer: (A) (x + 3)² = 4

    Explanation: x² + 6x = −5. Adding (6/2)² = 9 to both sides gives x² + 6x + 9 = 4, i.e. (x + 3)² = 4. So x + 3 = ±2, giving x = −1 or −5.

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  24. What must be added to x² − 8x to make it a perfect square?

    1. (A) 64
    2. (B) 16
    3. (C) 8
    4. (D) 4
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    Correct answer: (B) 16

    Explanation: Take half the coefficient of x and add its square: (8/2)² = 16. Then x² − 8x + 16 = (x − 4)².

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  25. Assertion (A): The equation x² + x + 1 = 0 has no real roots. Reason (R): The discriminant of this equation is negative.

    1. (A) Both A and R are true, and R is the correct explanation of A.
    2. (B) Both A and R are true, but R is not the correct explanation of A.
    3. (C) A is true, but R is false.
    4. (D) A is false, but R is true.
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    Correct answer: (A) Both A and R are true, and R is the correct explanation of A.

    Explanation: D = 1 − 4 = −3 < 0. A negative discriminant means there are no real roots, so R correctly explains A.

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