Quadratic Equations MCQ Quiz

30 questions

1. Which of the following equations is a quadratic equation?

   1. (x + 1)³ = x³ - 4
   2. x² + 2/x = 3
   3. (x - 2)(x + 2) = x² + 5x
   4. x² + 3√x + 7 = 0

2. What is the discriminant of the quadratic equation 2√2x² - 5x + √2 = 0?

   1. 9
   2. 17
   3. 33
   4. -7

3. For what value of k does the quadratic equation x² - 2(k + 1)x + k² = 0 have equal real roots?

   1. k = 1/2
   2. k = -1/2
   3. k = -2
   4. k = 2

4. If one root of the quadratic equation 3x² - 10x + k = 0 is the reciprocal of the other, what is the value of k?

   1. 1/3
   2. -3
   3. 3
   4. 10/3

5. What are the roots of the quadratic equation p²x² - (p² - q²)x - q² = 0?

   1. 1 and -q²/p²
   2. -1 and q²/p²
   3. 1 and q²/p²
   4. -1 and -q²/p²

6. If the sum of the roots of kx² + 4x + 3k = 0 is equal to their product, what is the value of k?

   1. -3/4
   2. 4/3
   3. -4/3
   4. 3/4

7. Find the range of values of p for which the quadratic equation 2x² + px + 8 = 0 has no real roots.

   1. p > 8
   2. p < -8
   3. -8 < p < 8
   4. p = ± 8

8. The hypotenuse of a right-angled triangle is 1 cm more than twice the shortest side. If the third side is 7 cm more than the shortest side, what is the length of the shortest side?

   1. 6 cm
   2. 8 cm
   3. 10 cm
   4. 12 cm

9. If α and β are the roots of the equation x² - 6x + 4 = 0, what is the value of α² + β²?

   1. 28
   2. 32
   3. 44
   4. 20

10. Two natural numbers differ by 3 and their product is 504. Which quadratic equation represents this situation if x represents the smaller number?

    1. x² - 3x - 504 = 0
    2. x² + 3x - 504 = 0
    3. x² + 3x + 504 = 0
    4. x² - 3x + 504 = 0

11. A motorboat whose speed in still water is 15 km/h goes 30 km downstream and returns to the starting point in 4.5 hours. What is the speed of the stream?

    1. 3 km/h
    2. 4 km/h
    3. 5 km/h
    4. 6 km/h

12. If the quadratic equation (a² + b²)x² - 2(ac + bd)x + (c² + d²) = 0 has equal roots, which relation must hold true?

    1. ab = cd
    2. ad = bc
    3. ac = bd
    4. a + d = b + c

13. What is the nature of the roots of the quadratic equation 3x² - 4√3x + 4 = 0?

    1. Real and distinct
    2. Real and equal
    3. No real roots
    4. Imaginary and unequal

14. What is the valid real solution for x in the equation √(2x + 9) + x = 13?

    1. x = 8 and x = 20
    2. x = 8 only
    3. x = 20 only
    4. x = -8 and x = 20

15. If the roots of ax² + bx + c = 0 are equal in magnitude but opposite in sign, which condition must be met?

    1. a = 0
    2. b = 0
    3. c = 0
    4. b² = 4ac

16. If x = 3 is a root of the quadratic equation x² - 2kx - 6 = 0, what is the value of k?

    1. 1/2
    2. -1/2
    3. 1/3
    4. 2

17. Two water taps together can fill a tank in 1 (7/8) hours. The tap with larger diameter takes 2 hours less than the smaller one to fill the tank separately. How long does the smaller tap take to fill the tank alone?

    1. 3 hours
    2. 5 hours
    3. 8 hours
    4. 10 hours

18. The sum of the squares of two consecutive positive even integers is 340. What are the integers?

    1. 10 and 12
    2. 12 and 14
    3. 14 and 16
    4. 16 and 18

19. If α and β are the roots of 2x² + 5x + 3 = 0, what is the value of (1/α + 1/β)?

    1. -5/3
    2. 5/3
    3. -3/5
    4. 3/5

20. For what values of k will the equation x² - kx + 9 = 0 have real and distinct roots?

    1. -6 < k < 6
    2. k > 6 or k < -6
    3. k = ±6
    4. k ≥ 6

21. The difference of two positive numbers is 4 and the difference of their reciprocals is 4/21. What are the numbers?

    1. 3 and 7
    2. 4 and 8
    3. 2 and 6
    4. 5 and 9

22. If the roots of x² - bx + c = 0 differ by 1, which of the following expressions is correct?

    1. b² = 1 + 4c
    2. b² = 1 - 4c
    3. c² = 1 + 4b
    4. b² = 4c - 1

23. A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less. What was the original speed of the train?

    1. 35 km/h
    2. 40 km/h
    3. 45 km/h
    4. 50 km/h

24. What are the solutions to the equation x + 1/x = 2.5?

    1. 2 and 1/2
    2. -2 and -1/2
    3. 3 and 1/3
    4. 1 and 1/2

25. A rectangular garden 10 m long and 8 m wide is surrounded by a path of uniform width x meters. If the area of the path alone is 64 m², which quadratic equation determines x?

    1. x² + 9x - 16 = 0
    2. x² + 18x - 32 = 0
    3. 2x² + 9x - 16 = 0
    4. x² + 9x - 32 = 0

26. If a and b (both non-zero) are the roots of x² + ax + b = 0, what are the values of a and b?

    1. a = 1, b = -2
    2. a = -1, b = 2
    3. a = 2, b = -1
    4. a = 1, b = 2

27. What is the discriminant of x² - 4x + (k + 2) = 0, and for what value of k does the equation have equal roots?

    1. D = 8 - 4k, k = 2
    2. D = 16 - 4k, k = 4
    3. D = 8 + 4k, k = -2
    4. D = 24 - 4k, k = 6

28. If the sum of a positive number and its reciprocal is 10/3, what is the number?

    1. 3 or 1/3
    2. 2 or 1/2
    3. 4 or 1/4
    4. 5 or 1/5

29. Which of the following quadratic equations has no real roots?

    1. x² - 4x + 5 = 0
    2. x² - 4x - 5 = 0
    3. x² + 4x + 4 = 0
    4. 2x² - 6x + 3 = 0

30. A cottage industry produces a certain number of articles per day. The cost of production per article was ₹3 more than twice the number of articles produced. If total cost was ₹90, how many articles were produced?

    1. 6
    2. 8
    3. 10
    4. 12

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