Quadratic Equation (Previous Year Questions)
1. The roots of the quadratic equation x2 – 0.04 = 0 are
a) ± 0.2 b) ± 0.02 c) 0.4 d) 2
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± 0.2
2. If x = − 1/2 is a solution of the quadratic equation 3x2 + 2kx – 3 = 0, find the value of k.
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3. If x = 2/3 and x = –3 are roots of the quadratic equations ax2 + 7x + b = 0, find the values of a and b.
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4. Solve the following quadratic equations by the factorisation method.
3x2 – 2√6x + 2 = 0
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5. Solve the equation 4/x − 3 = 5/ (2x + 3); x ≠ 0, -3/2, for x
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6. Solve for x: (16/x) – 1 = 15/ (x + 1); x ≠ 0, -1
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7. Solve for x: (x – 4)/ (x – 5) + (x – 6)/ (x – 7) = 10/3; x ≠ 5, 7
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8. The sum of the squares of two consecutive odd numbers is 394. Find the numbers.
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9. In the centre of a rectangular lawn of dimensions 50 m × 40 m, a rectangular pond has to be constructed so that the area of the grass surrounding the pond would be 1184 m2. Find the length and breadth of the pond.
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10. The difference of two natural numbers is 5 and the difference of their reciprocals is 1/10. Find the numbers.
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11. If Zeba was younger by 5 years than what she really is, then the square of her age (in years) would have been 11 more than five times her actual age. What is her age now?
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12. A train travelling at a uniform speed for 360 km would have taken 48 minutes less to travel the same distance if its speed were 5 km/hour more. Find the original speed of the train.
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13. A plane left 30 minutes late than its scheduled time and in order to reach the destination 1500 km away in time, it had to increase its speed by 100 km/h from the usual speed. Find its usual speed.
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14. Find the roots of the equation x2 + 7x + 10 = 0.
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15. Solve for x:
a) 4√3x2 + 5x – 2√3 = 0
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b) x2 – (√2 + 1) x + √2 = 0
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c) √ (2x + 9) + x = 13
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d) √ (6x + 7) – (2x – 7) = 0
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e) √3x2 – 2√2x – 2√3 = 0
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16. The difference of two natural numbers is 3 and the difference of their reciprocals is 3/28. Find the numbers.
View AnswerAns. 4, 7
17. The difference of two natural numbers is 5 and the difference of their reciprocals is 5/14. Find the numbers.
View AnswerAns. 7, 2
18. Solve the equation for x: (3x – 4)/7 + 7/ (3x – 4) = 5/2, x ≠ 4/3
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19. Solve the equation for x: 1/ (x + 1) + 2 (x + 2) = 5/ (x + 4), x ≠ -1, -2, -4.
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20. Some students planned a picnic. The total budget for food was `2,000. But 5 students failed to attend the picnic and thus the cost of food for each member increased by `20. How many students attended the picnic and how much did each student pay for the food?
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21. A two-digit number is such that the product of its digits is 14. When 45 is added to the number, the digits interchange their places. Find the number.
View AnswerAns. 27
22. Two water taps together can fill a tank in 6 hours. The tap of larger diameter takes 9 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.
View AnswerAns. 9 hrs
23. A pole has to be erected at a point on the boundary of a circular park of diameter 17 m in such a way that the differences of its distances from two diametrically opposite fixed gates A and B on the boundary is 7 metres. Find the distances from the two gates where the pole is to be erected.
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24. A motorboat whose speed in still water is 18 km/h, takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream.
View AnswerAns. 6 km/h.
25. At present Asha’s age (in years) is 2 more than the square of her daughter Nisha’s age. When Nisha grows to her mother’s present age, Asha’s age would be one year less than 10 times the present age of Nisha. Find the present ages of both Asha and Nisha.
View AnswerAns. Nisha’s age = 5 years, Asha’s age = 27 years
26. A train travels at a certain average speed for a distance of 63 km and then travels at a distance of 72 km at an average speed of 6 km/hr more than its original speed. If it takes 3 hours to complete total journey, what is the original average speed?
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27. Find two consecutive positive integers sum of whose squares is 365.
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28. A rectangular park is to be designed whose breadth is 3 m less than its length. Its area is to be 4 square metres more than the area of a park that has already been made in the shape of an isosceles triangle with its base as the breadth of the rectangular park and of altitude 12 m. Find the length and breadth of the park.
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29. In a flight of 600 km, an aircraft was slowed down due to bad weather. The average speed of the trip was reduced by 200 km/hr and the time of flight increased by 30 minutes. Find the duration of flight.
View AnswerAns. 1 hr
30. Find the roots of the equation ax2 + a = a2x + x
View AnswerAns. a, 1/a
31. Solve the following quadratic equation for x: 4x2 – 4a2 x + (a4 – b4) = 0.
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32. Solve the following quadratic equation for x: 9x2 – 6b2x – (a4 – b4) = 0
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33. Solve the following quadratic equation for x: 4x 2 + 4bx – (a2 – b2) = 0.
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34. Solve the following quadratic equation for x: x2 – 2ax – (4b2 – a2) = 0
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35. Two water taps together can fill a tank in 9 hours 36 minutes. The tap of large diameter takes 8 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.
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36. A rectangular field is 20 m long and 14 m wide. There is a path of equal width all around it, having an area of 111 sq m. Find the width of the path.
View AnswerAns. 1.5 m
37. The difference of squares of two numbers is 88. If the larger number is 5 less than twice the smaller number, then find the two numbers.
View AnswerAns. 9 and 13
38. Write the set of values of k for which the quadratic equation 2x2 + kx + 8 has real roots.
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39. If –5 is a root of the quadratic equation 2x2 + px – 15 = 0 and the quadratic equation p (x2 + x) + k = 0 has equal roots, then find the value of k.
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40. Find the value of p, for which one root of the quadratic equation px2 – 14x + 8 = 0 is 6 times the other.
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41. Find the value(s) of k for which the equation x2 + 5kx + 16 = 0 has real and equal roots.
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42. If the equation (1 + m2) x2 + 2mcx + c2 – a2 = 0 has equal roots, show that c2 = a2 (1 + m2).
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43. Find the value of p, so that the quadratic equation px(x – 3) + 9 = 0 has equal roots.
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44. For what values of k, the roots of the equation x2 + 4x + k = 0 are real?
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45. Find the value of k for which the roots of the equation 3x2 – 10x + k = 0 are reciprocal of each other.
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46. Find the discriminant of the quadratic equation 2x2 – 4x + 3 = 0, hence find the nature of its roots.
View AnswerAns. –8, no real roots
47. If x = 3 is one root of the quadratic equation x2 – 2kx– 6 = 0, then find the value of k.
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48. For what values of k, the equation 9x2 + 6kx + 4 = 0 has equal roots?
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49. For what value(s) of ‘a’ quadratic equation 3ax2 – 6x + 1 = 0 has no real roots?
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50. If 2 is a root of the quadratic equation 3x2 + px – 8 = 0 and the quadratic equation 4x2 – 2px + k = 0 has equal roots, find the value of k.
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51. The roots α and β of the quadratic equation x2 – 5x + 3(k – 1) = 0 are such that α – β = 1. Find the value k.
View AnswerAns. k = 3