Class 10 Mathematics: Quadratic Equations important questions

Quadratic Equations is a chapter of Class 10 Mathematics for the CBSE. These 12 questions are marked Very Important, Important or Practice, and each answer has been checked by our team. Use them to revise, not as a prediction of what will come in the exam.

1. Practice · 1 marks

   1. Let p be a prime number. The quadratic equation whose roots are the factors of p is: (a) x² − px + p = 0 (b) x² − (p + 1)x + p = 0 (c) x² + (p + 1)x + p = 0 (d) x² − px + p + 1 = 0

   (b) x² − (p + 1)x + p = 0 The factors of a prime p are 1 and p: sum p + 1, product p.

2. Very Important · 5 marks

   2. Two pipes fill a swimming pool. If the larger pipe runs for 4 hours and the smaller for 9 hours, only half the pool is filled. The smaller pipe takes 10 hours more than the larger to fill the pool alone. How long does each take alone?

   Let the larger pipe take x hours; the smaller takes x + 10. 4/x + 9/(x + 10) = 1/2, which simplifies to x² − 16x − 80 = 0, i.e. (x − 20)(x + 4) = 0. x = 20. Larger pipe: 20 hours; smaller pipe: 30 hours.

3. Very Important · 5 marks

   3. On a 600 km flight, an aircraft was slowed down by bad weather: its average speed fell by 200 km/h and the flight took 30 minutes longer. Find the scheduled duration of the flight.

   Let the usual speed be x km/h. 600/(x − 200) − 600/x = 1/2. 240000 = x² − 200x, so x² − 200x − 240000 = 0 and (x − 600)(x + 400) = 0. x = 600 km/h, so the scheduled duration is 600/600 = 1 hour.

4. Practice · 1 marks

   4. The nature of the roots of 9x² − 6x − 2 = 0 is: (a) no real roots (b) 2 equal real roots (c) 2 distinct real roots (d) more than 2 real roots

   (c) 2 distinct real roots D = 36 + 72 = 108 > 0.

5. Very Important · 5 marks

   5. A motor boat 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.

   Let the stream's speed be x km/h. 24/(18 − x) − 24/(18 + x) = 1. 48x = 324 − x², so x² + 48x − 324 = 0 and (x + 54)(x − 6) = 0. x = 6 (speed cannot be negative). The stream flows at 6 km/h.

6. Very Important · 5 marks

   6. Two water taps together can fill a tank in 9⅜ hours. The larger tap takes 10 hours less than the smaller one to fill the tank alone. Find the time each tap takes alone.

   Let the smaller tap take x hours; the larger takes x − 10. 1/x + 1/(x − 10) = 8/75, so 75(2x − 10) = 8x(x − 10), i.e. 4x² − 115x + 375 = 0. x = (115 ± 85)/8 = 25 or 3.75. x = 3.75 makes x − 10 negative, so x = 25. Smaller tap: 25 hours; larger tap: 15 hours.

7. Practice · 1 marks

   7. The roots of the quadratic equation x² + x − 1 = 0 are: (A) irrational and distinct (B) not real (C) rational and distinct (D) real and equal

   (A) Irrational and distinct D = 1 + 4 = 5 > 0 and not a perfect square.

8. Important · 3 marks

   8. The sum of two numbers is 18 and the sum of their reciprocals is 9/40. Find the numbers.

   Let the numbers be x and 18 − x. 1/x + 1/(18 − x) = 9/40. 18 × 40 = 9x(18 − x), so x² − 18x + 80 = 0 and (x − 10)(x − 8) = 0. The numbers are 8 and 10.

9. Practice · 1 marks

   9. Which of these is not a quadratic equation? (A) (x + 2)² = 2(x + 3) (B) x² + 3x = (−1)(1 − 3x)² (C) x³ − x² + 2x + 1 = (x + 1)³ (D) (x + 2)(x + 1) = x² + 2x + 3

   (D) (x + 2)(x + 1) = x² + 3x + 2, so the equation becomes x² + 3x + 2 = x² + 2x + 3, i.e. x − 1 = 0, which is linear.

10. Very Important · 5 marks

    10. A train travels 63 km at a certain average speed, then 72 km at a speed 6 km/h more. The whole journey takes 3 hours. Find the original average speed.

    Let the speed be x km/h. 63/x + 72/(x + 6) = 3. 63(x + 6) + 72x = 3x(x + 6), so 3x² − 117x − 378 = 0, i.e. x² − 39x − 126 = 0 and (x − 42)(x + 3) = 0. x = 42 (speed cannot be negative). The original speed is 42 km/h.

11. Practice · 1 marks

    11. If (1 − p) is a root of x² + px + 1 − p = 0, then its roots are: (A) 0, 1 (B) −1, 1 (C) 0, −1 (D) −1, 2

    (C) 0, −1 (1 − p)² + p(1 − p) + 1 − p = 0 gives p = 1. The equation becomes x² + x = 0, i.e. x(x + 1) = 0, so the roots are 0 and −1.

12. Very Important · 5 marks

    12. A rectangular park is 50 m × 40 m. A rectangular swimming pool is built in the middle with a grass strip of uniform width all round. If the grass strip has area 1184 m², find the dimensions of the pool.

    Let the strip be x m wide. Pool = (50 − 2x) by (40 − 2x). 50 × 40 − (50 − 2x)(40 − 2x) = 1184, which gives x² − 45x + 296 = 0, i.e. (x − 8)(x − 37) = 0. x = 37 is impossible (the pool's sides would be negative), so x = 8. The pool is 34 m long and 24 m wide.