LIWA

Quadratic Equations

CBSE Class 10 · Maths · Chapter 4

Past board exam questions

CBSE 2027 sample paper · 1 mark

If (1 – p) is a root of the quadratic equation x2 + px + 1 - p = 0, then its roots are:

  1. 0, 1
  2. – 1, 1
  3. 0, – 1
  4. – 1, 2
Show answer
0, – 1

CBSE 2027 sample paper · 1 mark

If one root of the quadratic equation ax2 + bx + c = 0 is the reciprocal of the other, then

  1. b = c
  2. a = b
  3. ac = 1
  4. a = c
Show answer
a = c

CBSE 2027 sample paper · 5 marks

In a rectangular park measuring 50 m × 40 m, the gram panchayat plans to construct a rectangular swimming pool in the middle, surrounded by a uniform-width grass strip throughout the park. Find the dimensions of the swimming pool, if the area of the grassy strip is 1184 m².

Show answer
Let x m be the uniform width of the grass strip. Length of the pool = (50 – 2x) m, breadth = (40 – 2x) m. Area of park – area of pool = area of grass strip: 50 × 40 - (50 - 2x)(40 - 2x) = 1184 2000 - (2000 - 180x + 4x2) = 1184 ⇒ 4x2 - 180x + 1184 = 0 ⇒ x2 - 45x + 296 = 0 ⇒ (x - 8)(x - 37) = 0 ⇒ x = 8 or 37. x = 37 is rejected (50 – 74 and 40 – 74 would be negative), so x = 8. Length of the pool = 50 – 16 = 34 m; breadth = 40 – 16 = 24 m. The swimming pool measures 34 m × 24 m.

CBSE 2027 sample paper · 5 marks

A train travels a distance of 360 km at a uniform speed. If the speed had been 5 km/h more, then it would have taken 1 hour less to cover the same distance. Find the speed of the train.

Show answer
Let the speed of the train be x km/h; the increased speed is (x + 5) km/h. (360)/(x) - (360)/(x + 5) = 1 360(x + 5) - 360x = x(x + 5) ⇒ x2 + 5x - 1800 = 0 x = (-5 ± √(52 + 4(1800)))/(2) = (-5 ± 85)/(2), so x = 40 or x = -45. Speed cannot be negative, so x = 40. ∴ Speed of the train = 40 km/h.

CBSE 2027 sample paper · 5 marks

John and Jivanti together have 45 marbles. Both of them lost 5 marbles each, and the product of the number of marbles they now have is 124. Find the number of marbles they had with them in the beginning.

Show answer
Let John have x marbles in the beginning; then Jivanti had 45 - x. After losing 5 each: (x - 5)(40 - x) = 124 ⇒ -x2 + 45x - 200 = 124 ⇒ x2 - 45x + 324 = 0 x = (45 ± √(452 - 4(324)))/(2) = (45 ± 27)/(2), so x = 36 or x = 9. Case 1: John had 36 marbles and Jivanti had 45 - 36 = 9. Case 2: John had 9 marbles and Jivanti had 45 - 9 = 36.

CBSE 2026 board paper · 1 mark

The roots of the quadratic equation (x - 1)2 = 16 are :

  1. 5, 3
  2. 4, – 4
  3. 5, – 3
  4. – 5, 3
Show answer
5, – 3

CBSE 2026 board paper · 1 mark

The value of k for which the equation kx2 - 6x - 4 = 0 has real and equal roots, is

  1. (9)/(4)
  2. −4
  3. -(9)/(4)
  4. −2
Show answer
-(9)/(4)

CBSE 2026 board paper · 1 mark

If the roots of a quadratic equation ax2 + bx + c = 0 are equal but opposite in sign, then :

  1. a = 0
  2. b = 0
  3. b ≠ 0
  4. a < b
Show answer
b = 0