Class 10 Maths Chapter 4 Quadratic Equations Practice Paper 1

PRACTICE PAPER 01 - CHAPTER 04 QUADRATIC EQUATIONS (2025-26) SUBJECT: MATHEMATICS MAX. MARKS: 40 CLASS: X DURATION: 1½ hrs

General Instructions:

1. All questions are compulsory.

2. This question paper contains 20 questions divided into five Sections A, B, C, D and E.

3. Section A: 10 MCQs of 1 mark each. Section B: 4 questions of 2 marks each. Section C: 3 questions of 3 marks each. Section D: 1 question of 5 marks. Section E: 2 Case Studies of 4 marks each.

4. There is no overall choice.

5. Use of Calculators is not permitted. SECTION – A (Questions 1 to 10 carry 1 mark each)

1. Which of the following equations has the sum of roots equal to 5?
(a) x² - 5x + 6 = 0
(b) x² + 5x - 6 = 0
(c) 2x² - 10x + 3 = 0
(d) x² - 10x + 9 = 0

2. The product of roots of equation 3x² - 6x + 9 = 0 is:
(a) 2
(b) 3
(c) -3
(d) 9

3. If one root of equation x² - 7x + k = 0 is 3, then the value of k is:
(a) 10
(b) 12
(c) 15
(d) 21

4. A quadratic equation whose roots are 2 and -3 is:
(a) x² + x - 6 = 0
(b) x² - x - 6 = 0
(c) x² + x + 6 = 0
(d) x² - 5x - 6 = 0

5. If α and β are roots of x² - 3x + 2 = 0, then the value of α + β + αβ is:
(a) 5
(b) 4
(c) 3
(d) 6

6. The roots of equation 2x² - 5x + 3 = 0 are:
(a) 1, 3/2
(b) 2, 3
(c) 1, 2
(d) 3/2, 2

7. If roots of a quadratic equation are equal, then discriminant is:
(a) Greater than zero
(b) Less than zero
(c) Equal to zero
(d) Cannot be determined

8. The equation x² + 4x + 5 = 0 has:
(a) Two distinct real roots
(b) Two equal real roots
(c) No real roots
(d) More than two real roots

9. Assertion
(a) : The equation 2x² - 3x + 1 = 0 can be solved by factorization method. Reason (R): Every quadratic equation can be solved by factorization.
(a) Both A and R are true and R is the correct explanation of A
(b) Both A and R are true but R is not the correct explanation of A
(c) A is true but R is false
(d) A is false but R is true

10. Assertion
(a) : If the sum of roots is 6 and product is 8, the quadratic equation is x² - 6x + 8 = 0. Reason (R): A quadratic equation is given by x² - (sum of roots)x + (product of roots) = 0.
(a) Both A and R are true and R is the correct explanation of A
(b) Both A and R are true but R is not the correct explanation of A
(c) A is true but R is false
(d) A is false but R is true SECTION – B (Questions 11 to 14 carry 2 marks each)

11. Find the sum and product of roots of the quadratic equation 5x² - 7x + 2 = 0 without solving it.

12. Form a quadratic equation whose roots are 4 and -5.

13. Check whether the equation x² - 6x + 9 = 0 has equal roots. Justify your answer.

14. If α and β are roots of equation 2x² - 5x + 3 = 0, find the value of α² + β². SECTION – C (Questions 15 to 17 carry 3 marks each)

15. Solve the quadratic equation by factorization method: 6x² - 13x + 6 = 0

16. Find the discriminant of the quadratic equation 3x² - 4x + 2 = 0 and hence determine the nature of its roots.

17. If α and β are the roots of the equation x² - 5x + 6 = 0, form a quadratic equation whose roots are α² and β². SECTION – D (Question 18 carries 5 marks)

18. The sum of the ages of a father and his son is 45 years. Five years ago, the product of their ages (in years) was 124. Determine their present ages. SECTION – E (Questions 19 to 20 carry 4 marks each)

19. A school has a rectangular playground with length 20 m more than its breadth. The area of the playground is 2400 m². The school management wants to fence the playground with barbed wire.
(a) Form a quadratic equation to represent this situation.
(b) Find the breadth of the playground.
(c) Find the length of barbed wire needed to fence the playground.

20. In a cricket match, a batsman hits a ball which follows a parabolic path. The height h (in meters) of the ball at time t seconds is given by the equation h = -5t² + 20t + 1.
(a) At what time will the ball reach maximum height?
(b) What is the maximum height reached by the ball?
(c) After how many seconds will the ball hit the ground? DETAILED ANSWER KEY

SECTION A - ANSWERS

1. Answer:
(a) x² - 5x + 6 = 0

Solution: Sum of roots = -b/a = -(-5)/1 = 5 ✓

2. Answer:
(b) 3

Solution: Product of roots = c/a = 9/3 = 3

3. Answer:
(b) 12

Solution: x² - 7x + k = 0, one root = 3

3² - 7(3) + k = 0 → 9 - 21 + k = 0 → k = 12

4. Answer:
(a) x² + x - 6 = 0

Solution: Sum = 2 + (-3) = -1, Product = 2 × (-3) = -6

Equation: x² - (sum)x + (product) = 0 → x² - (-1)x + (-6) = 0 → x² + x - 6 = 0

5. Answer:
(a) 5

Solution: x² - 3x + 2 = 0 → α + β = 3, αβ = 2

α + β + αβ = 3 + 2 = 5

6. Answer:
(a) 1, 3/2

Solution: 2x² - 5x + 3 = 0

(2x - 3)(x - 1) = 0 → x = 3/2 or x = 1

7. Answer:
(c) Equal to zero

Solution: For equal roots, discriminant D = b² - 4ac = 0

8. Answer:
(c) No real roots

Solution: D = b² - 4ac = 16 - 4(1)(5) = 16 - 20 = -4 < 0

Discriminant is negative, so no real roots

9. Answer:
(c) A is true but R is false

Solution: 2x² - 3x + 1 = (2x - 1)(x - 1) = 0 can be factorized ✓

But not EVERY quadratic can be factorized (e.g., x² + x + 1 = 0)

10. Answer:
(a) Both A and R are true and R is the correct explanation of A

Solution: Sum = 6, Product = 8

x² - 6x + 8 = 0 ✓ (Both A and R correct, R explains A)

SECTION B - ANSWERS

11. Solution: 5x² - 7x + 2 = 0 Sum of roots = -b/a = -(-7)/5 = 7/5 Product of roots = c/a = 2/5 Answer: Sum = 7/5, Product = 2/5

12. Solution: Roots are 4 and -5 Sum = 4 + (-5) = -1 Product = 4 × (-5) = -20 Equation: x² - (sum)x + (product) = 0 Answer: x² + x - 20 = 0

13. Solution: x² - 6x + 9 = 0 D = b² - 4ac = 36 - 4(1)(9) = 36 - 36 = 0 Since D = 0, the equation has equal roots. Answer: Yes, it has equal roots (both roots = 3)

14. Solution: 2x² - 5x + 3 = 0 α + β = 5/2, αβ = 3/2 α² + β² = (α + β)² - 2αβ = (5/2)² - 2(3/2) = 25/4 - 3 = 25/4 - 12/4 Answer: α² + β² = 13/4

SECTION C - ANSWERS

15. Solution: 6x² - 13x + 6 = 0 6x² - 9x - 4x + 6 = 0 3x(2x - 3) - 2(2x - 3) = 0 (3x - 2)(2x - 3) = 0 Answer: x = 2/3 or x = 3/2

16. Solution: 3x² - 4x + 2 = 0 D = b² - 4ac = (-4)² - 4(3)(2) = 16 - 24 = -8 Since D < 0, the roots are not real (imaginary/complex). Answer: Discriminant = -8, Nature: No real roots

17. Solution: x² - 5x + 6 = 0 → Roots: α = 2, β = 3 α² = 4, β² = 9 New equation with roots 4 and 9: Sum = 4 + 9 = 13, Product = 4 × 9 = 36 Answer: x² - 13x + 36 = 0

SECTION D - ANSWER

18. Solution: Let son's present age = x years, father's present age = (45 - x) years Five years ago: Son = (x - 5), Father = (40 - x) Product: (x - 5)(40 - x) = 124 40x - x² - 200 + 5x = 124 -x² + 45x - 200 = 124 x² - 45x + 324 = 0 x² - 36x - 9x + 324 = 0 x(x - 36) - 9(x - 36) = 0 (x - 9)(x - 36) = 0 x = 9 or x = 36 If son = 36, father = 9 (not possible) Answer: Son's age = 9 years, Father's age = 36 years

SECTION E - ANSWERS

19. Solution: Let breadth = x m, then length = (x + 20) m
(a) Area = length × breadth x(x + 20) = 2400 Equation: x² + 20x - 2400 = 0
(b) x² + 20x - 2400 = 0 (x + 60)(x - 40) = 0 x = 40 (taking positive value) Breadth = 40 m
(c) Length = 60 m, Breadth = 40 m Perimeter = 2(60 + 40) = 200 m Length of barbed wire = 200 m

20. Solution: Height h = -5t² + 20t + 1
(a) For maximum height, t = -b/2a = -20/(2×(-5)) = 20/10 = 2 Time = 2 seconds
(b) h(2) = -5(4) + 20(2) + 1 = -20 + 40 + 1 = 21 Maximum height = 21 meters
(c) When ball hits ground, h = 0 -5t² + 20t + 1 = 0 5t² - 20t - 1 = 0 Using quadratic formula: t = [20 ± √(400 + 20)]/10 = [20 ± √420]/10 t ≈ 4.05 seconds (taking positive value) Time ≈ 4.05 seconds