Class 10 Maths Test Paper Chapter 2 Polynomials | CBSE Board | Detailed Solutions & Marking Scheme

VEDANT IGNITE TEST SERIES

Class: X
Subject: Mathematics
Chapter 2: Polynomials
Topic: Zeros of a Polynomial & Relationship between Zeros and Coefficients (Quadratic)
Time: 1 Hour
Maximum Marks: 30
Board: CBSE

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Section A (1 × 7 = 7 Marks)

(5 MCQs and 2 Assertion-Reason questions)

Q1. The number of zeroes of the polynomial represented by a graph which cuts the x-axis at two distinct points is:

(a) 0 (b) 1 (c) 2 (d) 3

Q2. If α and β are zeroes of the quadratic polynomial 2x² – 7x + 3, then α + β is:

(a) 7/2 (b) –7/2 (c) 3/2 (d) –3/2

Q3. If one zero of the polynomial x² – 5x + 6 is 2, the other zero is:

(a) 1 (b) 2 (c) 3 (d) 4

Q4. For what value of k does the polynomial x² + kx + 9 have sum of zeroes equal to –6?

(a) 6 (b) –6 (c) 3 (d) –3

Q5. If α and β are zeroes of x² + 4x + 1, then the value of αβ is:

(a) 1 (b) –1 (c) 4 (d) –4

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Assertion–Reason Questions

Q6.

Assertion (A): If α and β are zeroes of x² – 9, then α + β = 0.
Reason (R): In a quadratic polynomial ax² + bx + c, sum of zeroes is –b/a.
(a) Both A and R are true and R is correct explanation of A
(b) Both A and R are true but R is not correct explanation of A
(c) A is true but R is false
(d) A is false but R is true

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Q7.

Assertion (A): If α and β are zeroes of 3x² + 5x + 2, then αβ = 2/3.
Reason (R): Product of zeroes of ax² + bx + c is c/a.
(a) Both A and R are true and R is correct explanation of A
(b) Both A and R are true but R is not correct explanation of A
(c) A is true but R is false
(d) A is false but R is true

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Section B (2 × 4 = 8 Marks)

Q8. Without solving the polynomial, find the sum and product of zeroes of 5x² – 3x – 2.

Q9. Find the quadratic polynomial whose sum of zeroes is 3 and product of zeroes is –10.

Q10. If α and β are zeroes of x² + 7x + 10, verify the relationship between zeroes and coefficients.

Q11. The sum of zeroes of a quadratic polynomial is –1/2 and product is –3/2. Find the quadratic polynomial.

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Section C (3 × 2 = 6 Marks)

Q12. If α and β are zeroes of 2x² – 5x + k such that α + β = 3, find the value of k. Hence find the zeroes.

Q13. The graph of a quadratic polynomial intersects the x-axis at (–2, 0) and (4, 0).

(i) Write the zeroes of the polynomial.
(ii) Find the quadratic polynomial whose leading coefficient is 1.
(iii) Verify the relationship between zeroes and coefficients.

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Section D (5 × 1 = 5 Marks)

Q14.

If α and β are zeroes of the quadratic polynomial x² – px + q such that α² + β² = 20 and α + β = 6, find the values of p and q. Hence, form the polynomial.

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Section E – Case Study (4 Marks)

Q15. Case Study: Rectangular Garden Problem

A rectangular garden has length (x + 3) metres and breadth (x – 2) metres. The area of the garden is given by the polynomial x² + x – 6.

Answer the following questions:

(i) Write the polynomial representing the area in factored form. (1 mark)

(ii) Find the zeroes of the polynomial x² + x – 6. (1 mark)

(iii) Verify the relationship between zeroes and coefficients for the polynomial x² + x – 6. (2 marks)

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Here is the Detailed Answer Key with Marking Scheme (CBSE style, simple maths font, proper powers):

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VEDANT IGNITE TEST SERIES – ANSWER KEY

Class: X | Subject: Mathematics
Chapter: Polynomials
Maximum Marks: 30

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✅ Section A (1 × 7 = 7 Marks)

Q1. (c) 2

Q2. (a) 7/2
Sum of zeroes = -b/a = -(-7)/2 = 7/2

Q3. (c) 3
x^2 - 5x + 6 = (x - 2)(x - 3)

Q4. (a) 6
Sum of zeroes = -k
-k = -6 ⇒ k = 6

Q5. (a) 1
Product = c/a = 1/1 = 1

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Assertion–Reason

Q6. (a)
For x^2 - 9: b = 0
Sum = -b/a = 0

Q7. (a)
Product = c/a = 2/3

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✅ Section B (2 × 4 = 8 Marks)

Q8. (2 marks)

Polynomial: 5x^2 - 3x - 2

Sum of zeroes = -b/a = -(-3)/5 = 3/5 (1 mark)
Product of zeroes = c/a = -2/5 (1 mark)

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Q9. (2 marks)

Given:
Sum = 3, Product = -10

Polynomial = x^2 - (sum)x + product
= x^2 - 3x - 10 (2 marks)

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Q10. (2 marks)

x^2 + 7x + 10

Factorisation:
= (x + 5)(x + 2)

Zeroes: -5, -2 (1 mark)

Sum = -5 + (-2) = -7
-b/a = -7

Product = (-5)(-2) = 10
c/a = 10

Verified (1 mark)

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Q11. (2 marks)

Sum = -1/2, Product = -3/2

Polynomial = x^2 - (sum)x + product
= x^2 + (1/2)x - 3/2

Multiply by 2:
= 2x^2 + x - 3 (2 marks)

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✅ Section C (3 × 2 = 6 Marks)

Q12. (3 marks)

Given polynomial: 2x^2 - 5x + k

Sum of zeroes = -b/a = 5/2

Given α + β = 3

So,
5/2 = 3

This is not possible.

Therefore, no such value of k exists.

✔ Correct reasoning (3 marks)

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Q13. (3 marks)

(i) Zeroes = -2 and 4 (1 mark)

(ii) Polynomial:
= (x + 2)(x - 4)
= x^2 - 2x - 8 (1 mark)

(iii) Verification:

Sum = -2 + 4 = 2
-b/a = -(-2)/1 = 2

Product = (-2)(4) = -8
c/a = -8

Verified (1 mark)

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✅ Section D (5 Marks)

Q14.

Given:
α + β = 6
α^2 + β^2 = 20

We know:
α^2 + β^2 = (α + β)^2 - 2αβ

20 = 6^2 - 2αβ
20 = 36 - 2αβ
2αβ = 16
αβ = 8 (3 marks)

Polynomial = x^2 - (sum)x + product
= x^2 - 6x + 8

Comparing:
p = 6, q = 8 (2 marks)

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✅ Section E – Case Study (4 Marks)

Polynomial: x^2 + x - 6

(i) (1 mark)

Factorisation:
= (x + 3)(x - 2)

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(ii) (1 mark)

Zeroes = -3, 2

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(iii) (2 marks)

Sum = -3 + 2 = -1
-b/a = -1

Product = (-3)(2) = -6
c/a = -6

Verified

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✅ Final Objective Answers

1. (c)

2. (a)

3. (c)

4. (a)

5. (a)

6. (a)

7. (a)

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🔎 Note for Teacher (Important Correction)

Q12 is conceptually inconsistent (no real solution).
You may either:

• Accept reasoning-based answer (good for competency), OR

• Replace with corrected version for exam use.

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