Class 10 Maths MCQ Worksheet Chapter 2 Polynomials | Detailed Solutions & Marking Scheme

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VEDANT WORKSHEET SERIES 

Class X | Mathematics (041)

Chapter 2: Polynomials

Time Allowed: 60 Minutes
Maximum Marks: 30

SECTION A: MULTIPLE CHOICE QUESTIONS (15 Marks)

Each question carries 1 mark. Choose the correct option.

  1. If one zero of the quadratic polynomial p(x) = x² + 3x + k is 2, then the value of k is:
    (a) 10 (b) -10 (c) 5 (d) -5

  2. The number of polynomials having zeros -2 and 5 is:
    (a) 1 (b) 2 (c) 3 (d) More than 3

  3. If the zeros of the quadratic polynomial ax² + bx + c, c ≠ 0 are equal, then:
    (a) c and a have opposite signs (b) c and b have opposite signs (c) c and a have the same sign (d) c and b have the same sign

  4. The graph of a quadratic polynomial y = ax² + bx + c is a parabola which opens upwards if:
    (a) a > 0 (b) a < 0 (c) a = 0 (d) a ≥ 0

  5. If α and β are the zeros of the polynomial f(x) = x² - p(x + 1) - c, then (α + 1)(β + 1) is equal to:
    (a) c - 1 (b) 1 - c (c) c (d) 1 + c

  6. If the sum of the zeros of the polynomial f(x) = 2x³ - 3kx² + 4x - 5 is 6, then the value of k is:
    (a) 2 (b) 4 (c) -2 (d) -4

  7. A quadratic polynomial whose zeros are -3 and 4 is:
    (a) x² - x + 12 (b) x² + x + 12 (c) (1/2)x² - (1/2)x - 6 (d) 2x² + 2x - 24

  8. If α, β are the zeros of x² - 6x + k and 3α + 2β = 20, then the value of k is:
    (a) -8 (b) 16 (c) -16 (d) 8

  9. If one zero of the polynomial (a² + 9)x² + 13x + 6a is the reciprocal of the other, then a is:
    (a) 3 (b) -3 (c) 6 (d) 0

  10. If the zeros of the quadratic polynomial x² + (a + 1)x + b are 2 and -3, then:
    (a) a = -7, b = -1 (b) a = 5, b = -1 (c) a = 2, b = -6 (d) a = 0, b = -6

  11. The zeroes of the quadratic polynomial x² + 99x + 127 are:
    (a) Both positive (b) Both negative (c) One positive and one negative (d) Both equal

  12. If α and β are the zeros of 4x² + 3x + 7, then the value of (1/α + 1/β) is:
    (a) -3/7 (b) 3/7 (c) -7/3 (d) 7/3

  13. The maximum number of zeros a cubic polynomial can have is:
    (a) 1 (b) 4 (c) 2 (d) 3

  14. If the product of the zeros of ax² - 6x - 6 is 4, then the value of a is:
    (a) -3/2 (b) 3/2 (c) 2 (d) -2

  15. Given the graph of y = p(x) below, the number of zeros of p(x) is:
    (a) 0 (b) 1 (c) 2 (d) 3

SECTION B: ASSERTION – REASONING QUESTIONS (15 Marks)

Directions: Choose the correct option:
(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

Assertion (A): The degree of a non-zero constant polynomial is zero.
Reason (R): A constant number k can be written as kx⁰.

Assertion (A): x² + 4x + 5 has no real zeros.
Reason (R): A quadratic polynomial ax² + bx + c has no real zeros if b² - 4ac < 0.

Assertion (A): The polynomial p(x) = x² - 5x + 6 has two zeros.
Reason (R): A polynomial of degree n has at most n zeros.

Assertion (A): If the sum of zeros of kx² + 2x + 3k is equal to their product, then k = -2/3.
Reason (R): For ax² + bx + c, sum of zeros = -b/a and product of zeros = c/a.

Assertion (A): If 2 - √3 is a zero of a quadratic polynomial, then the other zero is 2 + √3.
Reason (R): Irrational zeros of a polynomial with rational coefficients occur in conjugate pairs.

Assertion (A): The graph of x² - 6x + 9 touches the x-axis at exactly one point.
Reason (R): A quadratic polynomial has equal zeros if its discriminant is zero.

Assertion (A): f(x) = 3x⁴ - 2x² + x - 1 is a quadratic polynomial.
Reason (R): The highest power of the variable in a polynomial is called its degree.

Assertion (A): If α, β are zeros of x² - x - 2, then α² + β² = 5.
Reason (R): (α + β)² = α² + β² + 2αβ.

Assertion (A): The value of k for which 3 is a zero of 2x² + x + k is -21.
Reason (R): If a is a zero of p(x), then p(a) = 0.

Assertion (A): A linear polynomial has exactly one zero.
Reason (R): The graph of y = ax + b is a straight line.

Assertion (A): x² + x + 1 has real zeros.
Reason (R): The discriminant of x² + x + 1 is -3.

Assertion (A): If the product of zeros of ax² + 2x + a² is 2, then a = 2.
Reason (R): Product of zeros = c/a.

Assertion (A): The zeros of x² - 16 are 4 and -4.
Reason (R): Zeros are values of x for which p(x) = 0.

Assertion (A): p(x) = 1/x + 2 is a polynomial.
Reason (R): In a polynomial, powers of the variable must be whole numbers.

Assertion (A): The sum of zeros of x² + 7x + 10 is -7.
Reason (R): Sum of zeros = -(coefficient of x)/(coefficient of x²).


Here is the detailed solved answer key with marking scheme in simple plain maths format (easy to copy for Vedant Classes use):

ANSWER KEY WITH MARKING SCHEME

Chapter 2: Polynomials | Class X

SECTION A – MCQs (1 mark each)

1.
p(2) = 0
⇒ 2² + 3(2) + k = 0
⇒ 4 + 6 + k = 0
⇒ k = -10
Answer: (b) -10
Marks: 1

2.
Infinite polynomials can be formed with given zeros (by multiplying constant)
Answer: (d) More than 3
Marks: 1

3.
Equal zeros ⇒ Discriminant = 0
⇒ b² - 4ac = 0 ⇒ b² = 4ac ⇒ a and c have same sign
Answer: (c)
Marks: 1

4.
Parabola opens upward when a > 0
Answer: (a)
Marks: 1

5.
(α + 1)(β + 1) = αβ + α + β + 1
Using relations ⇒ = c + 1
Answer: (d)
Marks: 1

6.
Sum of zeros = -b/a = 3k/2
⇒ 3k/2 = 6 ⇒ k = 4
Answer: (b)
Marks: 1

7.
Polynomial = x² - (sum)x + product
Sum = 1, Product = -12
⇒ x² - x - 12
Option (c) is equivalent
Answer: (c)
Marks: 1

8.
α + β = 6
Given: 3α + 2β = 20
Solve ⇒ α = 8, β = -2
k = αβ = -16
Answer: (c)
Marks: 1

9.
Product = 1 ⇒ c/a = 1
⇒ 6a/(a² + 9) = 1
⇒ 6a = a² + 9
⇒ (a - 3)² = 0 ⇒ a = 3
Answer: (a)
Marks: 1

10.
Sum = -1 ⇒ -(a + 1) = -1 ⇒ a = 0
Product = -6 ⇒ b = -6
Answer: (d)
Marks: 1

11.
Sum = -99, Product = 127
Both negative
Answer: (b)
Marks: 1

12.
(1/α + 1/β) = (α + β)/(αβ)
= (-3/4)/(7/4) = -3/7
Answer: (a)
Marks: 1

13.
Max zeros = degree
Answer: (d) 3
Marks: 1

14.
Product = c/a = (-6)/a = 4
⇒ a = -3/2
Answer: (a)
Marks: 1

15.
Depends on graph (assumed typical quadratic crossing twice)
Answer: (c) 2
Marks: 1

SECTION B – ASSERTION REASONING (1 mark each)

16.
Both true, correct explanation
Answer: (a)
Marks: 1

17.
Both true, correct explanation
Answer: (a)
Marks: 1

18.
Both true but R not explanation
Answer: (b)
Marks: 1

19.
Sum = -2/k, Product = 3
⇒ -2/k = 3 ⇒ k = -2/3
Both true, correct explanation
Answer: (a)
Marks: 1

20.
Both true, correct explanation
Answer: (a)
Marks: 1

21.
Both true, correct explanation
Answer: (a)
Marks: 1

22.
A false (degree 4), R true
Answer: (d)
Marks: 1

23.
α + β = 1, αβ = -2
α² + β² = (α + β)² - 2αβ = 1 + 4 = 5
Both true, correct explanation
Answer: (a)
Marks: 1

24.
p(3) = 0 ⇒ 18 + 3 + k = 0 ⇒ k = -21
Both true, correct explanation
Answer: (a)
Marks: 1

25.
Both true but R not explanation
Answer: (b)
Marks: 1

26.
A false, R true
Answer: (d)
Marks: 1

27.
Product = a²/a = a ⇒ a = 2
Both true, correct explanation
Answer: (a)
Marks: 1

28.
Both true but R not explanation
Answer: (b)
Marks: 1

29.
A false, R true
Answer: (d)
Marks: 1

30.
Both true, correct explanation
Answer: (a)
Marks: 1

FINAL MARKING SUMMARY

Section A: 15 × 1 = 15
Section B: 15 × 1 = 15

Total = 30 Marks


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