Class 10 Maths Polynomials MCQ

Class 10 Maths Polynomials MCQ

Class 10 Mathematics — Polynomials

Question Bank

A. Multiple Choice Questions (MCQs)

1. If the graph of a polynomial intersects the x-axis at exactly two distinct points, which statement must be true?

A. The polynomial must be linear
B. The polynomial must have exactly two zeroes
C. The polynomial must have degree exactly 2
D. The polynomial has no zeroes

2. For the quadratic polynomial (2x^2-7x+3), the sum of its zeroes is:

A. (\frac72)
B. (-\frac72)
C. (\frac32)
D. (-\frac32)

3. If (\alpha,\beta) are the zeroes of (5x^2+3x-2), then (\alpha\beta) is:

A. (-\frac25)
B. (\frac25)
C. (-\frac35)
D. (\frac35)

4. A quadratic polynomial has two equal zeroes when its graph:

A. cuts the x-axis twice
B. does not meet the x-axis
C. touches the x-axis at exactly one point
D. is a straight line

5. Which polynomial can have three distinct zeroes?

A. (x^2+1)
B. (3x+5)
C. (x^3-4x)
D. (2x^2-3x+1)

6. If the zeroes of (x^2+px+q) have sum (4) and product (-5), then (p) and (q) are:

A. (p=4,\ q=-5)
B. (p=-4,\ q=-5)
C. (p=-4,\ q=5)
D. (p=4,\ q=5)

7. If (\alpha,\beta,\gamma) are zeroes of (2x^3-5x^2+7x-4), then

[
\alpha+\beta+\gamma=
]

A. (-\frac52)
B. (\frac52)
C. (\frac72)
D. (-\frac72)

8. For (ax^3+bx^2+cx+d), the sum (\alpha\beta+\beta\gamma+\gamma\alpha) is:

A. (-\frac ba)
B. (\frac ca)
C. (-\frac da)
D. (\frac da)

9. A polynomial of degree 4 can have at most:

A. 2 zeroes
B. 3 zeroes
C. 4 zeroes
D. 5 zeroes

10. If the graph of a polynomial touches the x-axis at one point and does not cross it there, the corresponding zero is:

A. necessarily absent
B. repeated/equal in the quadratic case
C. always irrational
D. necessarily positive

11. If the zeroes of a quadratic polynomial are (2) and (-3), which is a suitable polynomial?

A. (x^2-x-6)
B. (x^2+x-6)
C. (x^2-5x+6)
D. (x^2+5x-6)

12. If a quadratic polynomial has zeroes (\alpha,\beta), then the polynomial with the same zeroes can be changed by:

A. changing its zeroes
B. multiplying it by any non-zero constant
C. adding any constant
D. changing its degree arbitrarily

Answer Key — MCQs

  1. B
  2. A
  3. A
  4. C
  5. C
  6. B
  7. B
  8. B
  9. C
  10. B
  11. B
  12. B

B. Assertion–Reason Questions

Choose:

A. Both A and R are true, and R correctly explains A.
B. Both A and R are true, but R does not correctly explain A.
C. A is true, but R is false.
D. A is false, but R is true.

1.
Assertion: A quadratic polynomial can have no real zeroes.
Reason: Its graph may lie completely above or completely below the x-axis.

2.
Assertion: Every cubic polynomial has exactly three zeroes.
Reason: A polynomial of degree 3 can have at most three zeroes.

3.
Assertion: The zeroes of a polynomial are related to its graph.
Reason: The zeroes are the x-coordinates of the points where its graph intersects the x-axis.

4.
Assertion: For (ax^2+bx+c), the product of zeroes is (\frac ca).
Reason: If the zeroes are (\alpha,\beta), then (c=a\alpha\beta).

5.
Assertion: Multiplying a quadratic polynomial by a non-zero constant changes its zeroes.
Reason: The equation (kp(x)=0), for (k\neq0), has the same solutions as (p(x)=0).

6.
Assertion: A polynomial of degree (n) can have more than (n) zeroes.
Reason: Its graph can intersect the x-axis at at most (n) points.

Answers

  1. A
  2. D
  3. A
  4. A
  5. D
  6. D

C. Fill in the Blanks

1. The zeroes of a polynomial are the __________ coordinates of the points where its graph intersects the x-axis.

2. A polynomial of degree 2 is called a __________ polynomial.

3. A quadratic polynomial can have at most __________ zeroes.

4. If (\alpha,\beta) are the zeroes of (ax^2+bx+c), then (\alpha+\beta=) __________.

5. If (\alpha,\beta) are the zeroes of (ax^2+bx+c), then (\alpha\beta=) __________.

6. For (ax^3+bx^2+cx+d), the product of the three zeroes is __________.

7. The graph of (y=ax^2+bx+c) is called a __________.

8. If (a>0) in (y=ax^2+bx+c), the parabola opens __________.

9. If a quadratic graph does not intersect the x-axis, the polynomial has __________ real zeroes.

10. A polynomial of degree (n) has at most __________ zeroes.

Answers

  1. x
  2. quadratic
  3. 2
  4. (-\frac ba)
  5. (\frac ca)
  6. (-\frac da)
  7. parabola
  8. upwards
  9. no
  10. (n)

D. True or False

1. A quadratic polynomial may have two equal zeroes.

2. A cubic polynomial must have three distinct real zeroes.

3. If the graph does not meet the x-axis, the polynomial has no real zero.

4. The degree of a polynomial gives an upper limit for the number of its zeroes.

5. The sum of the zeroes of (ax^2+bx+c) is (\frac ba).

6. The product of the zeroes of (ax^2+bx+c) is (\frac ca).

7. A non-zero constant multiple of a polynomial has the same zeroes.

8. A linear polynomial has exactly one zero.

9. A cubic polynomial can have only one real zero.

10. The x-coordinate of an x-axis intersection gives a zero of the polynomial.

Answers

  1. True
  2. False
  3. True
  4. True
  5. False
  6. True
  7. True
  8. True
  9. True
  10. True

E. Match the Following

Column AColumn B
1. (ax+b)A. (-\frac da)
2. (ax^2+bx+c)B. At most 3 zeroes
3. (ax^3+bx^2+cx+d)C. Exactly one zero
4. (\alpha\beta) for a quadraticD. (\frac ca)
5. (\alpha\beta\gamma) for a cubicE. At most 2 zeroes

Answers

1–C, 2–E, 3–B, 4–D, 5–A

F. One-Word / Very Short Answer

1. What is the maximum number of zeroes of a polynomial of degree 5?

2. What is the geometrical meaning of a zero of a polynomial?

3. What type of graph represents a quadratic polynomial?

4. What is the zero of (7x-4)?

5. If a quadratic graph intersects the x-axis at two points, how many distinct zeroes does it have?

6. If the graph of a quadratic polynomial is completely above the x-axis, how many real zeroes does it have?

7. If the zeroes of a quadratic are (\alpha,\beta), what does (\alpha+\beta) equal in terms of coefficients?

8. What is the maximum number of zeroes of a cubic polynomial?

Answers

  1. 5
  2. x-coordinate of an x-axis intersection
  3. Parabola
  4. (\frac47)
  5. Two
  6. None
  7. (-\frac ba)
  8. Three

G. Case-Based / Competency Questions

Case 1: Quadratic Polynomial

A student studies the polynomial

[
p(x)=2x^2-8x+6.
]

It can be factorised as

[
p(x)=2(x-1)(x-3).
]

1. The zeroes of (p(x)) are:

A. (-1,-3)
B. (1,3)
C. (2,6)
D. (-2,-6)

2. The sum of the zeroes is:

A. 2
B. 3
C. 4
D. 6

3. The product of the zeroes is:

A. 2
B. 3
C. 4
D. 6

4. Which coefficient relationship is verified?

A. Sum (=\frac ca)
B. Product (=-\frac ba)
C. Sum (=-\frac ba)
D. Product (=-\frac da)

Answers: 1-B, 2-C, 3-C, 4-C

Case 2: Graph Interpretation

The graph of a polynomial intersects the x-axis at three distinct points.

1. The polynomial has:

A. one zero
B. two zeroes
C. three zeroes
D. no zeroes

2. Its degree must be at least:

A. 1
B. 2
C. 3
D. 4

3. If the polynomial is known to be cubic, then the number of its zeroes is:

A. exactly 1
B. exactly 2
C. exactly 3
D. at most 2

Answers: 1-C, 2-C, 3-C

H. Higher-Order / Application Questions

1. The zeroes of a quadratic polynomial have sum (5) and product (6). Construct one such polynomial.

2. A quadratic polynomial has zeroes (-2) and (5). Find the sum and product of its zeroes and construct a corresponding polynomial.

3. The sum and product of the zeroes of a quadratic polynomial are (-4) and (3), respectively. Find a suitable quadratic polynomial.

4. A cubic polynomial has zeroes (2,-1) and (\frac12). Determine the sum of the zeroes and the sum of their pairwise products.

5. For the cubic polynomial

[
3x^3-5x^2+7x-2,
]

find the sum of its zeroes, the sum of the products of its zeroes taken two at a time, and the product of its zeroes.

6. A quadratic polynomial has zeroes (\alpha,\beta), where

[
\alpha+\beta=7,\qquad \alpha\beta=10.
]

Find a quadratic polynomial having these zeroes.

7. Without finding the individual zeroes, determine the sum and product of the zeroes of

[
7x^2-9x+4.
]

8. A polynomial’s graph intersects the x-axis at four distinct points. What can you conclude about its degree?

9. Explain why a quadratic polynomial cannot have three distinct real zeroes using its geometrical interpretation.

10. If the zeroes of a cubic polynomial are (\alpha,\beta,\gamma), express the coefficient relationships for

[
ax^3+bx^2+cx+d
]

in terms of (\alpha,\beta,\gamma).

Answers / Key Results

  1. (x^2-5x+6)
  2. Sum (=3), product (=-10); polynomial (x^2-3x-10)
  3. (x^2+4x+3)
  4. Sum (=\frac32); pairwise product (=-\frac52)
  5. Sum (=\frac53); pairwise product (=\frac73); product (=\frac23)
  6. (x^2-7x+10)
  7. Sum (=\frac97); product (=\frac47)
  8. Its degree must be at least 4
  9. A quadratic graph is a parabola and can intersect the x-axis at at most two points
  10. (\alpha+\beta+\gamma=-\frac ba), (\alpha\beta+\beta\gamma+\gamma\alpha=\frac ca), (\alpha\beta\gamma=-\frac da)

I. Challenge MCQs

1. A quadratic polynomial has zeroes whose sum is zero and product is (-9). Which polynomial can represent it?

A. (x^2+9)
B. (x^2-9)
C. (x^2+9x)
D. (x^2-9x)

2. If the zeroes of (2x^2+px+8) have product (4), then (p) can be:

A. any real number
B. only 4
C. only (-4)
D. no such value exists

3. If (\alpha,\beta) are the zeroes of (x^2-6x+8), which statement is correct?

A. (\alpha+\beta=8,\ \alpha\beta=6)
B. (\alpha+\beta=6,\ \alpha\beta=8)
C. (\alpha+\beta=-6,\ \alpha\beta=8)
D. (\alpha+\beta=6,\ \alpha\beta=-8)

4. A quadratic polynomial has one real zero. Its graph:

A. cuts the x-axis at two distinct points
B. touches the x-axis at one point
C. never meets the x-axis
D. is necessarily a straight line

5. For a cubic polynomial (ax^3+bx^2+cx+d), which coefficient determines the negative of the product of its zeroes?

A. (a)
B. (b)
C. (c)
D. (d)

Answers

  1. B
  2. A
  3. B
  4. B
  5. D