Polynomials

CBSE Class 10 — Maths — Previous Year Questions

58 questions100 total marksMCQ: 29VSA: 8Assertion-Reason: 3Case-Study: 6SA: 8LA: 4

Q1.MCQ1 mark(2020)Easy

If one of the zeroes of the quadratic polynomial

x^{2} + 3 x + k

2

, then the value of

k

is:

Q2.MCQ1 mark(2020)Easy

The quadratic polynomial, the sum of whose zeroes is

- 5

and their product is

6

, is:

Q3.MCQ1 mark(2020)Medium

The zeroes of the polynomial

x^{2} - 3 x - m (m + 3)

are

Q4.MCQ1 mark(2020)Easy

In Fig. 1, the graph of the polynomial

p (x)

is given. The number of zeroes of the polynomial is

$Graph of polynomial p(x)$

Q5.MCQ1 mark(2020)Easy

The number of zeroes for a polynomial

p (x)

where graph of

y = p (x)

is given in Figure-1, is

$Graph of polynomial p(x)$

Q6.MCQ1 mark(2020)Easy

On dividing a polynomial

p (x)

x^{2} - 4

, quotient and remainder are found to be

x

and

3

respectively. The polynomial

p (x)

Q7.VSA1 mark(2020)Easy

Form a quadratic polynomial, the sum and product of whose zeroes are

(- 3)

and

2

respectively.

Q8.VSA1 mark(2020)Easy

Can

(x^{2} - 1)

be a remainder while dividing

x^{4} - 3 x^{2} + 5 x - 9

(x^{2} + 3)

? Justify your answer with reason.

Q9.MCQ1 mark(2023)Easy

The graph of

y = p (x)

is given, for a polynomial

p (x)

. The number of zeroes of

p (x)

from the graph is
(A)

3

(B)

1

(C)

2

(D)

0

Q10.MCQ1 mark(2023)Easy

α, β

are the zeroes of a polynomial

p (x) = x^{2} + x - 1

, then

\frac{1}{α} + \frac{1}{β}

equals to
(A)

1

(B)

2

(C)

- 1

(D)

\frac{- 1}{2}

Q11.MCQ1 mark(2023)Medium

Which of the following is a quadratic polynomial having zeroes

\frac{- 2}{3}

and

\frac{2}{3}

?
(A)

4 x^{2} - 9

(B)

\frac{4}{9} (9 x^{2} + 4)

(C)

x^{2} + \frac{9}{4}

(D)

5 (9 x^{2} - 4)

Q12.MCQ1 mark(2023)Medium

If one zero of the polynomial

6 x^{2} + 37 x - (k - 2)

is reciprocal of the other, then what is the value of

k

?
(a)

- 4

(b)

- 6

(c)

6

(d)

4

Q13.MCQ1 mark(2023)Easy

The zeroes of the polynomial

p (x) = x^{2} + 4 x + 3

are given by:
(a)

1, 3

(b)

- 1, 3

(c)

1, - 3

(d)

- 1, - 3

Q14.MCQ1 mark(2023)Medium

α

and

β

are the zeroes of the quadratic polynomial

p (x) = x^{2} - a x - b

, then the value of

α^{2} + β^{2}

is:
(a)

a^{2} - 2 b

(b)

a^{2} + 2 b

(c)

b^{2} - 2 a

(d)

b^{2} + 2 a

Q15.Assertion-Reason1 mark(2023)Medium

Assertion (A): The polynomial

p (x) = x^{2} + 3 x + 3

has two real zeroes.
Reason (R): A quadratic polynomial can have at most two real zeroes.
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
(b) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
(c) Assertion (A) is true, but Reason (R) is false.
(d) Assertion (A) is false, but Reason (R) is true.

Q16.MCQ1 mark(2023)Easy

α

β

are zeroes of the polynomial

x^{2} - 1

, then value of

(α + β)

is:

Q17.MCQ1 mark(2023)Easy

α

β

are the zeroes of the polynomial

p (x) = 4 x^{2} - 3 x - 7

, then

(\frac{1}{α} + \frac{1}{β})

is equal to:

Q18.MCQ1 mark(2023)Easy

The number of polynomials having zeroes

- 3

and

5

is:

Q19.Case-Study1 mark(2023)Easy

In a pool at an aquarium, a dolphin jumps out of the water travelling at

20

cm per second. Its height above water level after

t

seconds is given by

h = 20 t - 16 t^{2}

.

$Dolphin jumping$

Based on the above, answer the following questions:

(i) Find zeroes of polynomial

p (t) = 20 t - 16 t^{2}

Q20.Case-Study1 mark(2023)Easy

(ii) Which of the following types of graph represents

p (t)

?

$Graph options$

Q21.MCQ1 mark(2023)Easy

The number of quadratic polynomials having zeroes

- 5

and

- 3

Q22.MCQ1 mark(2023)Easy

α

and

β

are the zeroes of the polynomial

x^{2} - 1

, then the value of

(α + β)

Q23.MCQ1 mark(2023)Easy

If the zeroes of the quadratic polynomial

x^{2} + (a + 1) x + b

are

2

and

- 3

, then

Q24.MCQ1 mark(2024)Easy

If the sum of zeroes of the polynomial

p (x) = 2 x^{2} - k 2 x + 1

2

, then the value of

k

is:

Q25.MCQ1 mark(2024)Medium

The zeroes of a polynomial

x^{2} + p x + q

are twice the zeroes of the polynomial

4 x^{2} - 5 x - 6

. The value of

p

is:

Q26.Assertion-Reason1 mark(2024)Medium

**Assertion (A):** If the graph of a polynomial touches

x

-axis at only one point, then the polynomial cannot be a quadratic polynomial.

**Reason (R):** A polynomial of degree

n (n > 1)

can have at most

n

zeroes.

Q27.MCQ1 mark(2024)Easy

α

and

β

are zeroes of the polynomial

5 x^{2} + 3 x - 7

, the value of

\frac{1}{α} + \frac{1}{β}

Q28.MCQ1 mark(2024)Easy

If a polynomial

p (x)

is given by

p (x) = x^{2} - 5 x + 6

, then the value of

p (1) + p (4)

is:

Q29.MCQ1 mark(2024)Medium

α

and

β

are the zeroes of the polynomial

p (x) = k x^{2} - 30 x + 45 k

and

α + β = α β

, then the value of

k

is:

Q30.Assertion-Reason1 mark(2024)Easy

Assertion (A): Degree of a zero polynomial is not defined.

Reason (R): Degree of a non-zero constant polynomial is 0.

Q31.MCQ1 mark(2025)Easy

α

and

β

are the zeroes of polynomial

3 x^{2} + 6 x + k

such that

α + β + α β = - \frac{2}{3}

, then the value of

k

is:

Q32.MCQ1 mark(2025)Easy

Two polynomials are shown in the graph below. The number of distinct zeroes of both the polynomials is:

$Graph of two polynomials$

Q33.MCQ1 mark(2025)Easy

- 4

is a zero of the polynomial

p (x) = x^{2} - x - (2 + 2 k)

, then the value of

k

is:

Q34.MCQ1 mark(2025)Easy

α

and

β

are the zeroes of the polynomial

p (x) = x^{2} - a x - b

, then the value of

(α + β + α β)

is equal to:

Q35.MCQ1 mark(2025)Easy

Zeroes of the polynomial

p (x) = x^{2} - 32 x + 4

are:

Q36.MCQ1 mark(2025)Easy

Which of the following statements is true for a polynomial

p (x)

of degree 3?

Q37.VSA2 marks(2020)Easy

A teacher asked 10 of his students to write a polynomial in one variable on a paper and then to handover the paper. The following were the answers given by the students:

2 x + 3

3 x^{2} + 7 x + 2

4 x^{3} + 3 x^{2} + 2

x^{3} - 3 x + 7

7 x + 7

5 x^{3} - 7 x + 2

2 x^{2} + 3 - \frac{5}{x}

5 x - \frac{1}{2}

a x^{3} + b x^{2} + c x + d

x + \frac{1}{x}

Answer the following questions:
(i) How many of the above ten, are not polynomials?
(ii) How many of the above ten, are quadratic polynomials?

Q38.VSA2 marks(2023)Medium

If one zero of the polynomial

p (x) = 6 x^{2} + 37 x - (k - 2)

is reciprocal of the other, then find the value of

k

Q39.Case-Study2 marks(2023)Medium

(iii) (a) What would be the value of

h

t = \frac{3}{2}

? Interpret the result.

Q40.Case-Study2 marks(2023)Medium

(iii) (b) How much distance has the dolphin covered before hitting the water level again?

Q41.VSA2 marks(2024)Easy

α

β

are zeroes of the polynomial

p (x) = 5 x^{2} - 6 x + 1

, then find the value of

α + β + α β

Q42.VSA2 marks(2025)Easy

Find the zeroes of the polynomial

p (x) = x^{2} + \frac{8}{5} x - \frac{4}{5}

Q43.VSA2 marks(2025)Medium

p

and

q

are zeroes of the polynomial

p (y) = 21 y^{2} - y - 2

, then find the value of

(1 - p) \cdot (1 - q)

Q44.VSA2 marks(2025)Medium

If '

α

' and '

β

' are the zeroes of the polynomial

p (y) = y^{2} - 5 y + 3

, then find the value of

α^{4} β^{3} + α^{3} β^{4}

Q45.SA3 marks(2020)Medium

Find the quadratic polynomial whose zeroes are reciprocals of the zeroes of the polynomial

f (x) = a x^{2} + b x + c

a \neq = 0

c \neq = 0

Q46.SA3 marks(2020)Medium

Divide the polynomial

f (x) = 3 x^{4} - x^{3} - 3 x + 5

by the polynomial

g (x) = x - 1 - x^{2}

and verify the division algorithm.

Q47.SA3 marks(2020)Medium

4

is the zero of the cubic polynomial

x^{3} - 3 x^{2} - 10 x + 24

, find its other two zeroes.

Q48.SA3 marks(2020)Medium

If the sum of the squares of zeroes of the quadratic polynomial

f (x) = x^{2} - 8 x + k

is 40, find the value of

k

Q49.SA3 marks(2020)Medium

On dividing

x^{3} - 3 x^{2} + x + 2

by a polynomial

g (x)

, the quotient and remainder were

x - 2

and

- 2 x + 4

respectively. Find

g (x)

Q50.SA3 marks(2024)Medium

Find the zeroes of the polynomial

4 x^{2} + 4 x - 3

and verify the relationship between zeroes and coefficients of the polynomial.

Q51.SA3 marks(2024)Medium

α

and

β

are the zeroes of the polynomial

x^{2} + x - 2

, then find the value of

\frac{α}{β} + \frac{β}{α}

Q52.SA3 marks(2025)Medium

α

and

β

are zeroes of a quadratic polynomial

p x^{2} + q x + 1

. Form a quadratic polynomial whose zeroes are

\frac{2}{α}

and

\frac{2}{β}

Q53.LA4 marks(2020)Medium

Obtain other zeroes of the polynomial

f (x) = 2 x^{4} + 3 x^{3} - 5 x^{2} - 9 x - 3

if two of its zeroes are

3

and

- 3

Q54.LA4 marks(2020)Medium

Without actually calculating the zeroes, form a quadratic polynomial whose zeroes are reciprocals of the zeroes of the polynomial

5 x^{2} + 2 x - 3

Q55.LA4 marks(2020)Medium

Obtain other zeroes of the polynomial

P (x) = 2 x^{4} - x^{3} - 11 x^{2} + 5 x + 5

if two of its zeroes are

5

and

- 5

Q56.LA4 marks(2020)Hard

What minimum must be added to

2 x^{3} - 3 x^{2} + 6 x + 7

so that the resulting polynomial will be divisible by

x^{2} - 4 x + 8

Q57.Case-Study4 marks(2024)Medium

A ball is thrown in the air so that

t

seconds after it is thrown, its height

h

metre above its starting point is given by the polynomial

h = 25 t - 5 t^{2}

.

$Graph of polynomial h(t) showing parabola$

Observe the graph of the polynomial and answer the following questions:
(i) Write zeroes of the given polynomial.
(ii) Find the maximum height achieved by ball.
(iii) (a) After throwing upward, how much time did the ball take to reach the height of

30

m?

**OR**

(iii) (b) Find the two different values of

t

when the height of the ball was

20

Q58.Case-Study4 marks(2024)Medium

A ball is thrown in the air so that

t

seconds after it is thrown, its height

h

metre above its starting point is given by the polynomial

h = 25 t - 5 t^{2}

t

when the height of the ball was

20

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