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Class 10 Maths Previous Year Questions - Polynomials

Previous Year Questions 2025

Q1: Zeroes of the polynomial p(y) Class 10 Maths Previous Year Questions - Polynomials    (2025)
(a) Class 10 Maths Previous Year Questions - Polynomials
(b) Class 10 Maths Previous Year Questions - Polynomials
(c) Class 10 Maths Previous Year Questions - Polynomials
(d) Class 10 Maths Previous Year Questions - Polynomials

Class 10 Maths Previous Year Questions - PolynomialsView Answer  Class 10 Maths Previous Year Questions - Polynomials

Ans: (d)
The given polynomial is p(y) Class 10 Maths Previous Year Questions - Polynomials
Class 10 Maths Previous Year Questions - Polynomials
For zeroes of the polynomial, put p(y) = 0 
Class 10 Maths Previous Year Questions - Polynomials

Q2: Zeroes of the polynomial p(x) = x2 - 3√2x + 4 are:     (2025)
(a) 2,√2 
(b) 2√2, √2 
(c) 4√2, -√2 
(d) √2, 2 

Class 10 Maths Previous Year Questions - PolynomialsView Answer  Class 10 Maths Previous Year Questions - Polynomials

Ans: (b)
We have, p(x) = x2 - 3√2x + 4 
⇒ p(x) = x2 - 2√2x -√2x + 4 
= x(x - 2√2) - √2(x - 2√2) 
= (x - 2√2)(x - √2) 
For zeroes of the polynomial, put p(x) = 0 
⇒ (x - 2√2)(x -√2) = 0 
∴ x = 2√2, √2 are zeroes of p(x).

Q3: Find the zeros of the polynomial Class 10 Maths Previous Year Questions - Polynomials    (2025)

Class 10 Maths Previous Year Questions - PolynomialsView Answer  Class 10 Maths Previous Year Questions - Polynomials

Ans: 
Class 10 Maths Previous Year Questions - Polynomials
Class 10 Maths Previous Year Questions - Polynomials
⇒ 3x - 2 = 0 ⇒ x = 2/3 and x + 2 = 0 ⇒ x = -2 

x = -2 and x = 2/3 are zeroes of the given polynomial.

Q4: Two polynomials are shown in the graph below. The number of distinct zeroes of both the polynomials is:     (2025)

Class 10 Maths Previous Year Questions - Polynomials(a) 3
(b) 5
(c) 2
(d) 4

Class 10 Maths Previous Year Questions - PolynomialsView Answer  Class 10 Maths Previous Year Questions - Polynomials

Ans: (c)
Since both polynomials cut the x-axis at two distinct points each, the total number of distinct zeroes of both the polynomials combined is 2.

Q5: If α and β are the zeroes of the polynomial p(x) = x2 - ax - b, then the value of (α + β + αβ) is equal to:     (2025)
(a) a + b 
(c) a- b 
(b) -a - b 
(d) -a + b

Class 10 Maths Previous Year Questions - PolynomialsView Answer  Class 10 Maths Previous Year Questions - Polynomials

Ans: (c)
Given polynomial is p(x) = x2 - ax - b, and α and β are zeroes of p(x) 
∴ Sum of zeroes = α + β =a 
Product of zeroes = αβ = - b  
Now, α + β + αβ = a - b 
Concept Applied
If α and β are the zeroes of quadratic polynomial p(x) = ax2 + bx + c, then α + β = -b/a, αβ = c/a.

Q6: If α and β are zeroes of the polynomial p(x) = kx2 - 30x + 45k and α +β = αβ, then the value of 'k' is:     (2025)
(a) Class 10 Maths Previous Year Questions - Polynomials
(b) Class 10 Maths Previous Year Questions - Polynomials
(c) 3/2
(d) 2/3

Class 10 Maths Previous Year Questions - PolynomialsView Answer  Class 10 Maths Previous Year Questions - Polynomials

Ans: (d)
Given, α and β are zeroes of the polynomial p(x) = kx2 - 30x + 45k 
Here, a = k, b = -30, c = 45 k 
Class 10 Maths Previous Year Questions - Polynomials
∴ α + β = αβ     [Given]
Class 10 Maths Previous Year Questions - Polynomials

Q7: If α and β are the zeroes of the polynomial 3x2 + 6x + k such that Class 10 Maths Previous Year Questions - Polynomialsthen the value of k is:     (2025)
(a) -8 
(b) 8 
(c) -4 
(d) 4

Class 10 Maths Previous Year Questions - PolynomialsView Answer  Class 10 Maths Previous Year Questions - Polynomials

Ans: (d)
Compare 3x2 + 6x + k with ax2 + bx + c, we get a = 3, b = 6 and c = k 
Class 10 Maths Previous Year Questions - Polynomials

Q8: If the zeroes of the polynomialClass 10 Maths Previous Year Questions - Polynomialsare reciprocals of each other, then the value of bis     (2025)
(a) 2 
(b) 1/2
(c) -2
(d) Class 10 Maths Previous Year Questions - Polynomials

Class 10 Maths Previous Year Questions - PolynomialsView Answer  Class 10 Maths Previous Year Questions - Polynomials

Ans: (a)
Let α and β be the zeroes of the given polynomial Class 10 Maths Previous Year Questions - Polynomials
Class 10 Maths Previous Year Questions - Polynomials

Q9:  If the sum of the zeroes of the polynomial p(x) = (p + 1)x2 + (2p + 3) x + (3p + 4) is - 1, then find the value of 'p'.     (2025)

Class 10 Maths Previous Year Questions - PolynomialsView Answer  Class 10 Maths Previous Year Questions - Polynomials

Ans: The given polynomial is  p(x) = (p + 1)x2 + (2p + 3)x + (3p + 4) 
Let α and β are zeroes of given polynomial 
Class 10 Maths Previous Year Questions - Polynomials
⇒ p + 1 = 2p + 3 
⇒ p = -2 
Hence, the value of p is - 2.

Q10: If α and β are zeroes of the polynomial p(x) = x2 - 2x - 1, then find the value of Class 10 Maths Previous Year Questions - Polynomials    (2025)

Class 10 Maths Previous Year Questions - PolynomialsView Answer  Class 10 Maths Previous Year Questions - Polynomials

Ans: The given polynomial is p(x) = x2 - 2x - 1; 
If α and β are zeroes of given polynomial 
Class 10 Maths Previous Year Questions - Polynomials

Q11: If 'α' and 'β' are the zeroes of the polynomial p(y) = y2 - 5y+ 3, then find the value of α4β3 + α3β4 .     (2025)

Class 10 Maths Previous Year Questions - PolynomialsView Answer  Class 10 Maths Previous Year Questions - Polynomials

Ans: 
We have p(y) = y2 - 5y + 3 
Let α and β be zeroes of p(y). 
Given, sum of zeroes= α + β = 5 
Product of zeroes = αβ = 3 
α⁴β³ + α³β⁴ 
= α³β³ (α + β) 
= (αβ)³ (α + β) 
= (3)³ (5) = 27 × 5 = 135

Q12: If the zeroes of the polynomial x2 + ax + b are in the ratio 3 : 4, then prove that 12a2 = 49b.     (2025)

Class 10 Maths Previous Year Questions - PolynomialsView Answer  Class 10 Maths Previous Year Questions - Polynomials

Ans: Let the zeroes of the polynomial x2 + ax + b be 3x and 4x. 
Class 10 Maths Previous Year Questions - Polynomials
⇒ 49b = 12a2. Hence proved. 

Q13: Find the zeroes of the polynomial p(x) = 3x2 - 4x - 4. Hence, write a polynomial whose each of the zeroes is 2 more than the zeroes of p(x).     (2025)

Class 10 Maths Previous Year Questions - PolynomialsView Answer  Class 10 Maths Previous Year Questions - Polynomials

Ans: The polynomial is p(x) = 3x2 - 4x - 4. 
The zeroes are given by p(x) = 0 
⇒ 3x2 - 4x - 4 = 0  
⇒ 3x2 - 6x + 2x - 4 = 0 
⇒ 3x(x - 2) + 2(x - 2) = 0  
⇒ (x - 2)(3x + 2) = 0 
⇒ x - 2 = 0 or 3x + 2 = 0 
⇒  x = 2 and -2/3
Thus, zeroes of new polynomial are 
Class 10 Maths Previous Year Questions - Polynomials
Hence, new polynomial is 
Class 10 Maths Previous Year Questions - Polynomials
Taking a= 3, r(x) = 3x2 - 16x + 16 
Thus, r(x) = 3x2 - 16x + 16 is the new polynomial with zeroes x = 4 and x = 4/3.

Previous Year Questions 2024

Q1: What should be added from the polynomial x2 – 5x + 4, so that 3 is the zero of the resulting polynomial? (2024)
(a) 1
(b) 2
(c) 4
(d) 5

Class 10 Maths Previous Year Questions - PolynomialsView Answer  Class 10 Maths Previous Year Questions - Polynomials

Ans: (b)
Let, f(x) = x2 – 5x + 4 
Let p should be added to f(x) then 3 becomes zero of polynomial.
So, f(3) + p = 0 
⇒ (3)2 – 5 × (3) + 4 + p = 0 
⇒ 9 + 4 – 15 + p = 0 
⇒ – 2 + p = 0 
⇒ p = 2 

So, 2 should be added.

Q2: Find the zeroes of the quadratic polynomial x2 – 15 and verify the relationship between the zeroes and the coefficients of the polynomial.   (2024)

Class 10 Maths Previous Year Questions - PolynomialsView Answer  Class 10 Maths Previous Year Questions - Polynomials

Ans:
x– 15 = 0
x2 = 15
x = ± √15
Zeroes will be  α = √15 , β = – √15
Verification: Given polynomial is x– 15
On comparing above polynomial with
ax2 + bx + c, we have
a = 1, b = 0, c = –15
sum of zeros = α + β 
Class 10 Maths Previous Year Questions - Polynomials
Product of zeros = αβ 
Class 10 Maths Previous Year Questions - Polynomials
Hence, verified.

Previous Year Questions 2023


Q1: The graph of y = p(x) is given, for a polynomial p(x). The number of zeroes of p(x) from the graph is  (2023)

Class 10 Maths Previous Year Questions - Polynomials(a) 3
(b) 1
(c) 2
(d) 0

Class 10 Maths Previous Year Questions - PolynomialsView Answer  Class 10 Maths Previous Year Questions - Polynomials

Ans: (b)
 Here, y = p(x) touches the x-axis at one point
So, number of zeros is one.

Q2: If α, β are the zeroes of a polynomial p(x) = x2 + x - 1, then 1/α + 1/β equals to (2023)
(a) 1
(b) 2
(c) -1
(d) -1/2

Class 10 Maths Previous Year Questions - PolynomialsView Answer  Class 10 Maths Previous Year Questions - Polynomials

Ans: (a)

The polynomial is p(x) = x2 + x - 1.

Step 1: The relationships between the zeroes and coefficients:

Sum of zeroes (α + β): - ba = - 11 = -1

Product of zeroes (αβ): ca = -11 = -1

Step 2: Simplify 1α + 1β:

1α + 1β = α + βαβ

Substitute the values:

α + βαβ = -1-1 = 1

Final Answer: (a) 1

Q3: If α, β are the zeroes of a polynomial p(x) = x2 - 1,  then the value of (α + β) is  (2023)
(a) 1
(b) 2
(c) -1
(d) 0 

Class 10 Maths Previous Year Questions - PolynomialsView Answer  Class 10 Maths Previous Year Questions - Polynomials

Ans: (d)

The polynomial is p(x) = x2 - 1.

Step 1:  Sum of zeroes (α + β): - ba = - 01

Step 2: Simplify:

- 01 = 0

Final Answer: (d) 0

Q4: If α, β are the zeroes of a polynomial p(x) = 4x2 - 3x - 7, then (1/α + 1/β) is equal to  (2023)
(a) 7/3
(b) -7/3
(c) 3/7
(d) -3/7

Class 10 Maths Previous Year Questions - PolynomialsView Answer  Class 10 Maths Previous Year Questions - Polynomials

Ans: (d) 

The polynomial is p(x) = 4x2 - 3x - 7.

Step 1: calculating sum and product of zeroes 

Sum of zeroes (α + β): - ba = - (-3)4  = 34

Product of zeroes (αβ):  ca =  -74

Step 2: Simplify 1α + 1β:

α + βαβ  =  34-74  =  -37

Final Answer: (d) - 37

Q5: If one zero of the polynomial p(x) = 6x2 + 37x – (k – 2) is reciprocal of the other, then find the value of k. (CBSE 2023)

Class 10 Maths Previous Year Questions - PolynomialsView Answer  Class 10 Maths Previous Year Questions - Polynomials

Ans: We have,

The polynomial is p(x) = 6x2 + 37x - (k - 2).

Step 1: The relationship between the product of zeroes and coefficients:

Product of zeroes (αβ) =  ca = -(k - 2)6

It is given that αβ = 1. Substitute this:

-(k - 2)6 = 1

Step 2: Solve for k:

Multiply both sides by 6:

-(k - 2) = 6

Simplify:

k - 2 = -6

k = -4

Final Answer: k = - 4

Previous Year Questions 2022

Q1: If one of the zeroes of a quadratic polynomial ( k - 1 )x+ kx + 1 is - 3, then the value of k is   (2022)
(a) 4/3
(b) -4/3
(c) 2/3
(d) -2/3

Class 10 Maths Previous Year Questions - PolynomialsView Answer  Class 10 Maths Previous Year Questions - Polynomials

Ans: (a)
 Given that -3 is a zero of quadratic polynomial (k - 1)2+ kx + 1.
⇒ Putting x = -3 in above equation, we get
∴ (k - 1) (-3)2 + k(-3) +1 = 0
⇒ 9k - 9 - 3k + 1 = 0 ⇒ 6k - 8 = 0
⇒ k = 8/6
⇒ k = 4/3

Q2: If the path traced by the car has zeroes at -1 and 2, then it is given by   (2022)
(a) x2 + x + 2
(b) x2 - x + 2
(c) x- x - 2
(d) x2 + x - 2

Class 10 Maths Previous Year Questions - PolynomialsView Answer  Class 10 Maths Previous Year Questions - Polynomials

Ans: (c)

The zeroes of the polynomial are -1 and 2.

Step 1: The polynomial with given zeroes is:

p(x) = a(x - α)(x - β)

Substitute the zeroes α = -1 and β = 2:

p(x) = a(x - (-1))(x - 2) = p(x) = a(x + 1)(x - 2)

Step 2: Expand the polynomial:

p(x) = a[(x)(x) + (x)(-2) + (1)(x) + (1)(-2)]

p(x) = a[x2 - x - 2]

Step 3: Assuming a = 1:

p(x) = x2 - x - 2

Final Answer: (c) x2 - x - 2

Q3: The number of zeroes of the polynomial representing the whole curve, is   (2022)
Class 10 Maths Previous Year Questions - Polynomials(a) 4
(b) 3
(c) 2
(d) 1 

Class 10 Maths Previous Year Questions - PolynomialsView Answer  Class 10 Maths Previous Year Questions - Polynomials

Ans: (a)
 Given curve cuts the x-axis at four distinct points.
So, number of zeroes will be 4 .

Q4: The distance between C and G is   (2022)
Class 10 Maths Previous Year Questions - Polynomials
(a) 4 units
(b) 6 units
(c) 8 units
(d) 7 units

Class 10 Maths Previous Year Questions - PolynomialsView Answer  Class 10 Maths Previous Year Questions - Polynomials

Ans: (b)
The distance between point C and G is 6 units.

Q5: The quadratic polynomial, the sum of whose zeroes is -5 and their product is 6.   (2022)
(a) x2 + 5x + 6
(b) x2 - 5x + 6
(c) x2 - 5 x - 6
(d) - x2 + 5x + 6 

Class 10 Maths Previous Year Questions - PolynomialsView Answer  Class 10 Maths Previous Year Questions - Polynomials

Ans: (a)
 Let α, β be the zeroes of required polynomial p(x).
Given, α + β=-5 and α.β=6
p(x) = x- (Sum of zeros)x + (Product of zeros)
∴ p(x)=k[x- (-5)x + 6] = k[x+ 5x + 6]  
Thus, one of the polynomial which satisfy the given condition is x2+ 5x + 6

Previous Year Questions 2021

Q1: If one zero of the quadratic polynomial x2 + 3x + k is 2 then find the value of k.   (2021)

Class 10 Maths Previous Year Questions - PolynomialsView Answer  Class 10 Maths Previous Year Questions - Polynomials

Ans: Given, polynomial is f(x) =x2 + 3x + k
Since, 2 is zero of the polynomial f(x).
∴ f(2) = 0
⇒ f(2) =(2)+ 3 x 2 + k
⇒  4 + 6 + k = 0
⇒ k = -10

Previous Year Questions 2020


Q1: The degree of polynomial having zeroes -3 and 4 only is   (2020)
(a) 2
(b) 1
(c) more than 3
(d) 3 

Class 10 Maths Previous Year Questions - PolynomialsView Answer  Class 10 Maths Previous Year Questions - Polynomials

Ans: (a)
 Since, the polynomial has two zeroes only. So. the degree of the polynomial is 2.

Q2: If one of the zeroes of the quadratic polynomial x2 + 3x + k is 2. then the value of k is   (2020)
(a) 10
(b) - 10
(c) -7
(d) -2

Class 10 Maths Previous Year Questions - PolynomialsView Answer  Class 10 Maths Previous Year Questions - Polynomials

Ans: (b)
 Given, 2 is a zero of the polynomial
p(x) = x2 + 3x + k
∴ p (2) = 0
⇒ (2)2 + 3(2) + k = 0
⇒ 4 + 6 + k = 0 
⇒ 10 + k = 0
⇒ k= -10

Q3: The quadratic polynomial, the sum of whose zeroes is -5 and their product is 6 is ________.  (2020)
(a) x2 + 5x + 6
(b) x2 - 5x + 6
(c) x2- 5x - 6
(d) -x2 + 5x + 6

Class 10 Maths Previous Year Questions - PolynomialsView Answer  Class 10 Maths Previous Year Questions - Polynomials

Ans: (a)
 Let α, β be the zeroes of required polynomial p(x)
Given, α+ β = -5 and αβ = 6
p(x) = k[x2 - (- 5)x + 6]
= k[x2 + 5x + 6]
Thus, one of the polynomial which satisfy the given condition is x2 + 5x + 6.

Q4: Form a quadratic polynomial, the sum and product of whose zeroes are (-3) and 2 respectively.   (CBSE 2020) 

Class 10 Maths Previous Year Questions - PolynomialsView Answer  Class 10 Maths Previous Year Questions - Polynomials

Ans: Let α, β be the zeroes of required polynomial Given, α + β = -3 and αβ = 2
∴ p(x) = k[x2 - (-3)x + 2] = k(x2 + 3x + 2)
For k = 1 , p (x) = x2 + 3x + 2
Hence, one of the polynomial which satisfy the given condition is x2 + 3x + 2.

Q5: The zeroes of the polynomial x2 – 3x – m(m + 3) are: 
(a) m, m + 3 
(b) –m, m + 3 
(c) m, – (m + 3) 
(d) –m, – (m + 3) (CBSE 2020)

Class 10 Maths Previous Year Questions - PolynomialsView Answer  Class 10 Maths Previous Year Questions - Polynomials

Ans: (b)
Given:

x^2 - 3x - m(m + 3) = 0x2 − 3x − m(m + 3) = 0
Let's find the zeroes by applying the quadratic formula:
Class 10 Maths Previous Year Questions - Polynomials

Substitute into the formula:

Class 10 Maths Previous Year Questions - Polynomials

Simplify under the square root:

Class 10 Maths Previous Year Questions - Polynomials

Taking the square root:

Class 10 Maths Previous Year Questions - Polynomials

So, the zeroes are –m and m + 3.
Thus, the correct answer is (b) –m, m + 3.

Previous Year Questions 2019

Q1: Find the value of k such that the polynomial x2 - (k + 6)x + 2(2k - 1) has the sum of its zeroes equal to half of its product.    [Year 2019, 3 Marks] 

Class 10 Maths Previous Year Questions - PolynomialsView Answer  Class 10 Maths Previous Year Questions - Polynomials

Ans: 7
The given polynomial is x2 -(k + 6)x + 2(2k - 1)
According to the question
Sum of zeroes = 1/2(Product of Zeroes ):
⇒ k + 6 = 1/2 x 2 (2k - 1)
⇒ k + 6 = 2k - 1
⇒ k = 7

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FAQs on Class 10 Maths Previous Year Questions - Polynomials

1. What are polynomials in mathematics?
Ans. Polynomials are algebraic expressions that consist of variables and coefficients, combined using addition, subtraction, multiplication, and non-negative integer exponents. A polynomial can have one or more terms, where each term is a product of a coefficient and a variable raised to a whole number power.
2. How do you identify the degree of a polynomial?
Ans. The degree of a polynomial is determined by the highest exponent of the variable in the expression. For example, in the polynomial \(3x^4 + 2x^2 - 5\), the degree is 4 because the highest exponent of \(x\) is 4.
3. Can you give examples of different types of polynomials?
Ans. Yes! Some examples of different types of polynomials include: - Monomial: \(5x\) (one term) - Binomial: \(3x^2 + 4\) (two terms) - Trinomial: \(x^2 - 3x + 2\) (three terms) - Polynomial with more than three terms: \(2x^3 - x + 7 + 4x^2\)
4. What are the common operations performed on polynomials?
Ans. Common operations on polynomials include addition, subtraction, multiplication, and division. For example, to add two polynomials, you combine like terms, while for multiplication, you apply the distributive property to multiply each term in one polynomial by each term in the other.
5. Why is it important to study polynomials in school?
Ans. Studying polynomials is important because they are fundamental to algebra and are used in various areas of mathematics, such as calculus, statistics, and applied mathematics. Understanding polynomials helps students solve equations, model real-world situations, and develop critical problem-solving skills.
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