CHAPTER 2 : POLYNOMIALS
CBSE Class 10 Mathematics — PYQs 2026
BASIC QUESTION PAPER 2026
SECTION A : ASSERTION–REASON QUESTIONS (1 Mark Each)
1.
Assertion (A):
Every quadratic equation has two real roots.
Reason (R):
A quadratic polynomial can have at most two zeroes.
[Sets: 430/3/1 Q20, 430/3/2 Q19, 430/3/3 Q20]
SECTION B : SHORT ANSWER QUESTIONS (3 Marks Each)
TYPE A : CONSTRUCTING A QUADRATIC POLYNOMIAL
2.
Find a quadratic polynomial whose sum and product of zeroes are
\(0\) and \(-9\), respectively. Also, find the zeroes of the polynomial so obtained.
[Set 430/1/1 Q27]
3.
Find a quadratic polynomial, sum and product of whose zeroes are
\(5\) and \(-6\), respectively. Also, find the zeroes of the polynomial so obtained.
[Set 430/1/2 Q31]
4.
Determine a quadratic polynomial, sum and product of whose zeroes are
\(-10\) and \(24\), respectively. Also, determine the zeroes of the polynomial so obtained.
[Set 430/1/3 Q29]
TYPE B : FINDING ZEROES AND VERIFYING THE RELATIONSHIP
5.
Find the zeroes of the polynomial
\(p(x)=3x^2-2x-1\)
and verify the relationship between the zeroes and the coefficients.
[Set 430/2/1 Q27]
6.
Find the zeroes of the polynomial
\(p(x)=2x^2+5x+2\)
and verify the relationship between the zeroes and its coefficients.
[Set 430/2/2 Q31]
7.
Find the zeroes of the polynomial
\(q(x)=6x^2-5x-1\)
and verify the relationship between the zeroes and its coefficients.
[Set 430/2/3 Q29]
8.
Find the zeroes of the polynomial
\(9s^2-6s+1\)
and verify the relationship between the zeroes and the coefficients.
[Set 430/3/1 Q27]
9.
Find the zeroes of the polynomial
\(4x^2+4x+1\)
and verify the relationship between the zeroes and the coefficients.
[Set 430/3/2 Q31]
10.
Find the zeroes of the polynomial
\(25a^2-10a+1\)
and verify the relationship between the zeroes and the coefficients.
[Set 430/3/3 Q29]
STANDARD QUESTION PAPER 2026
SECTION A : MULTIPLE CHOICE QUESTIONS (1 Mark Each)
GRAPHS AND NUMBER OF ZEROES
11.
The number of distinct zeroes of \(y=f(x)\) is:
(A) \(0\)
(B) \(1\)
(C) \(2\)
(D) \(4\)
[Set 30/1/1 Q4, Set 30/1/2 Q5, Set 30/4/1 Q5, Set 30/4/2 Q12]
12.
The number of distinct zeroes of \(y=f(x)\) is:
(A) \(0\)
(B) \(1\)
(C) \(2\)
(D) \(3\)
[Set 30/2/1 Q4, Set 30/2/2 Q1]
13.
The zeroes of the polynomial \(p(x)\) are:
(A) \(-2,0,2.5\)
(B) \(-2,2.5\)
(C) \(0,4\)
(D) \(-2,0\)
[Set 30/4/3 Q7]
14.
How many zeroes does
\(p(x)=(x-2)(x+3)\)
have?
(A) Zero
(B) One
(C) Two
(D) Three
[Set 30/2/3 Q8]
ZEROES AND COEFFICIENT RELATIONSHIPS
15.
If \(\alpha\) and \(\beta\) are two zeroes of a polynomial
\(f(x)=px^2-2x+3p\)
and
\(\alpha+\beta=\alpha\beta\),
then find the value of \(p\).
(A) \(-\frac{2}{3}\)
(B) \(\frac{2}{3}\)
(C) \(\frac{1}{3}\)
(D) \(-\frac{1}{3}\)
[Set 30/2/1 Q5, Set 30/2/2 Q8, Set 30/2/3 Q10]
16.
A polynomial \(p(x)\), which has sum of its zeroes equal to their product, is:
(A) \(3x^2+2x+2\)
(B) \(3x^2-2x-3\)
(C) \(3x^2-2x+2\)
(D) \(x^2-3x+2\)
[Set 30/3/3 Q6]
FORMING POLYNOMIALS
17.
If the zeroes of a polynomial \(p(x)\) are \(-3\) and \(8\), then \(p(x)\) equals:
(A) \(x^2+5x-4\)
(B) \((x+3)(-x+8)\)
(C) \(a(x^2+5x-24)\)
(D) \(x^2-24\)
[Set 30/5/1 Q3, Set 30/5/3 Q15]
18.
The sum and product of zeroes of a quadratic polynomial \(p(x)\)
are \(-\frac{1}{3}\) and \(2\), respectively. The polynomial \(p(x)\) is:
(A) \(3x^2-x+6\)
(B) \(x^2+\frac{1}{3}x-2\)
(C) \(3x^2-x+2\)
(D) \(-3x^2-x-6\)
[Set 30/3/1 Q14, Set 30/3/2 Q6]
19.
If sum and product of zeroes of a polynomial are \(-3\) and \(-2\),
respectively, then a polynomial is:
(A) \(x^2-3x-2\)
(B) \(x^2+3x-2\)
(C) \(x^2+3x+2\)
(D) \(x^2-3x+2\)
[Set 30/5/2 Q11]
SECTION B : ASSERTION–REASON QUESTIONS (1 Mark Each)
20.
Assertion (A):
The polynomial \(p(y)=y^2+4y+3\) has two zeroes.
Reason (R):
A quadratic polynomial can have at most two zeroes.
[Set 30/1/1 Q20, Set 30/1/2 Q20, Set 30/1/3 Q19]
SECTION C : VERY SHORT ANSWER QUESTIONS (2 Marks Each)
EVALUATING EXPRESSIONS WITH ZEROES
21.
If \(\alpha,\beta\) are the zeroes of the polynomial
\(p(x)=x^2-3x-1\),
then find the value of
\[
\frac{1}{\alpha}+\frac{1}{\beta}.
\]
[Set 30/1/1 Q21]
22.
\(\alpha\) and \(\beta\) are the zeroes of the polynomial
\(5x^2-16x-10\).
Find the value of
\[
\frac{\alpha}{\beta}+\frac{\beta}{\alpha}.
\]
[Set 30/4/1 Q25, Set 30/4/3 Q22, Set 30/5/1 Q22, Set 30/5/3 Q22]
23.
\(\alpha,\beta\) are zeroes of the polynomial
\(p(x)=3x^2-6x-5\).
Find the value of
\[
\frac{1}{\alpha^2}+\frac{1}{\beta^2}.
\]
[Set 30/4/2 Q24, Set 30/5/2 Q24]
24.
If \(\alpha,\beta\) are the zeroes of the quadratic polynomial
\(px^2+qx+r\),
then find the value of
\[
\alpha^3\beta+\beta^3\alpha.
\]
[Set 30/2/1 Q21, Set 30/2/3 Q23]
FINDING ZEROES AND COEFFICIENTS
25.
Find the zeroes of the quadratic polynomial
\(x^2+7x+10\),
and verify the relationship between the zeroes and its coefficients.
[Set 30/1/2 Q25]
26.
Find the value of \(p\), for which one zero of the quadratic polynomial
\(px^2-14x+8\)
is \(6\) times the other.
[Set 30/2/2 Q24]
27.
Find a quadratic polynomial whose zeroes are
\(5-2\sqrt{3}\) and \(5+2\sqrt{3}\).
[Set 30/1/3 Q24]
SECTION D : CASE STUDY QUESTIONS (4 Marks Each)
28.
The Theatre Drama Backdrop:
The shape of the curve shown can be represented by the polynomial
\[
p(x)=-x^2+2x+8.
\]
(i)
Determine the height of the arch. (1 Mark)
(ii)
Find the zeroes of the polynomial \(p(x)\).
Which points on the graph represent the zeroes?
(2 Marks)
OR
(ii)
Find the span of the arch on the stage floor.
(2 Marks)
(iii)
Write the coordinates of the point of intersection of the curve
with the \(y\)-axis.
(1 Mark)
[Set 30/3/1 Q36, Set 30/3/2 Q38]
29.
The Railway Bridge Arch:
The parabolic curve is represented by the polynomial
\[
p(x)=-0.0025x^2-0.025x+136.
\]
(i)
Write the coordinates of point \(A\).
(1 Mark)
(ii)
Find the span of the arch.
(1 Mark)
(iii)(a)
Write the zeroes of the polynomial using the diagram
and verify the relationship between the sum of zeroes
and the coefficients of the polynomial.
(2 Marks)
OR
(iii)(b)
Find the values of \(p(x)\) at \(x=100\) and \(x=-100\).
Are they same?
(2 Marks)
[Set 30/5/1 Q36, Set 30/5/2 Q37, Set 30/5/3 Q37]