CHAPTER 4 : QUADRATIC EQUATIONS
CBSE Class 10 Mathematics — PYQs 2026 | Basic + Standard
BASIC QUESTION PAPER 2026
SECTION A : MULTIPLE CHOICE QUESTIONS (1 Mark Each)
A. Finding the Discriminant
1.
The discriminant of the quadratic equation
x² − 3x − 2 = 0 is:
(A) 1
(B) 17
(C) √17
(D) −√17
[Set 430/1/1]
2.
The discriminant of the quadratic equation
x² − 5x + 6 = 0 is:
(A) 1
(B) −1
(C) 49
(D) 7
[Set 430/1/2]
3.
The discriminant of the quadratic equation
2x² − 3x − 5 = 0 is:
(A) −31
(B) 49
(C) 7
(D) √−31
[Set 430/1/3]
B. Standard Form Formulation
4.
The equation
x + 1/x = 3
(
x ≠ 0) is expressed as a quadratic equation in the form
ax² + bx + c = 0.
The value of
a − b + c is:
(A) 5
(B) 2
(C) 1
(D) −1
[Sets: 430/1/1, 430/1/2, 430/1/3]
5.
If
x = √x
(Note: Source text contains a typo; the original expression should be verified from the source paper),
x ≠ 0, is expressed as a quadratic equation in the form
ax² + bx + c = 0,
then the value of
a + b + c is:
(A) 0
(B) 1
(C) 2
(D) 3
[Sets: 430/3/1, 430/3/2, 430/3/3]
C. Identifying Quadratic Equations
6.
Which of the following equations is a quadratic equation?
(A) x² = (x + 1)²
(B) (x − 1)(x + 2) = 2x + 1
(C) (x + 2)³ = 2x(x² − 1)
(D) √x = x²
[Sets: 430/2/1, 430/2/2, 430/2/3]
D. Condition for Equal Roots
7.
Find the value of
a for which
ax² + 3x + 1 = 0
has real and equal roots.
(A) 4/9
(B) 9/4
(C) 3/2
(D) 2/3
[Sets: 430/2/1, 430/2/2]
8.
Find the value of
p for which
px² + 4x + p = 0
has real and equal roots.
(A) 4
(B) −4
(C) 2
(D) 0
[Set 430/2/3]
SECTION B : ASSERTION–REASON QUESTIONS (1 Mark Each)
9.
Assertion (A):
Every quadratic equation has two real roots.
Reason (R):
A quadratic polynomial can have at most two zeroes.
[Sets: 430/3/1, 430/3/2, 430/3/3]
SECTION C : LONG ANSWER QUESTIONS (5 Marks Each)
Type A : Number Word Problems
10.
The difference of the squares of two positive numbers is
180.
The square of the smaller number is
8 times the greater number.
Find the two numbers.
[Sets: 430/1/1, 430/1/2, 430/1/3]
11.
Find two consecutive odd integers, sum of whose squares is
290.
[Sets: 430/3/1, 430/3/2, 430/3/3]
Type B : Geometry Word Problems
12.
The sum of areas of two squares is
2650 cm².
If the sum of their perimeters is
280 cm,
find the sides of the two given squares.
[Sets: 430/2/1, 430/2/2, 430/2/3]
13.
A charity trust decides to build a rectangular hall having an area of
300 m².
The length of the hall is one metre more than twice its width.
Find the length and breadth of the hall.
[Sets: 430/3/1, 430/3/2, 430/3/3]
Type C : Parameter Calculation & Root Finding
14.
Find the value(s) of
k for which the equation
2x² + kx + 3 = 0
has real and equal roots.
Hence, find the roots of the equations so obtained.
[Sets: 430/1/1, 430/1/2, 430/1/3]
Type D : Solving Algebraic Equations
15.
Express the equation
1/x − 1/(x − 2) = 3
(
x ≠ 0, 2)
as a quadratic equation in standard form.
Hence, find the roots of the quadratic equation so obtained.
[Sets: 430/2/1, 430/2/2, 430/2/3]
STANDARD QUESTION PAPER 2026
SECTION A : MULTIPLE CHOICE QUESTIONS (1 Mark Each)
A. Condition for Real and Equal Roots
1.
If the quadratic equation
9x² + 8kx + 16 = 0
has real and equal roots, then the value of
k is:
(A) 3
(B) −3
(C) −4
(D) 2√3
[Set 30/4/1 Q1; Set 30/4/3 Q14]
2.
If roots of the quadratic equation
x² − k√3x + 2 = 0
are real and equal, then the value of
k is:
(A) −2
(B) √(3/8)
(C) 1
(D) 2
[Set 30/4/2 Q11]
3.
The value of
k for which the equation
kx² − 6x − 4 = 0
has real and equal roots, is:
(A) 9/4
(B) −4
(C) −9/4
(D) −2
[Set 30/5/1 Q1; Set 30/5/2 Q18]
4.
The value of
m for which the quadratic equation
3x² − 7x + m = 0
has real and equal roots, is:
(A) 7
(B) 49/12
(C) 49/3
(D) 4
[Set 30/5/3 Q14]
5.
If the roots of the quadratic equation
√3x² − kx + 2√3 = 0
are real and equal, then the value(s) of
k is/are:
(A) ±√24
(B) 0
(C) 4
(D) −5
[Set 30/3/1 Q10; Set 30/3/2 Q4; Set 30/3/3 Q18]
B. Solving and Rational Roots
6.
The roots of the quadratic equation
(x − 1)² = 16
are:
(A) 5, 3
(B) 4, −4
(C) 5, −3
(D) −5, 3
[Set 30/3/1 Q1; Set 30/3/3 Q17]
7.
The roots of the quadratic equation
4x² − (a − 1)² = 0
are:
(A) a − 1, a + 1
(B) (a − 1)/2, (−a + 1)/2
(C) (a − 1)/2, (−a − 1)/2
(D) ±(a − 1)
[Set 30/3/2 Q7]
8.
The value of
p for which roots of the quadratic equation
x² − px + 6 = 0
are rational, is:
(A) 1
(B) −5
(C) 25
(D) 5
[Set 30/5/1 Q9; Set 30/5/3 Q16]
SECTION B : VERY SHORT ANSWER QUESTIONS (2 Marks Each)
9.
Verify that roots of the quadratic equation
(p − q)x² + (q − r)x + (r − p) = 0
are equal when
q + r = 2p.
[Set 30/4/1 Q21; Set 30/4/2 Q23; Set 30/4/3 Q24]
SECTION C : SHORT ANSWER QUESTIONS (3 Marks Each)
10.
Find two consecutive negative integers, sum of whose squares is
481.
[Set 30/3/3 Q31]
SECTION D : LONG ANSWER QUESTIONS (5 Marks Each)
A. Distance, Speed and Time Problems
11.
A person on tour has
₹5,400
for his expenses. If he extends his tour by 5 days, he has to cut down his daily expenses by
₹180.
Find the original duration of the tour and daily expense.
[Set 30/4/1 Q32(a); Set 30/4/2 Q34(a); Set 30/4/3 Q33(a); Set 30/2/1 Q33(a)]
12.
A faster train takes one hour less than a slower train for a journey of
200 km.
If the speed of the slower train is
10 km/hr
less than that of the faster train, find the speeds of the two trains.
[Set 30/1/1 Q33(a); Set 30/1/2 Q32(a); Set 30/1/3 Q35(a)]
13.
In a flight of
600 km,
an aircraft slowed down its speed due to bad weather.
Its average speed for the trip reduced by
200 km/h
from its usual speed and time of flight increased by
30 minutes.
Find the scheduled duration of the flight.
[Set 30/2/2 Q34(a)]
14.
Venkat can row a boat in still water at the speed of
12 km/h.
He ferries tourists
15 km
upstream and
18 km
downstream in 3 hours.
Find the speed of the stream.
[Set 30/5/1 Q32]
B. Work and Pipe Problems
15.
Two pipes are used to fill a swimming pool.
If the pipe of the larger diameter is used for 4 hours and the pipe of the smaller diameter for 9 hours,
only half of the pool can be filled.
Find how long it would take for each pipe to fill the pool, separately,
if the pipe of smaller diameter takes 10 hours more than the pipe of larger diameter to fill the pool.
[Set 30/2/2 Q34(b)]
16.
Two water taps together can fill a tank in
40/13 hours.
The tap of larger diameter takes 4 hours less than the smaller one to fill the tank separately.
Find the time in which each tap can separately fill the tank.
[Set 30/5/3 Q35]
C. Geometric and Commercial Word Problems
17.
The sum of the areas of two squares is
640 m².
If the difference in their perimeters is
64 m,
find the sides of the two squares.
[Set 30/1/1 Q33(b)]
18.
The area of a right-angled triangle is
600 cm².
If the base of the triangle exceeds the altitude by
10 cm,
find all the three dimensions of the triangle.
[Set 30/2/1 Q33(b)]
19.
By selling an article for
₹48,
a trader loses as much percent as half of the cost price of the article.
Calculate the cost price and loss amount of the article.
[Set 30/5/2 Q32]
20.
The total cost of certain piece of cloth was
₹2,100.
During special sale time, the shopkeeper offered 2 m extra cloth for free,
thus reducing the price of cloth per metre by
₹120.
What was the original per metre price of cloth and its length?
[Set 30/4/1 Q32(b); Set 30/4/2 Q34(b); Set 30/4/3 Q33(b)]