JEE Main 2026: Maths Sample Paper with Answer Key, Download PDF

Related: 

JEE Main 2026: Is 75% Rule Applicable This Year? NTA Clarifies the Criteria

JEE Main 2026: Best 60 Days Preparation Plan to Score 99 Percentile

Question Type

Total Questions in Section

Questions to be Attempted

Multiple Choice Questions (MCQs)

20

20 (Compulsory)

Numerical Value Questions (Non-MCQs)

10

5 (Any 5 to be attempted)

Total

30

25

SECTION A: Multiple Choice Questions: This section contains 20 multiple choice questions. Each question has 4 choices (1), (2), (3) and (4), out of which ONLY ONE is correct.

SL. No.

Questions

1.

Mean of first 15 numbers is 12 and variance is 14. Mean of next 15 numbers is 14 and variance is a. If variance of all 30 numbers is 13, then a is equal to 

(1) 12 

(2) 14 

(3) 10 

(4) 3

2.

If a≠b and are purely real, z  complex number, Re(az2 + bz) = a and Re(bz2 + az) = b then number of value of z possible is 

(1) 0 

(2) 1 

(3) 2 

(4) 3

3.

A rectangle is drawn by lines x = 0, x = 2, y = 0 and y = 5. Points A and B lie on coordinate axes. If line AB divides the area of rectangle in 4 : 1, then the locus of mid-point of AB is 

(1) Circle 

(2) Hyperbola 

(3) Ellipse 

(4) Straight line

4.

Sum of first 20 turns of the series 5, 11, 19, 29, 41, …… is 

(1) 3130 

(2) 3520 

(3) 2790 

(4) 1880

5.

3, 8, 13, ……, 373 are in arithmetic series. The sum of numbers not divisible by three is 

(1) 9310 

(2) 8340 

(3) 9525 

(4) 7325

6.

Let statement 1 : (2002)^2023 – (1919)^2002 is divisible by 8. Statement 2 : 13.13n – 12n – 13 is divisible by 144  ∀ n∈N, then 

(1) Statement-1 and statement-2 both are true 

(2) Only statement-1 is true 

(3) Only statement-2 is true 

(4) Neither statement-1 nor statement-2 are true

7.

From a square of side 30 cm the squares of side x cm is cut off to make a cuboid of maximum volume. The surface area of cuboid with open top is 

(1) 400 cm2 

(2) 464 cm2 

(3) 800 cm2 

(4) 900 cm2

8.

A parabola with focus (3, 0) and directrix x = –3. Points P and Q lie on the parabola and their ordinates are in the ratio 3 : 1. The point of intersection of tangents drawn at points P and Q lies on the parabola 

(1) y 2 = 16x 

(2) y 2 = 4x 

(3) y 2 = 8x 

(4) x 2 = 4y

9.

10.

If V is volume of parallelepiped whose three coterminous edges are 𝑎⃗, 𝑏⃗, 𝑐⃗, then volume of a parallelopiped whose coterminous edges are 𝑎⃗,𝑏⃗+𝑐⃗,𝑎⃗ 𝑏⃗+ 2𝑏⃗+3𝑐⃗ is (1) 6V 

(2) V 

(3) 2V 

(4) 3V

11.

If the order of matrix A is 3 × 3 and |A| = 2, then the value of |3adj (|3A|A^2)| is

(1) 3^10. 2^21

(2) 2^10.  3^21

(3) 2^12.  3^15

(4) 3^12.  2^15

12.

In probability distribution for discrete variables x = 0, 1, 2 … P(x = x) = k(x + 1).3–x . The probability of P(x>2) is equal to 

(1) 5 /18 

(2) 10 /18 

(3) 20/ 27 

(4) 7/ 27

13.

If x + y + z = 17 and x, y, z are non-negative integers, then find the number of integral solutions is 

(1) 136 

(2) 171 

(3) 90 

(4) 130

14.

Let f(x) = {1, 2, 3, 4, 5, 6, 7} the relation R = {(x, y)  ∈ A × A, x + y = 7} is

(1) Symmetric 

(2) Reflexive 

(3) Transitive 

(4) Equivalence

15.

If a 2 + (ar) 2 + (ar2 ) 2 = 33033, (a, r ∈ N), then the value of a + ar + ar2 is

(1) 148 

(2) 249 

(3) 230 

(4) 231

16.

Area of region enclosed by curve y = x 3 and its tangent at (–1, –1) 

(1) 4 

(2) 27 

(3) 4/ 27 

(4) 27/4

17.

If the lengths of the sides of a triangle are in A.P. and the greatest angle is double the smallest, then a ratio of lengths of the sides of this triangle is:

A. 5:9:13 

B. 5:6:7 

C. 3:4:5 

D. 4:5:6

18.

Find the number of integral values of x which satisfy the inequality x^2 – 10x + 19 < 6

(1) 5 

(2) 11 

(3) 7 

(4) 8

19.

The minimum number of times one has to toss a fair coin so that the probability of observing at least one head is at least 90% is:

A. 5 

B. 4 

C. 3 

D. 2

20.

The number of words with or without meaning can be formed from the word MATHEMATICS where C, S not come together is

(1) 9/8×10! 

(2) 1/8×10!

(3) 5/8×10!

(4) 1/2×10!

SECTION B: Numerical Value Type Questions: In Section B, attempt ANY FIVE questions. The answer to each question is a NUMERICAL VALUE.

21.

22.

Matrix A is 2 × 2 matrix and A2 = I, no elements of the matrix is zero, let sum of diagonal elements is a and det(A) = b, then the value of 3a 2 + b 2 is

23.

24.

All the letters of the word GTWENTY are written in all possible ways with or without meaning and these words are written as in a dictionary. The serial number of the word GTWENTY is___________.

25.

A group of students appeared in an examination of subjects - Mathematics, Physics & Chemistry. It was found that all students passed in at least one of the subjects, students passed in Mathematics, students passed in Physics, students passed in Chemistry, at most students passed in both Mathematics and Physics, at most students passed in both Physics and Chemistry, at most students passed in both

Mathematics and Chemistry. The maximum number of students passed in all the three subjects is _____.

26.

27.

The number of points of non-differentiability of the function f(x) = [4 + 13sinx] in (0,2𝜋) is ____.

28.

29.

A line with direction ratio 2, 1, 2 meets the lines x = y + 2 = z and x + 2 = 2y = 2z respectively at the point P and Q. If the length of the perpendicular from the point (1, 2, 12) to the line PQ is l, then l^2 is ___________.

30.

SECTION A

Sl. No.

Answers

1.

(3) 10

2.

(1) 0

3.

(2) Hyperbola

4.

(2) 3520

5.

(3) 9525

6.

(3) Only statement-2 is true

7.

(3) 800 cm2

8.

(1) y 2 = 16x

9.

(3)

10.

(2) V 

11.

(2) 2^10.  3^21

12.

(4) 7/ 27

13.

(2) 171

14.

(1) Symmetric

15.

(4) 231

16.

(4) 27/4

17.

D. 4:5:6

18.

(3) 7

19

B. 4

20.

(1) 9/8×10!

SECTION B

21.

2890

22.

1

23.

29

24.

553

25.

10

26.

202

27.

50

28.

5

29.

65

30.

12

JEE Main 2026: Maths Sample Paper with Answer Key, Download PDF