Previous Year Questions

JEE Main Maths PYQs (2018-2026)

Every previous year question across all chapters. Filter by chapter, year, or difficulty. Each question has a detailed solution.

241 PYQs · 20 chapters · 8 years

241 questions

Complex Numbers & Quadratic Equations

31 PYQs

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Q1JEE 2024EasyVisual Solution

z = 3 + 4i

, find

|z|

(A)

5

(B)

7

(C)

\sqrt{7}

(D)

25

Q2JEE 2024EasyVisual Solution

The conjugate of

2 - 3i

is:

(A)

-2 + 3i

(B)

2 + 3i

(C)

-2 - 3i

(D)

3 - 2i

Q3JEE 2024EasyVisual Solution

z = 1 + i

, then

z \cdot \bar{z}

equals:

(A)

2i

(B)

0

(C)

2

(D)

1 + i

Q4JEE 2024StandardVisual Solution

|z - 2| = |z + 2|

, then

z

lies on:

(A)The real axis

(B)The imaginary axis

(C)A circle of radius 2

(D)A circle of radius 4

Q5JEE 2024StandardVisual Solution

The value of

(1 + i)^8

is:

(A)

16

(B)

-16

(C)

16i

(D)

-16i

Q6JEE 2024StandardVisual Solution

\omega

is a cube root of unity, then

1 + \omega + \omega^2

equals:

(A)

1

(B)

-1

(C)

0

(D)

\omega

Q7JEE 2023 Jan Session 1StandardVisual Solution

z = \frac{1+2i}{1-i}

, then

\text{Im}(z)

equals:

(A)

\frac{3}{2}

(B)

\frac{1}{2}

(C)

-\frac{1}{2}

(D)

\frac{-3}{2}

Q8JEE 2024TrickyVisual Solution

The locus of

z

such that

\arg\left(\frac{z-1}{z+1}\right) = \frac{\pi}{2}

is:

(A)A straight line

(B)A circle passing through

1

and

-1

(C)The imaginary axis

(D)The real axis

Q9JEE 2024TrickyVisual Solution

|z| = 2

, the maximum value of

|z + 3 + 4i|

is:

(A)

5

(B)

7

(C)

3

(D)

9

Q10JEE 2024 Jan Session 2Tricky

\frac{z - i}{z + i}

is purely real, then

z

lies on:

(A)The imaginary axis

(B)The real axis

(C)The unit circle

(D)The line

y = x

Q11JEE 2025 Jan 22 Shift 1TrickyVisual Solution

Let

z_1, z_2

and

z_3

be three complex numbers on the circle

|z| = 1

with

\arg(z_1) = -\frac{\pi}{4}

\arg(z_2) = 0

and

\arg(z_3) = \frac{\pi}{4}

. If

|z_1 \bar{z}_2 + z_2 \bar{z}_3 + z_3 \bar{z}_1|^2 = \alpha + \beta\sqrt{2}

\alpha, \beta \in \mathbb{Z}

, then the value of

\alpha^2 + \beta^2

is:

(A)

24

(B)

29

(C)

41

(D)

31

Q12JEE 2020 Jan Shift 1Easy

z = \frac{1}{1-i}

, then

|z|

is equal to:

(A)

\frac{1}{\sqrt{2}}

(B)

\sqrt{2}

(C)

1

(D)

2

Q13JEE 2019 Jan Shift 2Easy

z = 3 + 4i

, then

z \cdot \bar{z}

is equal to:

(A)

7

(B)

25

(C)

5

(D)

12 + 7i

Q14JEE 2018 Apr Shift 1Easy

The value of

i^{2018} + i^{2019} + i^{2020} + i^{2021}

is:

(A)

1

(B)

-1

(C)

i

(D)

0

Q15JEE 2021 Feb Shift 1Easy

The argument of the complex number

-1 - i\sqrt{3}

is:

(A)

\frac{2\pi}{3}

(B)

-\frac{2\pi}{3}

(C)

\frac{\pi}{3}

(D)

-\frac{\pi}{3}

Q16JEE 2019 Apr Shift 1Easy

\alpha

and

\beta

are roots of

x^2 + x + 1 = 0

, then

\alpha^2 + \beta^2

equals:

(A)

1

(B)

-1

(C)

2

(D)

0

Q17JEE 2022 Jun Shift 1Standard

z = x + iy

and

\frac{z - 5i}{z + 5i}

is purely real, then the locus of

z

is:

(A)

y = 5

(B)The imaginary axis

(C)The real axis

(D)

x^2 + y^2 = 25

Q18JEE 2022 Jul Shift 2Standard

\omega

is a non-real cube root of unity, then

(1 - \omega + \omega^2)^5 + (1 + \omega - \omega^2)^5

is equal to:

(A)

0

(B)

32

(C)

-32

(D)

64

Q19JEE 2020 Sep Shift 1Standard

z_1 = 2 + 3i

and

z_2 = 3 - 2i

, then

\frac{z_1}{z_2}

equals:

(A)

i

(B)

-i

(C)

\frac{12 + 13i}{13}

(D)

1 + i

Q20JEE 2021 Mar Shift 2Standard

|z - 1| = |z - i|

, then the locus of

z

is:

(A)A circle

(B)The line

y = x

(C)The line

x + y = 0

(D)The imaginary axis

Q21JEE 2023 Apr Shift 2Standard

The value of

\left(\frac{1+i}{\sqrt{2}}\right)^{20} + \left(\frac{1-i}{\sqrt{2}}\right)^{20}

is:

(A)

0

(B)

2

(C)

-2

(D)

1

Q22JEE 2022 Jun Shift 2Standard

z^2 + |z| = 0

, then

z

can be:

(A)Only purely real

(B)Only purely imaginary

(C)Either purely real or purely imaginary

(D)Neither real nor imaginary

Q23JEE 2024 Jan Shift 2Standard

The number of complex numbers

z

satisfying

|z| = 1

and

\left|\frac{z}{\bar{z}} + \frac{\bar{z}}{z}\right| = 1

is:

Q24JEE 2019 Jan Shift 1Standard

|z| = 3

, then the minimum value of

|z + 5 + 12i|

is:

(A)

10

(B)

16

(C)

8

(D)

13

Q25JEE 2024 Jan Shift 1TrickyVisual Solution

z

satisfies

|z - 1| + |z + 1| = 4

, then the eccentricity of the locus of

z

is:

(A)

\frac{1}{2}

(B)

\frac{\sqrt{3}}{2}

(C)

\frac{1}{\sqrt{2}}

(D)

\frac{1}{3}

Q26JEE 2020 Sep Shift 2Tricky

\omega \neq 1

is a cube root of unity, then the value of

(1 + \omega)^3 - (1 + \omega^2)^3

is:

(A)

0

(B)

2

(C)

-2

(D)

2\omega

Q27JEE 2022 Jun Shift 1Tricky

Let

z_1

and

z_2

satisfy

|z_1| = |z_2| = 1

and

\arg(z_1) + \arg(z_2) = \pi

. Then

z_1

equals:

(A)

\bar{z}_2

(B)

-\bar{z}_2

(C)

z_2

(D)

-z_2

Q28JEE 2021 Feb Shift 2Tricky

|z_1| = 1

|z_2| = 2

|z_3| = 3

and

|9z_1 z_2 + 4z_1 z_3 + z_2 z_3| = 12

, then the value of

|z_1 + z_2 + z_3|

is:

Q29JEE 2023 Jan Shift 1Tricky

z = \cos\theta + i\sin\theta

, then

z^n + \frac{1}{z^n}

equals:

(A)

2\cos n\theta

(B)

2\sin n\theta

(C)

2i\sin n\theta

(D)

\cos n\theta + \sin n\theta

Q30JEE 2023 Apr Shift 1Hard

The value of

\sum_{k=1}^{6} \left(\sin\frac{2k\pi}{7} - i\cos\frac{2k\pi}{7}\right)

is:

(A)

0

(B)

-1

(C)

-i

(D)

i

Q31JEE 2024 Apr Shift 2Hard

Let

z_1, z_2

be complex numbers with

|z_1| = 1

|z_2| = 2

and

\text{Re}(z_1 \bar{z}_2) = 0

. If

w_1 = z_1 + \frac{z_2}{2}

and

w_2 = z_1 - \frac{z_2}{2}

, then

|w_1|^2 + |w_2|^2

equals:

Sequence & Series

30 PYQs

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Q1JEE 2024 Jan 29 Shift 1EasyVisual Solution

If in a G.P. of 64 terms, the sum of all the terms is 7 times the sum of the odd terms of the G.P., then the common ratio of the G.P. is equal to

(A)

7

(B)

4

(C)

5

(D)

6

Q2JEE 2024 Jan (1 Feb) Shift 1StandardVisual Solution

Let

3, a, b, c

be in A.P. and

3, a-1, b+1, c+9

be in G.P. Then, the arithmetic mean of

a

b

and

c

is:

(A)

-4

(B)

-1

(C)

13

(D)

11

Q3JEE 2024 Jan (1 Feb) Shift 2StandardVisual Solution

Let

S_n

denote the sum of the first

n

terms of an arithmetic progression. If

S_{10} = 390

and the ratio of the tenth and the fifth terms is

15 : 7

, then

S_{15} - S_5

is equal to:

(A)

800

(B)

890

(C)

790

(D)

690

Q4JEE 2024 Jan 27 Shift 2StandardVisual Solution

The

20^{\text{th}}

term from the end of the progression

20, 19\frac{1}{4}, 18\frac{1}{2}, 17\frac{3}{4}, \ldots, -129\frac{1}{4}

is:

(A)

-118

(B)

-110

(C)

-115

(D)

-100

Q5JEE 2023 Apr 6 Shift 1StandardVisual Solution

The sum of the first 20 terms of the series

5 + 11 + 19 + 29 + 41 + \ldots

(A)

3520

(B)

3450

(C)

3250

(D)

3420

Q6JEE 2023 Apr 10 Shift 1StandardVisual Solution

Let the first term

a

and the common ratio

r

of a geometric progression be positive integers. If the sum of squares of its first three terms is

33033

, then the sum of these three terms is equal to

(A)

241

(B)

231

(C)

210

(D)

220

Q7JEE 2023 Apr 13 Shift 1StandardVisual Solution

Let

s_1, s_2, \ldots, s_{10}

respectively be the sum of 12 terms of 10 A.P.s whose first terms are

1, 2, 3, \ldots, 10

and the common differences are

1, 3, 5, \ldots, 19

respectively. Then

\sum_{i=1}^{10} s_i

is equal to

(A)

7220

(B)

7360

(C)

7260

(D)

7380

Q8JEE 2024 Jan 31 Shift 1TrickyVisual Solution

The sum of the series

\frac{1}{1-3 \cdot 1^2+1^4} + \frac{2}{1-3 \cdot 2^2+2^4} + \frac{3}{1-3 \cdot 3^2+3^4} + \ldots

up to 10 terms is

(A)

\frac{45}{109}

(B)

-\frac{45}{109}

(C)

\frac{55}{109}

(D)

-\frac{55}{109}

Q9JEE 2023 Apr 13 Shift 2TrickyVisual Solution

Let

a_1, a_2, a_3, \ldots

be a G.P. of increasing positive numbers. Let the sum of its

6^{\text{th}}

and

8^{\text{th}}

terms be 2 and the product of its

3^{\text{rd}}

and

5^{\text{th}}

terms be

\frac{1}{9}

. Then

6(a_2 + a_4)(a_4 + a_6)

is equal to

(A)

3

(B)

3\sqrt{3}

(C)

2

(D)

2\sqrt{2}

Q10JEE 2024 Apr 9 Shift 2TrickyVisual Solution

Let

a, ar, ar^2, \ldots

be an infinite G.P. If

\sum_{n=0}^{\infty} ar^n = 57

and

\sum_{n=0}^{\infty} a^3 r^{3n} = 9747

, then

a + 18r

is equal to

(A)

46

(B)

38

(C)

31

(D)

27

Q11JEE 2019 Jan Shift 1Easy

If the 9th term of an A.P. is zero, then the ratio of its 29th and 19th terms is:

(A)

1:2

(B)

2:1

(C)

1:3

(D)

3:1

Q12JEE 2018 Apr Shift 1Easy

If the third term of a G.P. is 4, then the product of its first five terms is:

(A)

4^3

(B)

4^4

(C)

4^5

(D)

4^2

Q13JEE 2020 Jan Shift 2Easy

If the sum of first

n

terms of an A.P. is

3n^2 + 5n

, then the common difference is:

(A)

5

(B)

6

(C)

3

(D)

8

Q14JEE 2021 Feb Shift 1Easy

The sum of the infinite G.P.

1, \frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \ldots

is:

(A)

\frac{3}{2}

(B)

\frac{2}{3}

(C)

3

(D)

\frac{1}{2}

Q15JEE 2022 Jun Shift 1Easy

a, b, c

are in A.P. and

a + b + c = 12

, then

b

is equal to:

(A)

3

(B)

4

(C)

6

(D)

5

Q16JEE 2023 Jan Shift 1StandardVisual Solution

The sum of the series

1 + 2 \cdot 2 + 3 \cdot 2^2 + 4 \cdot 2^3 + \ldots + 100 \cdot 2^{99}

is:

(A)

99 \cdot 2^{100} + 1

(B)

100 \cdot 2^{100} + 1

(C)

99 \cdot 2^{100} - 1

(D)

100 \cdot 2^{99} + 1

Q17JEE 2022 Jul Shift 1Standard

The value of

\sum_{r=1}^{n} r(r+1)

is:

(A)

\frac{n(n+1)(n+2)}{3}

(B)

\frac{n(n+1)(2n+1)}{6}

(C)

\frac{n^2(n+1)^2}{4}

(D)

\frac{n(n+1)(n+2)}{6}

Q18JEE 2022 Jun Shift 2Standard

a, b, c

are in A.P. and

a^2, b^2, c^2

are in G.P. with

a < b < c

and

a + b + c = \frac{3}{2}

, then

a

is equal to:

(A)

\frac{1}{2} - \frac{1}{\sqrt{2}}

(B)

\frac{1}{2} - \frac{1}{2\sqrt{2}}

(C)

\frac{1}{2} + \frac{1}{\sqrt{2}}

(D)

\frac{1}{4}

Q19JEE 2019 Apr Shift 2Standard

a_1, a_2, a_3, \ldots, a_n

are in A.P. with

a_1 = 3

and

a_n = 48

, and the sum of all terms is 459, then

n

is:

Q20JEE 2020 Sep Shift 1Standard

If three terms of a G.P. are

x, x+2, x+6

(in some order), then

x

equals:

(A)

-2

(B)

2

(C)

1

(D)

-6

Q21JEE 2021 Mar Shift 1Standard

The value of

1^2 + 3^2 + 5^2 + \ldots + (2n-1)^2

is:

(A)

\frac{n(2n-1)(2n+1)}{3}

(B)

\frac{n(n+1)(2n+1)}{6}

(C)

\frac{n^2(2n+1)}{3}

(D)

n(2n^2+1)

Q22JEE 2023 Apr Shift 2Standard

The sum

\sum_{k=1}^{10} k \cdot k!

is equal to:

(A)

11! - 1

(B)

10! - 1

(C)

11!

(D)

10! + 1

Q23JEE 2018 Apr Shift 2Standard

If the 2nd, 5th and 9th terms of a non-constant A.P. are in G.P., then the common ratio of this G.P. is:

(A)

\frac{7}{4}

(B)

\frac{4}{3}

(C)

\frac{8}{5}

(D)

1

Q24JEE 2022 Jun Shift 1Tricky

The sum

\sum_{r=1}^{20} \frac{1}{r(r+1)}

is equal to:

(A)

\frac{20}{21}

(B)

\frac{19}{20}

(C)

\frac{21}{22}

(D)

\frac{10}{11}

Q25JEE 2024 Jan Shift 1Tricky

S_n = 1 + \frac{1}{2} + \frac{1}{4} + \ldots + \frac{1}{2^{n-1}}

and

2 - S_n < \frac{1}{100}

, then the least value of

n

is:

Q26JEE 2023 Jan Shift 2Tricky

The value of

\sum_{k=1}^{n} \frac{1}{k(k+1)(k+2)}

is:

(A)

\frac{n(n+3)}{4(n+1)(n+2)}

(B)

\frac{n}{2(n+1)(n+2)}

(C)

\frac{n+3}{4(n+2)}

(D)

\frac{1}{2} - \frac{1}{(n+1)(n+2)}

Q27JEE 2021 Mar Shift 2Tricky

If the sum of the infinite series

1 + \frac{2}{3} + \frac{3}{3^2} + \frac{4}{3^3} + \ldots

S

, then

S

equals:

(A)

\frac{9}{4}

(B)

\frac{3}{2}

(C)

\frac{3}{4}

(D)

3

Q28JEE 2020 Jan Shift 1Tricky

The sum

\sum_{r=1}^{n} \frac{1}{(2r-1)(2r+1)}

is equal to:

(A)

\frac{n}{2n+1}

(B)

\frac{n}{2n-1}

(C)

\frac{2n}{2n+1}

(D)

\frac{n+1}{2n+1}

Q29JEE 2024 Apr Shift 1Hard

If the sum of the first

n

terms of the series

\frac{3}{1^2 \cdot 2^2} + \frac{5}{2^2 \cdot 3^2} + \frac{7}{3^2 \cdot 4^2} + \ldots

\frac{n^2 + an + b}{(n+1)^2}

, then

a + b

equals:

Q30JEE 2022 Jul Shift 2Hard

Let

|x| < 1

. The sum of the series

1 + 2x + 3x^2 + 4x^3 + \ldots

is:

(A)

\frac{1}{(1-x)^2}

(B)

\frac{1}{1-x^2}

(C)

\frac{1+x}{(1-x)^2}

(D)

\frac{2}{(1-x)^2}

Permutations & Combinations

30 PYQs

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Q1JEE 2024 Jan 31 Shift 2EasyVisual Solution

The number of ways in which 21 identical apples can be distributed among three children such that each child gets at least 2 apples, is

(A)

406

(B)

130

(C)

142

(D)

136

Q2JEE 2024 Jan (1 Feb) Shift 1StandardVisual Solution

n

is the number of ways five different employees can sit into four indistinguishable offices where any office may have any number of persons including zero, then

n

is equal to:

(A)

47

(B)

53

(C)

51

(D)

43

Q3JEE 2023 Jan 24 Shift 2StandardVisual Solution

The number of integers, greater than 7000 that can be formed, using the digits 3, 5, 6, 7, 8 without repetition, is

(A)

120

(B)

168

(C)

220

(D)

48

Q4JEE 2023 Apr 6 Shift 2StandardVisual Solution

All the letters of the word PUBLIC are written in all possible orders and these words are written as in a dictionary with serial numbers. Then the serial number of the word PUBLIC is

(A)

576

(B)

578

(C)

580

(D)

582

Q5JEE 2024 Apr 6 Shift 1StandardVisual Solution

The number of triangles whose vertices are at the vertices of a regular octagon but none of whose sides is a side of the octagon is

(A)

48

(B)

56

(C)

24

(D)

16

Q6JEE 2023 Apr 10 Shift 2StandardVisual Solution

Eight persons are to be transported from city A to city B in three cars of different makes. If each car can accommodate at most three persons, then the number of ways, in which they can be transported, is

(A)

1120

(B)

3360

(C)

1680

(D)

560

Q7JEE 2023 Apr 15 Shift 1StandardVisual Solution

The total number of three-digit numbers, divisible by 3, which can be formed using the digits 1, 3, 5, 8, if repetition of digits is allowed, is

(A)

21

(B)

20

(C)

22

(D)

18

Q8JEE 2023 Apr 8 Shift 1TrickyVisual Solution

The number of ways, in which 5 girls and 7 boys can be seated at a round table so that no two girls sit together, is

(A)

720

(B)

126(5!)^2

(C)

7(360)^2

(D)

7(720)^2

Q9JEE 2024 Apr 4 Shift 1TrickyVisual Solution

There are 5 points

P_1, P_2, P_3, P_4, P_5

on the side

AB

, excluding

A

and

B

, of a triangle

ABC

. Similarly there are 6 points

P_6, P_7, \ldots, P_{11}

on side

BC

and 7 points

P_{12}, P_{13}, \ldots, P_{18}

on side

CA

. The number of triangles that can be formed using the points

P_1, P_2, \ldots, P_{18}

as vertices, is:

(A)

776

(B)

796

(C)

751

(D)

771

Q10JEE 2023 Apr 8 Shift 1TrickyVisual Solution

The number of arrangements of the letters of the word "INDEPENDENCE" in which all the vowels always occur together is

(A)

16800

(B)

33600

(C)

18000

(D)

14800

Q11JEE 2019 Jan Shift 1Easy

The value of

{}^{10}C_3 + {}^{10}C_4

is equal to:

(A)

{}^{11}C_4

(B)

{}^{11}C_3

(C)

{}^{10}C_7

(D)

{}^{20}C_7

Q12JEE 2018 Apr Shift 1Easy

The number of ways to arrange the letters of the word APPLE is:

(A)

120

(B)

60

(C)

30

(D)

24

Q13JEE 2020 Jan Shift 1Easy

A committee of 3 members is to be selected from 5 men and 4 women. The number of ways in which this can be done if at least one woman is selected is:

(A)

74

(B)

84

(C)

64

(D)

70

Q14JEE 2021 Feb Shift 1Easy

The number of ways to distribute 8 identical balls into 3 distinct boxes is:

(A)

45

(B)

56

(C)

36

(D)

28

Q15JEE 2019 Apr Shift 1Easy

The number of 3-digit even numbers that can be formed using the digits 1, 2, 3, 4, 5 without repetition is:

(A)

24

(B)

36

(C)

30

(D)

12

Q16JEE 2024 Jan Shift 2Standard

The number of non-negative integer solutions of

x + y + z = 10

where

x \leq 5

is:

(A)

51

(B)

66

(C)

45

(D)

56

Q17JEE 2022 Jun Shift 1Standard

The number of ways to arrange 4 boys and 3 girls in a row such that no two girls are adjacent is:

(A)

4! \times {}^5P_3

(B)

4! \times {}^5C_3 \times 3!

(C)

7! - 4! \times 3!

(D)

4! \times {}^4C_3 \times 3!

Q18JEE 2023 Jan Shift 1Standard

The number of diagonals of a regular polygon with 12 sides is:

(A)

48

(B)

54

(C)

60

(D)

66

Q19JEE 2022 Jul Shift 1Standard

All the words formed using the letters A, H, L, R, U are arranged in a dictionary. The rank of the word RAHUL is:

Q20JEE 2021 Mar Shift 1Standard

The total number of 4-digit numbers that are divisible by 2 and formed using the digits 0, 1, 2, 3, 4 without repetition is:

(A)

48

(B)

60

(C)

72

(D)

54

Q21JEE 2020 Sep Shift 1Standard

The number of ways to select 4 cards from a pack of 52 cards such that all 4 cards are of the same suit is:

(A)

4 \times {}^{13}C_4

(B)

{}^{52}C_4

(C)

{}^{13}C_4

(D)

4 \times 13^4

Q22JEE 2023 Apr Shift 1Standard

The number of ways to distribute 5 distinct balls into 3 distinct boxes such that no box is empty is:

(A)

150

(B)

120

(C)

180

(D)

243

Q23JEE 2024 Apr Shift 2Standard

The number of words (with or without meaning) that can be formed using all letters of the word EQUATION such that all vowels are together is:

(A)

2880

(B)

1440

(C)

5760

(D)

720

Q24JEE 2022 Jun Shift 2Tricky

The number of ways 6 people can sit around a circular table such that 2 particular people are never adjacent is:

(A)

72

(B)

48

(C)

96

(D)

60

Q25JEE 2024 Jan Shift 1Tricky

The number of non-negative integer solutions of

x_1 + x_2 + x_3 + x_4 = 20

where each

x_i \leq 8

is:

Q26JEE 2021 Mar Shift 2Tricky

The number of rectangles that can be formed using the lines in a grid of 8 horizontal and 6 vertical lines is:

(A)

315

(B)

420

(C)

480

(D)

350

Q27JEE 2023 Apr Shift 2Tricky

The number of 5-digit numbers divisible by 5 that can be formed using digits 0, 1, 2, 3, 4, 5, 6 without repetition is:

Q28JEE 2020 Jan Shift 2Tricky

The number of 6-letter words that can be formed using the letters of the word PEPPER is:

(A)

120

(B)

60

(C)

30

(D)

90

Q29JEE 2024 Apr Shift 1Hard

The number of ways to distribute 10 identical chocolates among 4 children such that each child gets at least 1 and at most 4 chocolates is:

Q30JEE 2023 Jan Shift 2Hard

The number of integral points (both coordinates integers) strictly inside the triangle with vertices

(0,0)

(0,21)

and

(21,0)

is:

Binomial Theorem

30 PYQs

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Q1JEE 2024 Jan 27 Shift 1EasyVisual Solution

A

denotes the sum of all the coefficients in the expansion of

(1 - 3x + 10x^2)^n

and

B

denotes the sum of all the coefficients in the expansion of

(1 + x^2)^n

, then:

(A)

A = B^3

(B)

3A = B

(C)

B = A^3

(D)

A = 3B

Q2JEE 2024 Jan 29 Shift 2EasyVisual Solution

Remainder when

64^{32^{32}}

is divided by 9 is equal to _____.

Q3JEE 2024 Jan (1 Feb) Shift 2StandardVisual Solution

Let

m

and

n

be the coefficients of seventh and thirteenth terms respectively in the expansion of

\left(\frac{1}{3}x^{\frac{1}{3}} + \frac{1}{2x^{\frac{2}{3}}}\right)^{18}

. Then

\left(\frac{n}{m}\right)^{\frac{1}{3}}

is:

(A)

\frac{4}{9}

(B)

\frac{1}{9}

(C)

\frac{1}{4}

(D)

\frac{9}{4}

Q4JEE 2024 Jan 30 Shift 1StandardVisual Solution

Number of integral terms in the expansion of

\left\{7^{\frac{1}{2}} + 11^{\frac{1}{6}}\right\}^{824}

is equal to _____.

Q5JEE 2024 Apr 4 Shift 2StandardVisual Solution

If the coefficients of

x^4, x^5

and

x^6

in the expansion of

(1 + x)^n

are in the arithmetic progression, then the maximum value of

n

is:

(A)

7

(B)

21

(C)

28

(D)

14

Q6JEE 2024 Apr 8 Shift 2StandardVisual Solution

If the term independent of

x

in the expansion of

\left(\sqrt{a}\,x^2 + \frac{1}{2x^3}\right)^{10}

is 105, then

a^2

is equal to:

(A)

2

(B)

4

(C)

6

(D)

9

Q7JEE 2024 Apr 4 Shift 1StandardVisual Solution

The sum of all rational terms in the expansion of

\left(2^{\frac{1}{5}} + 5^{\frac{1}{3}}\right)^{15}

is equal to:

(A)

3133

(B)

931

(C)

6131

(D)

633

Q8JEE 2024 Jan 29 Shift 1TrickyVisual Solution

\frac{^{11}C_1}{2} + \frac{^{11}C_2}{3} + \ldots + \frac{^{11}C_9}{10} = \frac{n}{m}

with

\gcd(n, m) = 1

, then

n + m

is equal to _____.

Q9JEE 2024 Apr 9 Shift 1TrickyVisual Solution

The remainder when

428^{2024}

is divided by 21 is _____.

Q10JEE 2024 Jan 31 Shift 1TrickyVisual Solution

In the expansion of

(1+x)(1-x^2)\left(1 + \frac{3}{x} + \frac{3}{x^2} + \frac{1}{x^3}\right)^5

x \neq 0

, the sum of the coefficient of

x^3

and

x^{-13}

is equal to _____.

Q11JEE 2020 Jan 7 Shift 1Easy

The sum of the coefficients of all even powers of

x

in the expansion of

(1 + x + x^2 + x^3)^5

is:

(A)

256

(B)

512

(C)

1024

(D)

128

Q12JEE 2019 Jan 9 Shift 1Easy

The remainder when

3^{248}

is divided by 13 is:

(A)

1

(B)

3

(C)

9

(D)

12

Q13JEE 2021 Feb 24 Shift 1Easy

The coefficient of

x^4

in the expansion of

(1 + x + x^2)^{10}

is:

(A)

615

(B)

660

(C)

210

(D)

\binom{10}{4} + \binom{10}{3}\binom{7}{1} + \binom{10}{2}

Q14JEE 2022 Jun 25 Shift 1Easy

If the sum of all the coefficients in the expansion of

(1 - 2x + 3x^2)^{10}

\alpha

and the sum of all the coefficients in the expansion of

(1 + x)^{10}

\beta

, then:

(A)

\alpha = \beta^2

(B)

\alpha = 2\beta

(C)

\alpha = \beta

(D)

\alpha = \frac{\beta}{2}

Q15JEE 2018 Apr 15 Shift 1Easy

The value of

{}^{10}C_0 - {}^{10}C_1 + {}^{10}C_2 - {}^{10}C_3 + \ldots + {}^{10}C_{10}

is:

(A)

1024

(B)

0

(C)

-1

(D)

1

Q16JEE 2023 Jan 24 Shift 2StandardVisual Solution

The coefficient of

x^{18}

in the expansion of

\left(x^4 - \frac{1}{x^3}\right)^{15}

is:

(A)

{}^{15}C_6

(B)

-{}^{15}C_6

(C)

{}^{15}C_9

(D)

-{}^{15}C_9

Q17JEE 2022 Jun 26 Shift 2Standard

The number of integral terms in the expansion of

\left(3^{\frac{1}{2}} + 5^{\frac{1}{4}}\right)^{680}

is:

(A)

171

(B)

172

(C)

170

(D)

341

Q18JEE 2023 Jan 29 Shift 1Standard

If the constant term in the expansion of

\left(\frac{x^3}{2} - \frac{3}{x^2}\right)^{10}

p

, then

\frac{p}{2^6 \cdot 3^5}

is equal to:

(A)

252

(B)

-252

(C)

126

(D)

-126

Q19JEE 2021 Mar 16 Shift 1Standard

The remainder when

7^{103}

is divided by 25 is _____.

Q20JEE 2022 Jun 29 Shift 1Standard

If the coefficient of the middle term in the expansion of

(1 + x)^{2n}

\alpha

and the coefficients of two middle terms in the expansion of

(1 + x)^{2n-1}

are

\beta

and

\gamma

, then

\frac{\alpha}{\beta + \gamma}

is:

(A)

1

(B)

2

(C)

\frac{1}{2}

(D)

\frac{2n}{n+1}

Q21JEE 2020 Sep 2 Shift 1Standard

If the term independent of

x

in the expansion of

\left(\frac{3}{2}x^2 - \frac{1}{3x}\right)^9

k

, then

18k

is equal to:

(A)

5

(B)

7

(C)

9

(D)

11

Q22JEE 2023 Jan 31 Shift 2Standard

{}^{20}C_1 + (2^2) {}^{20}C_3 + (3^2){}^{20}C_5 + \ldots + (10^2){}^{20}C_{19}

equals

\alpha \cdot 2^{18}

, then

\alpha

is:

(A)

100

(B)

200

(C)

50

(D)

400

Q23JEE 2020 Jan 8 Shift 2Standard

The value of

\sum_{r=0}^{6} \left({}^{6}C_r \cdot {}^{6}C_{6-r}\right)

is equal to:

(A)

{}^{12}C_6

(B)

{}^{6}C_6

(C)

2^{12}

(D)

2^6

Q24JEE 2023 Apr 6 Shift 1Tricky

Let the sixth term in the expansion of

\left(x + \frac{x^2}{\binom{x}{2}}\right)^n

, where

n \in \mathbb{N}

, and

\binom{x}{2} = \frac{x(x-1)}{2}

, when

x = 2

and

n = 12

, be

\alpha

. Then

\alpha

is equal to _____.

Q25JEE 2022 Jul 25 Shift 1Tricky

The remainder when

(2023)^{2023}

is divided by 35 is _____.

Q26JEE 2021 Feb 25 Shift 2Tricky

\sum_{r=0}^{n} \frac{{}^{n}C_r}{{}^{n}C_{r-1}} = \frac{(n+1)(n+2)}{2k}

, then

k

is equal to:

(A)

n

(B)

n+1

(C)

2n

(D)

2

Q27JEE 2019 Jan 11 Shift 1Tricky

The value of

r

for which

{}^{20}C_r \cdot {}^{20}C_0 + {}^{20}C_{r-1} \cdot {}^{20}C_1 + {}^{20}C_{r-2} \cdot {}^{20}C_2 + \ldots + {}^{20}C_0 \cdot {}^{20}C_r

is maximum, is:

(A)

15

(B)

10

(C)

20

(D)

11

Q28JEE 2018 Apr 16 Shift 2Tricky

If the third term in the expansion of

\left(\frac{1}{x} + x^{\log_2 x}\right)^5

2560

, then a possible value of

x

is:

(A)

4\sqrt{2}

(B)

\frac{1}{4}

(C)

2\sqrt{2}

(D)

\frac{1}{8}

Q29JEE 2023 Apr 8 Shift 1Hard

Let

S = \sum_{k=0}^{10} {}^{10}C_k \cdot {}^{20}C_{10-k}

and

T = \sum_{k=0}^{10} \left({}^{10}C_k\right)^2

, then

\frac{S}{T}

is equal to _____.

Q30JEE 2023 Jan 30 Shift 1Hard

\sum_{k=1}^{15} k \cdot {}^{15}C_k = 2^n \cdot m

where

m

is odd, then

\frac{n}{m}

is equal to _____.

Matrices & Determinants

30 PYQs

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Q1JEE 2024 Jan 27 Shift 2EasyVisual Solution

Let

A

be a

2 \times 2

real matrix and

I

be the identity matrix of order 2. If the roots of the equation

|A - xI| = 0

are

-1

and

3

, then the sum of the diagonal elements of the matrix

A^2

is:

Q2JEE 2024 Jan 27 Shift 1EasyVisual Solution

Consider the matrix

f(x) = \begin{bmatrix} \cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1 \end{bmatrix}

.\n\nStatement I:

f(-x)

is the inverse of

f(x)

.\nStatement II:

f(x) \cdot f(y) = f(x + y)

.\n\nWhich of the following is correct?

(A)Statement I is false but Statement II is true

(B)Both Statement I and Statement II are false

(C)Statement I is true but Statement II is false

(D)Both Statement I and Statement II are true

Q3JEE 2024 Jan (1 Feb) Shift 1StandardVisual Solution

A = \begin{bmatrix} \sqrt{2} & 1 \\ -1 & \sqrt{2} \end{bmatrix}

B = \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}

C = ABA^T

and

X = A^TC^2A

, then

\det X

is equal to:

(A)

243

(B)

729

(C)

27

(D)

891

Q4JEE 2024 Jan (1 Feb) Shift 2StandardVisual Solution

Let the system of equations

x + 2y + 3z = 5

2x + 3y + z = 9

4x + 3y + \lambda z = \mu

have infinite number of solutions. Then

\lambda + 2\mu

is equal to:

(A)

28

(B)

17

(C)

22

(D)

15

Q5JEE 2024 Jan 29 Shift 1StandardVisual Solution

Let

A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \alpha & \beta \\ 0 & \beta & \alpha \end{bmatrix}

and

|2A|^3 = 2^{21}

where

\alpha, \beta \in \mathbb{Z}

. Then a value of

\alpha

is:

(A)

3

(B)

5

(C)

17

(D)

9

Q6JEE 2024 Apr 8 Shift 2StandardVisual Solution

\alpha \neq a, \beta \neq b, \gamma \neq c

and

\begin{vmatrix} \alpha & b & c \\ a & \beta & c \\ a & b & \gamma \end{vmatrix} = 0

, then

\frac{a}{\alpha - a} + \frac{b}{\beta - b} + \frac{\gamma}{\gamma - c}

is equal to:

(A)

3

(B)

0

(C)

1

(D)

2

Q7JEE 2024 Apr 4 Shift 2StandardVisual Solution

Let

A = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}

and

B = I + \text{adj}(A) + (\text{adj}\,A)^2 + \ldots + (\text{adj}\,A)^{10}

. Then the sum of all elements of the matrix

B

is:

(A)

-124

(B)

22

(C)

-88

(D)

-110

Q8JEE 2024 Apr 4 Shift 2Tricky

Let

A

be a

2 \times 2

symmetric matrix such that

A\begin{bmatrix} 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 3 \\ 7 \end{bmatrix}

and the determinant of

A

be 1. If

A^{-1} = \alpha A + \beta I

, where

I

is the identity matrix, then

\alpha + \beta

equals _____.

Q9JEE 2024 Jan 31 Shift 2Tricky

Let

A

be a

3 \times 3

matrix and

\det(A) = 2

. If

n = \det(\underbrace{\text{adj}(\text{adj}(\ldots(\text{adj}\,A)))}_{2024 \text{ times}})

, then the remainder when

n

is divided by 9 is equal to _____.

Q10JEE 2024 Jan 31 Shift 1TrickyVisual Solution

If the system of linear equations\n

x - 2y + z = -4

2x + \alpha y + 3z = 5

3x - y + \beta z = 3

\nhas infinitely many solutions, then

12\alpha + 13\beta

is equal to:

(A)

60

(B)

64

(C)

54

(D)

58

Q11JEE 2023 Jan 24 Shift 1Easy

A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}

, then

A^{10}

is equal to:

(A)

\begin{bmatrix} 1 & 10 \\ 0 & 1 \end{bmatrix}

(B)

\begin{bmatrix} 1 & 1 \\ 0 & 10 \end{bmatrix}

(C)

\begin{bmatrix} 10 & 1 \\ 0 & 1 \end{bmatrix}

(D)

\begin{bmatrix} 1 & 0 \\ 10 & 1 \end{bmatrix}

Q12JEE 2022 Jun 24 Shift 2Easy

A

is a

3 \times 3

matrix such that

|A| = 5

, then

|3A|

is equal to:

(A)

45

(B)

15

(C)

135

(D)

125

Q13JEE 2021 Aug 26 Shift 1Easy

A

is a

3 \times 3

skew-symmetric matrix, then the value of

|A|

is:

(A)

1

(B)

3

(C)

-1

(D)

0

Q14JEE 2020 Jan 7 Shift 1Easy

A

is a

3 \times 3

non-singular matrix such that

|A| = 4

, then

|\text{adj}(A)|

is equal to:

(A)

4

(B)

8

(C)

16

(D)

64

Q15JEE 2023 Jan 31 Shift 2Easy

If the eigenvalues of a

2 \times 2

matrix

A

are

2

and

5

, then

\text{tr}(A) + |A|

is equal to:

Q16JEE 2023 Jan 29 Shift 2Standard

The system of equations

x + y + z = 6

x + 2y + 3z = 10

x + 2y + \lambda z = \mu

has no solution for:

(A)

\lambda = 3

and

\mu = 10

(B)

\lambda = 3

and

\mu \neq 10

(C)

\lambda \neq 3

and

\mu = 10

(D)

\lambda \neq 3

and

\mu \neq 10

Q17JEE 2022 Jun 25 Shift 1Standard

A = \begin{bmatrix} 2 & -1 \\ 1 & 1 \end{bmatrix}

and

B = \begin{bmatrix} 1 & 1 \\ -1 & 1 \end{bmatrix}

, then

(B^{-1}AB)^{10}

is equal to:

(A)

B^{-1}A^{10}B

(B)

A^{10}

(C)

B^{10}A^{10}

(D)

A^{10}B^{10}

Q18JEE 2022 Jul 25 Shift 1Standard

The value of

\begin{vmatrix} a-b & b-c & c-a \\ b-c & c-a & a-b \\ c-a & a-b & b-c \end{vmatrix}

is:

(A)

abc

(B)

a + b + c

(C)

a^3 + b^3 + c^3 - 3abc

(D)

0

Q19JEE 2021 Feb 24 Shift 2Standard

A = \begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix}

and

A^{-1} = xA + yI

, then the value of

x + y

is:

Q20JEE 2023 Apr 6 Shift 1Standard

A

and

B

are square matrices of order 3 such that

|A| = -1

and

|B| = 3

, then

|3AB|

is equal to:

(A)

-81

(B)

81

(C)

-27

(D)

27

Q21JEE 2019 Jan 9 Shift 1Standard

A = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}

, then

A^{-1}

is:

(A)

-\text{adj}(A)

(B)

\text{adj}(A)

(C)

A

(D)

-A

Q22JEE 2020 Sep 2 Shift 1Standard

The values of

k

for which the system

x + ky + 3z = 0

3x + ky - 2z = 0

2x + 3y - 4z = 0

has a non-trivial solution is:

Q23JEE 2018 Apr 15 Shift 1Standard

f(x) = \begin{vmatrix} x & \sin x & \cos x \\ x^2 & -\tan x & -x^3 \\ 2x & \sin 2x & 5x \end{vmatrix}

, then

\lim_{x \to 0} \frac{f(x)}{x^2}

is equal to:

(A)

0

(B)

-1

(C)

1

(D)

2

Q24JEE 2023 Jan 30 Shift 1Tricky

Let

A

be a

3 \times 3

matrix such that

A^2 - 5A + 7I = O

. Then

(A^{-1})

can be expressed as

\frac{1}{p}(qI - A)

, where

p + q

is equal to:

Q25JEE 2022 Jun 29 Shift 2Tricky

A = \begin{bmatrix} 1 & -1 & 1 \\ 2 & 1 & -3 \\ 1 & 1 & 1 \end{bmatrix}

and

10B = \begin{bmatrix} 4 & 2 & 2 \\ -5 & 0 & 5 \\ 1 & -2 & 3 \end{bmatrix}

, then

B

is:

(A)

A^{-1}

(B)

A^T

(C)

\text{adj}(A)

(D)

-A^{-1}

Q26JEE 2021 Mar 16 Shift 1Tricky

\begin{vmatrix} a+b+2c & a & b \\ c & b+c+2a & b \\ c & a & c+a+2b \end{vmatrix} = k(a+b+c)^3

, then

k

is equal to:

(A)

4

(B)

1

(C)

2

(D)

8

Q27JEE 2019 Jan 12 Shift 1Tricky

If the system of equations

2x + 3y - z = 0

x + ky - 2z = 0

2x - y + z = 0

has a non-trivial solution

(x, y, z)

, then

\frac{x}{y} + \frac{y}{z} + \frac{z}{x}

is equal to:

(A)

-4

(B)

\frac{1}{4}

(C)

4

(D)

-\frac{1}{4}

Q28JEE 2020 Sep 3 Shift 2Tricky

A

is a symmetric matrix and

B

is a skew-symmetric matrix such that

A + B = \begin{bmatrix} 2 & 3 \\ 5 & -1 \end{bmatrix}

, then

AB

is equal to:

(A)

\begin{bmatrix} -4 & 2 \\ 1 & 4 \end{bmatrix}

(B)

\begin{bmatrix} 4 & -2 \\ -1 & -4 \end{bmatrix}

(C)

\begin{bmatrix} -4 & -2 \\ -1 & 4 \end{bmatrix}

(D)

\begin{bmatrix} 4 & 2 \\ 1 & -4 \end{bmatrix}

Q29JEE 2023 Apr 8 Shift 1Hard

Let

A

be a non-singular matrix of order 3. If

\text{adj}(\text{adj}(\text{adj}\,A)) = |A|^n \cdot A

, then the value of

n

is:

Q30JEE 2022 Jun 26 Shift 1Hard

Let

A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 5 & 7 \\ 3 & 7 & 11 \end{bmatrix}

. If

\text{adj}(\text{adj}(A)) = \alpha A + \beta I

, then

\alpha + \beta

is equal to:

Probability

30 PYQs

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Q1JEE 2024 Jan Shift 1EasyVisual Solution

A bag contains 4 white and 6 black balls. Two balls are drawn at random one after the other without replacement. The probability that the first drawn ball is white and the second drawn ball is black is:

(A)

\frac{4}{15}

(B)

\frac{2}{7}

(C)

\frac{1}{3}

(D)

\frac{4}{9}

Q2JEE 2024 Jan Shift 1Standard

A box contains 3 red, 4 blue, and 5 green balls. If 3 balls are drawn at random, the probability that all three are of different colours, given that at least one is red, is:

(A)

\frac{18}{35}

(B)

\frac{12}{35}

(C)

\frac{3}{7}

(D)

\frac{9}{35}

Q3JEE 2024 Jan Shift 1Standard

If a random variable

X

follows binomial distribution with mean 3 and variance

\frac{3}{2}

, then

P(X \geq 5)

is equal to:

Q4JEE 2024 Jan Shift 2StandardVisual Solution

Three urns contain 2 white and 3 black, 3 white and 2 black, and 4 white and 1 black balls respectively. One ball is drawn from each urn. The probability of drawing exactly 2 white balls is:

(A)

\frac{3}{5}

(B)

\frac{2}{5}

(C)

\frac{3}{715}

(D)

\frac{19}{50}

Q5JEE 2024 Jan Shift 2StandardVisual Solution

Two dice are thrown simultaneously. The probability that the sum is a prime number, given that the sum is odd, is:

(A)

\frac{5}{11}

(B)

\frac{5}{18}

(C)

\frac{7}{9}

(D)

\frac{7}{18}

Q6JEE 2024 Jan Shift 2TrickyVisual Solution

A company has two factories. Factory I produces 60% of total items and Factory II produces 40%. The defective rates are 2% and 3% respectively. An item is found defective. The probability it came from Factory II is:

(A)

\frac{21}{50}

(B)

\frac{1}{2}

(C)

\frac{2}{5}

(D)

\frac{29}{50}

Q7JEE 2024 Apr Shift 1StandardVisual Solution

A coin is biased with

P(H) = \frac{3}{5}

. It is tossed 4 times. The probability of getting exactly 2 heads is:

(A)

\frac{216}{625}

(B)

\frac{54}{625}

(C)

\frac{12}{25}

(D)

\frac{96}{625}

Q8JEE 2024 Apr Shift 1Tricky

A random variable

X

has the distribution:

P(X=0) = 3k^3

P(X=1) = 4k - 10k^2

P(X=2) = 5k - 1

. If

E(X) = P(X=1) + P(X=2)

, then the value of

71k

is:

Q9JEE 2024 Apr Shift 2Tricky

From a group of 6 men and 5 women, a committee of 5 is formed with additional constraints. The number of ways this can be done is:

Q10JEE 2024 Apr Shift 2Tricky

A

and

B

are two events such that

P(A) = \frac{1}{3}

P(B) = \frac{1}{4}

, and

P(A \cap B) = \frac{1}{12}

, then

P(B'|A')

is:

(A)

\frac{5}{8}

(B)

\frac{5}{54}

(C)

\frac{1}{3}

(D)

\frac{17}{24}

Q11JEE 2023 Jan Shift 1Easy

A

and

B

are independent events with

P(A) = \frac{1}{2}

and

P(B) = \frac{1}{3}

, then

P(A \cup B)

is:

(A)

\frac{2}{3}

(B)

\frac{5}{6}

(C)

\frac{1}{6}

(D)

\frac{1}{3}

Q12JEE 2023 Jan Shift 1StandardVisual Solution

There are three bags. Bag I contains 2 red and 3 black balls, Bag II contains 3 red and 2 black balls, and Bag III contains 4 red and 1 black ball. A bag is chosen at random and a ball drawn from it is found to be red. The probability that the ball came from Bag III is:

(A)

\frac{4}{9}

(B)

\frac{1}{3}

(C)

\frac{2}{9}

(D)

\frac{5}{9}

Q13JEE 2023 Jan Shift 2Standard

A box contains 5 red and 10 blue balls. Two balls are drawn at random without replacement. The probability that both balls are red is:

(A)

\frac{2}{21}

(B)

\frac{1}{9}

(C)

\frac{10}{21}

(D)

\frac{1}{7}

Q14JEE 2023 Jan Shift 2Standard

In a binomial distribution

B(n, p)

, if

n = 5

and

P(X = 2) = 9 \cdot P(X = 3)

, then the value of

p

is:

(A)

\frac{1}{10}

(B)

\frac{1}{5}

(C)

\frac{9}{10}

(D)

\frac{3}{10}

Q15JEE 2023 Apr Shift 1Easy

The odds in favour of an event

A

are

3:5

. The odds against the event

B

are

4:3

. If

A

and

B

are independent, then

P(A' \cap B)

is:

(A)

\frac{15}{56}

(B)

\frac{9}{56}

(C)

\frac{3}{56}

(D)

\frac{25}{56}

Q16JEE 2022 Jun Shift 1Easy

P(A) = 0.4

P(B) = 0.5

, and

P(A \cap B) = 0.2

, then

P(A | B)

is:

(A)

\frac{2}{5}

(B)

\frac{1}{2}

(C)

\frac{4}{5}

(D)

\frac{1}{5}

Q17JEE 2022 Jun Shift 1Standard

Four cards are drawn at random from a well-shuffled pack of 52 playing cards. The probability that all four cards are of the same suit is:

(A)

\frac{44}{4165}

(B)

\frac{1}{4165}

(C)

\frac{13}{4165}

(D)

\frac{88}{4165}

Q18JEE 2022 Jun Shift 2Tricky

Let

X

be a random variable with the probability distribution:

P(X = -2) = \frac{1}{5}

P(X = -1) = \frac{3}{10}

P(X = 0) = \frac{1}{5}

P(X = 1) = \frac{1}{10}

P(X = 2) = \frac{1}{5}

. Then

\text{Var}(X)

is equal to:

Q19JEE 2022 Jun Shift 2Tricky

A laboratory blood test is 99% effective in detecting a disease when the disease is present. However, it gives a false positive 2% of the time. If 0.5% of the population has the disease, the probability that a person who tests positive actually has the disease is:

(A)

\frac{99}{496}

(B)

\frac{99}{100}

(C)

\frac{1}{2}

(D)

\frac{99}{200}

Q20JEE 2022 Jul Shift 1Standard

A fair die is thrown 6 times. The probability of getting at least one six is:

(A)

1 - \left(\frac{5}{6}\right)^6

(B)

\left(\frac{1}{6}\right)^6

(C)

6 \cdot \left(\frac{1}{6}\right) \cdot \left(\frac{5}{6}\right)^5

(D)

\left(\frac{5}{6}\right)^6

Q21JEE 2021 Feb Shift 1Easy

A

and

B

are two independent events such that

P(A) = \frac{1}{3}

and

P(A \cup B) = \frac{7}{12}

, then

P(B)

is:

(A)

\frac{3}{8}

(B)

\frac{1}{4}

(C)

\frac{5}{12}

(D)

\frac{1}{2}

Q22JEE 2021 Feb Shift 1Standard

An urn contains 8 white and 4 black balls. Three balls are drawn one by one without replacement. The probability that the third ball drawn is black is:

(A)

\frac{1}{3}

(B)

\frac{1}{4}

(C)

\frac{2}{5}

(D)

\frac{4}{11}

Q23JEE 2021 Feb Shift 2Hard

A word of 5 letters is formed using the letters of the word "MATHEMATICS". The probability that the word contains exactly one M and exactly one A is:

Q24JEE 2021 Mar Shift 1Standard

Two cards are drawn successively without replacement from a well-shuffled pack of 52 cards. The probability that the first card is a king and the second card is a queen is:

(A)

\frac{4}{663}

(B)

\frac{2}{663}

(C)

\frac{1}{169}

(D)

\frac{8}{663}

Q25JEE 2021 Mar Shift 1Standard

If the mean and variance of a binomial distribution are 4 and 2 respectively, then the probability of at most one success is:

(A)

\frac{9}{256}

(B)

\frac{7}{256}

(C)

\frac{3}{16}

(D)

\frac{1}{16}

Q26JEE 2020 Jan Shift 1Tricky

A person has undertaken a construction job. The probability that there will be a strike is

\frac{13}{20}

, that the construction will be completed on time if there is no strike is

\frac{4}{5}

, and that it will be completed on time if there is a strike is

\frac{1}{5}

. The probability that the construction was completed on time, given that there was no strike, using Bayes' theorem, is:

(A)

\frac{28}{41}

(B)

\frac{13}{41}

(C)

\frac{4}{5}

(D)

\frac{7}{20}

Q27JEE 2020 Jan Shift 2Hard

A random variable

X

takes values

0, 1, 2, 3, \ldots

with probability

P(X = k) = \frac{(k+1)}{2^{k+2}}

for

k = 0, 1, 2, \ldots

. The value of

4 \cdot E(X)

is:

Q28JEE 2019 Jan Shift 1Easy

Two events

A

and

B

are such that

P(A) = \frac{1}{4}

P(B) = \frac{1}{2}

, and

P(A \cap B) = \frac{1}{8}

. Then

P(A' \cap B')

equals:

(A)

\frac{3}{8}

(B)

\frac{5}{8}

(C)

\frac{1}{4}

(D)

\frac{3}{4}

Q29JEE 2019 Jan Shift 2Standard

Three numbers are chosen at random from the set

\{1, 2, 3, \ldots, 20\}

without replacement. The probability that the minimum of the chosen numbers is 5 or their maximum is 15 is:

(A)

\frac{71}{228}

(B)

\frac{37}{228}

(C)

\frac{1}{19}

(D)

\frac{5}{57}

Q30JEE 2018 Jan Shift 1Standard

X

follows a binomial distribution with parameters

n = 10

and

p = \frac{1}{4}

, then

P(X \geq 1)

is equal to:

(A)

1 - \left(\frac{3}{4}\right)^{10}

(B)

\left(\frac{3}{4}\right)^{10}

(C)

1 - \left(\frac{1}{4}\right)^{10}

(D)

10 \cdot \frac{1}{4} \cdot \left(\frac{3}{4}\right)^9

Straight Lines

30 PYQs

View all →

Q1JEE 2024 Jan Shift 1StandardVisual Solution

The portion of the line

4x + 5y = 20

in the first quadrant is trisected by the lines

L_1

and

L_2

passing through the origin. The tangent of the angle between the lines

L_1

and

L_2

is:

(A)

\frac{8}{5}

(B)

\frac{25}{41}

(C)

\frac{2}{5}

(D)

\frac{30}{41}

Q2JEE 2024 Jan Shift 2TrickyVisual Solution

Let R be the interior region between the lines

3x - y + 1 = 0

and

x + 2y - 5 = 0

containing the origin. The set of all values of

a

, for which the points

(a^2, a+1)

lie in

R

, is:

(A)

(-3,-1) \cup \left(-\frac{1}{3}, 1\right)

(B)

(-3,0) \cup \left(\frac{1}{3}, 1\right)

(C)

(-3,0) \cup \left(\frac{2}{3}, 1\right)

(D)

(-3,-1) \cup \left(\frac{1}{3}, 1\right)

Q3JEE 2024 Jan Shift 2StandardVisual Solution

Let A and B be two finite sets with m and n elements respectively. The total number of subsets of A is 56 more than the total number of subsets of B. Then the distance of the point

P(m, n)

from the point

Q(-2, -3)

is:

(A)

10

(B)

6

(C)

4

(D)

8

Q4JEE 2024 Jan Shift 1StandardVisual Solution

A line passing through the point

A(9, 0)

makes an angle of

30°

with the positive direction of the x-axis. If this line is rotated about A through an angle of

15°

in the clockwise direction, then its equation in the new position is:

(A)

\frac{y}{\sqrt{3}-2} + x = 9

(B)

\frac{x}{\sqrt{3}-2} + y = 9

(C)

\frac{x}{\sqrt{3}+2} + y = 9

(D)

\frac{y}{\sqrt{3}+2} + x = 9

Q5JEE 2024 Jan Shift 2StandardVisual Solution

Let A be the point of intersection of the lines

3x + 2y = 14

5x - y = 6

and B be the point of intersection of the lines

4x + 3y = 8

6x + y = 5

. The distance of the point

P(5, -2)

from the line AB is:

(A)

\frac{13}{2}

(B)

8

(C)

\frac{5}{2}

(D)

6

Q6JEE 2024 Jan Shift 1TrickyVisual Solution

In a

\triangle ABC

, suppose

y = x

is the equation of the bisector of angle B and the equation of the side AC is

2x - y = 2

. If

2AB = BC

and the point A and B are respectively

(4, 6)

and

(\alpha, \beta)

, then

\alpha + 2\beta

is equal to:

(A)

42

(B)

39

(C)

48

(D)

45

Q7JEE 2024 Jan Shift 2TrickyVisual Solution

If the sum of squares of all real values of

\alpha

, for which the lines

2x - y + 3 = 0

6x + 3y + 1 = 0

and

\alpha x + 2y - 2 = 0

do not form a triangle, is

p

, then the greatest integer less than or equal to

p

is:

Q8JEE 2024 Jan Shift 2TrickyVisual Solution

The distance of the point

(2, 3)

from the line

2x - 3y + 28 = 0

, measured parallel to the line

\sqrt{3}x - y + 1 = 0

, is equal to:

(A)

4\sqrt{2}

(B)

6\sqrt{3}

(C)

3 + 4\sqrt{2}

(D)

4 + 6\sqrt{3}

Q9JEE 2024 Apr Shift 2StandardVisual Solution

Let

A(-1, 1)

and

B(2, 3)

be two points and

P

be a variable point above the line

AB

such that the area of

\triangle PAB

is 10. If the locus of

P

ax + by = 15

, then

5a + 2b

is:

(A)

6

(B)

-\frac{6}{5}

(C)

4

(D)

-\frac{12}{5}

Q10JEE 2024 Apr Shift 1TrickyVisual Solution

A ray of light coming from the point

P(1, 2)

gets reflected from the point

Q

on the x-axis and then passes through the point

R(4, 3)

. If the point

S(h, k)

is such that

PQRS

is a parallelogram, then

hk^2

is equal to:

(A)

70

(B)

80

(C)

60

(D)

90

Q11JEE 2023 Jan Shift 1Easy

The equation of the line passing through the point

(1, 2)

and perpendicular to the line

x + y + 1 = 0

is:

(A)

y - x + 1 = 0

(B)

y - x - 1 = 0

(C)

y + x - 3 = 0

(D)

y + x + 1 = 0

Q12JEE 2023 Jan Shift 2Easy

The distance between the parallel lines

3x + 4y - 5 = 0

and

6x + 8y + 15 = 0

is:

(A)

\frac{5}{2}

(B)

\frac{7}{2}

(C)

\frac{3}{2}

(D)

\frac{15}{2}

Q13JEE 2023 Apr Shift 1Standard

The acute angle between the lines

y = 2x + 3

and

y = 3x - 1

is:

(A)

\tan^{-1}\left(\frac{1}{7}\right)

(B)

\tan^{-1}\left(\frac{1}{5}\right)

(C)

\tan^{-1}(1)

(D)

\tan^{-1}\left(\frac{5}{7}\right)

Q14JEE 2022 Jun Shift 1Easy

The point

P

which divides the line segment joining

A(2, -5)

and

B(5, 2)

in the ratio

2:3

internally has coordinates:

(A)

\left(\frac{16}{5}, \frac{-11}{5}\right)

(B)

\left(\frac{11}{5}, \frac{-16}{5}\right)

(C)

\left(\frac{19}{5}, \frac{-4}{5}\right)

(D)

\left(4, -1\right)

Q15JEE 2022 Jun Shift 2Standard

The line passing through the intersection of

x + y - 3 = 0

and

x - y + 1 = 0

and parallel to the line

x - y + 5 = 0

is:

(A)

x - y + 1 = 0

(B)

x - y - 1 = 0

(C)

x - y + 3 = 0

(D)

x - y - 3 = 0

Q16JEE 2022 Jul Shift 1Standard

The area of the triangle formed by the lines

y = x

y = 2x

and

y = 3x + 4

is:

Q17JEE 2021 Feb Shift 1Standard

The image of the point

(3, 8)

with respect to the line

x + 3y = 7

is:

(A)

(-1, -4)

(B)

(1, 4)

(C)

(-1, 4)

(D)

(1, -4)

Q18JEE 2021 Feb Shift 2Easy

If a line makes equal intercepts on the coordinate axes and passes through the point

(2, 3)

, then its equation is:

(A)

x + y = 5

(B)

x + y = 3

(C)

x - y = 5

(D)

x + y = 2

Q19JEE 2021 Mar Shift 1Tricky

A variable line passes through the fixed point

(2, 3)

. The locus of the foot of the perpendicular drawn from the origin to this line is:

(A)

x^2 + y^2 - 2x - 3y = 0

(B)

x^2 + y^2 - 3x - 2y = 0

(C)

x^2 + y^2 + 2x + 3y = 0

(D)

x^2 + y^2 - 2x + 3y = 0

Q20JEE 2021 Mar Shift 2Standard

The perpendicular distance from the origin to the line

3x + 4y - 10 = 0

is:

Q21JEE 2020 Jan Shift 1Easy

The angle between the lines

2x + 3y = 5

and

3x - 2y = 7

is:

(A)

30°

(B)

45°

(C)

60°

(D)

90°

Q22JEE 2020 Jan Shift 2Standard

If the line

3x + 4y - 24 = 0

intersects the x-axis at point

A

and the y-axis at point

B

, then the incentre of the triangle

OAB

, where

O

is the origin, is:

(A)

(1, 2)

(B)

(2, 2)

(C)

(3, 4)

(D)

(4, 3)

Q23JEE 2020 Sep Shift 1Tricky

The locus of the midpoint of the line segment joining the point

(2t, t)

on the line

x = 2y

and the foot of the perpendicular from

(2t, t)

to the line

x = y

is:

(A)

2x - 3y = 0

(B)

3x - 2y = 0

(C)

5x - 7y = 0

(D)

7x - 5y = 0

Q24JEE 2019 Jan Shift 1Standard

If a straight line passing through the point

P(-3, 4)

is such that its intercepted portion between the coordinate axes is bisected at

P

, then its equation is:

(A)

4x - 3y + 24 = 0

(B)

3x - 4y + 25 = 0

(C)

x - y + 7 = 0

(D)

4x + 3y = 0

Q25JEE 2019 Jan Shift 2Hard

Lines are drawn parallel to the line

4x - 3y + 2 = 0

, at a distance

\frac{3}{5}

from the origin. The equations of the lines are:

(A)

4x - 3y + 3 = 0

and

4x - 3y - 3 = 0

(B)

4x - 3y + 5 = 0

and

4x - 3y - 5 = 0

(C)

4x - 3y + 1 = 0

and

4x - 3y - 1 = 0

(D)

4x - 3y + 6 = 0

and

4x - 3y - 6 = 0

Q26JEE 2019 Apr Shift 1Standard

The value of

k

for which the points

(2, 3)

(4, k)

and

(6, 7)

are collinear is:

Q27JEE 2018 Apr Shift 1Hard

The perpendicular bisector of the line segment joining

A(1, 4)

and

B(k, 3)

has y-intercept

-4

. A possible value of

k

is:

Q28JEE 2018 Jan Shift 1Tricky

The foot of the perpendicular from the point

(2, 4)

upon

x + y = 1

is:

(A)

\left(-\frac{1}{2}, \frac{3}{2}\right)

(B)

\left(\frac{3}{2}, -\frac{1}{2}\right)

(C)

\left(\frac{1}{2}, \frac{1}{2}\right)

(D)

\left(-1, 2\right)

Q29JEE 2018 Jan Shift 2Tricky

The image of the point

(1, 3)

in the line

x + y - 6 = 0

is:

(A)

(3, 5)

(B)

(5, 3)

(C)

(5, 5)

(D)

(3, 3)

Q30JEE 2018 Apr Shift 2Easy

If the x-intercept of a line

L

is double that of the line

3x + 4y = 12

and the y-intercept of

L

is the same as that of

3x + 4y = 12

, then the equation of

L

is:

(A)

3x + 8y = 24

(B)

3x + 8y = 12

(C)

6x + 4y = 24

(D)

6x + 8y = 24

Definite & Indefinite Integrals

30 PYQs

View all →

Q1JEE 2024 Jan Shift 2StandardVisual Solution

The value of the integral

\int_0^{\pi/4} \frac{x \, dx}{\sin^4(2x) + \cos^4(2x)}

equals:

(A)

\frac{\sqrt{2}\pi^2}{8}

(B)

\frac{\sqrt{2}\pi^2}{16}

(C)

\frac{\sqrt{2}\pi^2}{32}

(D)

\frac{\sqrt{2}\pi^2}{64}

Q2JEE 2024 Jan Shift 2EasyVisual Solution

The value of

\int_0^1 (2x^3 - 3x^2 - x + 1)^{1/3} \, dx

is equal to:

(A)

0

(B)

1

(C)

2

(D)

-1

Q3JEE 2024 Jan Shift 2StandardVisual Solution

\int_0^{\pi/3} \cos^4 x \, dx = a\pi + b\sqrt{3}

, where

a

and

b

are rational numbers, then

9a + 8b

is equal to:

(A)

2

(B)

1

(C)

3

(D)

\frac{3}{2}

Q4JEE 2024 Jan Shift 2StandardVisual Solution

For

0 < a < 1

, the value of the integral

\int_0^{\pi} \frac{dx}{1 - 2a\cos x + a^2}

is:

(A)

\frac{\pi^2}{\pi + a^2}

(B)

\frac{\pi^2}{\pi - a^2}

(C)

\frac{\pi}{1-a^2}

(D)

\frac{\pi}{1+a^2}

Q5JEE 2024 Jan Shift 2TrickyVisual Solution

Let

f : (0,\infty) \to \mathbb{R}

and

F(x) = \int_0^x t f(t) \, dt

. If

F(x^2) = x^4 + x^5

, then

\sum_{r=1}^{12} f(r^2)

is equal to:

Q6JEE 2024 Apr Shift 1StandardVisual Solution

The value of

\int_{-\pi}^{\pi} \frac{2y(1+\sin y)}{1+\cos^2 y} \, dy

is:

(A)

2\pi^2

(B)

\frac{\pi^2}{2}

(C)

\frac{\pi^2}{2}

(D)

\pi^2

Q7JEE 2024 Apr Shift 1StandardVisual Solution

The integral

\int_0^{\pi/4} \frac{\cos^2 x \sin^2 x}{(\cos^3 x + \sin^3 x)^2} \, dx

is equal to:

(A)

\frac{1}{6}

(B)

\frac{1}{3}

(C)

\frac{1}{12}

(D)

\frac{1}{9}

Q8JEE 2024 Jan Shift 1StandardVisual Solution

The integral

\int \frac{(x^8 - x^2) \, dx}{(x^{12} + 3x^6 + 1) \tan^{-1}\left(x^3 + \frac{1}{x^3}\right)}

equal to:

(A)

\log_e\left(\left|\tan^{-1}\left(x^3+\frac{1}{x^3}\right)\right|\right)^{1/3} + C

(B)

\log_e\left(\left|\tan^{-1}\left(x^3+\frac{1}{x^3}\right)\right|\right)^{1/2} + C

(C)

\log_e\left|\tan^{-1}\left(x^3+\frac{1}{x^3}\right)\right| + C

(D)

\log_e\left(\left|\tan^{-1}\left(x^3+\frac{1}{x^3}\right)\right|\right)^{3} + C

Q9JEE 2024 Apr Shift 2StandardVisual Solution

If the value of the integral

\int_{-1}^{1} \frac{\cos \alpha x}{1+3^x} \, dx

\frac{2}{\pi}

, then a value of

\alpha

is:

(A)

\frac{\pi}{3}

(B)

\frac{\pi}{6}

(C)

\frac{\pi}{4}

(D)

\frac{\pi}{2}

Q10JEE 2024 Apr Shift 1TrickyVisual Solution

Let

I(x) = \int \frac{6}{\sin^2 x(1-\cot x)^2} \, dx

. If

I(0) = 3

, then

I\left(\frac{\pi}{12}\right)

is equal to:

(A)

2\sqrt{3}

(B)

\sqrt{3}

(C)

3\sqrt{3}

(D)

6\sqrt{3}

Q11JEE 2023 Jan Shift 1Easy

The value of

\int \frac{dx}{x(x^4 - 1)}

is:

(A)

\frac{1}{4}\log_e\left|\frac{x^4-1}{x^4}\right| + C

(B)

\frac{1}{4}\log_e\left|\frac{x^4}{x^4-1}\right| + C

(C)

\log_e\left|\frac{x^4-1}{x^4}\right| + C

(D)

\frac{1}{4}\log_e|x^4 - 1| + C

Q12JEE 2023 Jan Shift 2Standard

The value of

\int_0^1 x^2 e^{2x} \, dx

is:

(A)

\frac{e^2 - 1}{4}

(B)

\frac{e^2 + 1}{4}

(C)

\frac{3e^2 + 1}{4}

(D)

\frac{3e^2 - 1}{4}

Q13JEE 2023 Apr Shift 1StandardVisual Solution

The value of

\int_0^{\pi/2} \frac{\sin^2 x}{1 + \sin x \cos x} \, dx

is:

(A)

\frac{\pi}{3\sqrt{3}}

(B)

\frac{\pi}{2\sqrt{3}}

(C)

\frac{\pi}{\sqrt{3}}

(D)

\frac{2\pi}{3\sqrt{3}}

Q14JEE 2022 Jun Shift 1Easy

The value of

\int \frac{dx}{\sqrt{3x+4} + \sqrt{3x+1}}

is:

(A)

\frac{2}{27}\left[(3x+4)^{3/2} - (3x+1)^{3/2}\right] + C

(B)

\frac{1}{3}\left[(3x+4)^{3/2} - (3x+1)^{3/2}\right] + C

(C)

\frac{2}{9}\left[(3x+4)^{3/2} + (3x+1)^{3/2}\right] + C

(D)

\frac{1}{9}\left[(3x+4)^{3/2} - (3x+1)^{3/2}\right] + C

Q15JEE 2022 Jun Shift 2Standard

\int \frac{dx}{(x+2)(x^2+1)} = a \log_e|1+x^2| + b \tan^{-1}x + \frac{1}{5}\log_e|x+2| + C

, then the value of

10(a + b)

is equal to:

Q16JEE 2022 Jul Shift 1Tricky

The value of

\int_0^{\pi} \frac{x \sin^8 x}{\sin^8 x + \cos^8 x} \, dx

is:

(A)

\frac{\pi^2}{4}

(B)

\frac{\pi^2}{2}

(C)

\pi^2

(D)

2\pi^2

Q17JEE 2021 Feb Shift 1Standard

The value of

\int_0^{\pi} |\cos x|^3 \, dx

is:

Q18JEE 2021 Mar Shift 1Standard

\int \frac{5x+1}{(x+2)(x-1)} \, dx = a \log_e|x+2| + b \log_e|x-1| + C

, then

a + b

is equal to:

(A)

5

(B)

3

(C)

7

(D)

1

Q19JEE 2021 Feb Shift 2Easy

The value of

\int_0^{\pi} |\cos x| \, dx

is:

(A)

1

(B)

2

(C)

0

(D)

\frac{\pi}{2}

Q20JEE 2020 Jan Shift 1Standard

\int \frac{x^3}{\sqrt{1+x^2}} \, dx = a(1+x^2)^{3/2} + b\sqrt{1+x^2} + C

, then

a + b

is:

(A)

\frac{1}{3}

(B)

-\frac{2}{3}

(C)

\frac{2}{3}

(D)

-\frac{1}{3}

Q21JEE 2020 Jan Shift 2Tricky

The value of

\lim_{n \to \infty} \frac{1}{n} \sum_{r=0}^{n-1} \frac{1}{\sqrt{4-\left(\frac{r}{n}\right)^2}}

is:

(A)

\frac{\pi}{3}

(B)

\frac{\pi}{6}

(C)

\frac{\pi}{4}

(D)

\frac{\pi}{2}

Q22JEE 2020 Sep Shift 1Standard

The value of

\int_0^1 x \tan^{-1} x \, dx

is:

(A)

\frac{\pi}{4} - \frac{1}{2}

(B)

\frac{\pi}{4} + \frac{1}{2}

(C)

\frac{\pi - 1}{4}

(D)

\frac{\pi + 1}{4}

Q23JEE 2019 Jan Shift 1Easy

The value of

\int \frac{e^x(1+x)}{\cos^2(xe^x)} \, dx

is:

(A)

\tan(xe^x) + C

(B)

\sec(xe^x) + C

(C)

-\cot(xe^x) + C

(D)

\sin(xe^x) + C

Q24JEE 2019 Jan Shift 2Tricky

The value of

\int_0^{\pi/2} \frac{\cos^2 x}{1 + 3\sin^2 x} \, dx

is:

(A)

\frac{\pi}{8}

(B)

\frac{\pi}{6}

(C)

\frac{\pi}{4}

(D)

\frac{\pi}{3}

Q25JEE 2019 Apr Shift 1Hard

The value of

\int_0^1 x^4(1-x)^4 \, dx

is equal to

\frac{p}{q}

in lowest terms. The value of

p + q

is:

Q26JEE 2018 Jan Shift 1Standard

The value of

\int_{-\pi}^{\pi} \frac{\cos^2 x}{1 + a^x} \, dx

, where

a > 0

, is:

(A)

\frac{\pi}{2}

(B)

\pi

(C)

2\pi

(D)

\frac{\pi}{4}

Q27JEE 2018 Jan Shift 2Standard

The value of

\int \frac{\sin x}{\sin^3 x + \cos^3 x} \, dx

is best evaluated by substituting:

(A)

t = \tan x

(B)

t = \cos x

(C)

t = \sin x + \cos x

(D)

t = \sec x

Q28JEE 2018 Apr Shift 1Hard

The value of

\int_0^{\pi} \frac{x \, dx}{1 + \cos\alpha \sin x}

, where

0 < \alpha < \pi

, is:

(A)

\frac{\pi \alpha}{\sin \alpha}

(B)

\frac{\pi \alpha}{\cos \alpha}

(C)

\frac{\pi (\pi - \alpha)}{\sin \alpha}

(D)

\frac{\pi^2}{2\sin \alpha}

Q29JEE 2023 Apr Shift 2Standard

The value of

\int_0^1 e^x\left(\frac{x}{(x+1)^2}\right) dx

is:

(A)

\frac{e}{2} - 1

(B)

e - 2

(C)

\frac{e}{2} + 1

(D)

e - 1

Q30JEE 2022 Jul Shift 2Tricky

Let

[t]

denote the greatest integer less than or equal to

t

. The value of

\int_0^2 x [x^2] \, dx

is: