JEE Main 2024 question paper pdf with solutions- Download Jan 31 Shift 2 Mathematics Question Paper pdf

If \( a = \sin^{-1}(\sin 5), b = \cos^{-1}(\cos 5) \), then \( a^2 + b^2 \) is equal to:

• (1) \( 8\pi^2 - 40\pi + 50 \)
• (2) \( 4\pi^2 + 25 \)
• (3) \( 8\pi^2 - 50 \)
• (4) \( 8\pi^2 + 40\pi + 50 \)

A coin is biased such that head has two chances than tails, what is the probability of getting 2 heads and 1 tail?

• (1) \( \frac{1}{29} \)
• (2) \( \frac{2}{29} \)
• (3) \( \frac{1}{9} \)
• (4) \( \frac{4}{9} \)

Let mean and variance of 6 observations \( a, b, 68, 44, 40, 60 \) be 55 and 194. If \( a > b \), then find \( a + 3b \).

• (1) 211.83
• (2) 201.59
• (3) 189.57
• (4) 198.87

If the 2nd, 8th, and 44th terms of an A.P. are the 1st, 2nd, and 3rd terms respectively of a G.P. and the first term of A.P. is 1, then the sum of the first 20 terms of the A.P. is:

• (1) 970
• (2) 916
• (3) 980
• (4) 990

The area of the region enclosed by the parabolas \[ y = 4 - x^2 \quad and \quad 3y = (x - 4)^2 \quad is in (sq. units). \]

• (1) \( \frac{14}{3} \)
• (2) 4
• (3) \( \frac{32}{3} \)
• (4) 6

If \[ A = \begin{bmatrix} 1 & 0 & 1
2 & 1 & -1
-1 & 4 & 0 \end{bmatrix}, \quad A^{-1} = \begin{bmatrix} 0 & 2 & 1
1 & -1 & 0
0 & 0 & 1 \end{bmatrix}, \]
then the value of \( (A - 3I) \begin{bmatrix} x
y
z \end{bmatrix} = \begin{bmatrix} -1
2
3 \end{bmatrix} \) is:

• (1) \( (1, 2, 3) \)
• (2) \( (1, -2, 3) \)
• (3) \( (1, -2, -3) \)
• (4) \( (-1, -2, -3) \)

Let \( f: \mathbb{R} \to \infty \) be an increasing function such that \[ \lim_{x \to \infty} \frac{f(7x)}{f(x)} = 1. \quad Then \quad \lim_{x \to \infty} \frac{f(5x)}{f(x) - 1} \quad is equal to: \]

• (1) 0
• (2) 4
• (3) 1
• (4) \( \frac{4}{5} \)

Let \( z_1 \) and \( z_2 \) be two complex numbers such that \[ z_1 + z_2 = 5 \quad and \quad z_1^3 + z_2^3 = 20 + 15i, \quad then the value of \quad |z_1^4 + z_2^4| \quad is equal to: \]

• (1) 75
• (2) \( 25\sqrt{5} \)
• (3) \( 15\sqrt{15} \)
• (4) \( 30\sqrt{3} \)

The number of solutions to the equation \[ e^{\sin x} - 2e^{-\sin x} = 2 \]

• (1) More than 2
• (2) 2
• (3) 1
• (4) 0

The line passes through the center of the circle \[ x^2 + y^2 - 16x - 4y = 0, \quad it interacts with the positive coordinate axis at A \& B. Then find the minimum value of OA + OB, where O is the origin. \]

• (1) 20
• (2) 18
• (3) 12
• (4) 24

If for some \( m, n \): \[ 6C_m + 2 \left( 6C_{m+1} \right) + 6C_{m+2} > 8C_3 \]
and \[ n^{-1} P_3 : n P_4 = 1 : 8, then n P_{m+1} + n^{+1} C_m is equal to: \]

• (1) 6756
• (2) 7250
• (3) 6223
• (4) 6550

Let \( f: (-\infty, -1] \to (a, b) \) be defined as \[ f(x) = e^{x^3 - 3x + 1}, \quad if f is both one and onto, then the distance from a point P(2a + 4, b + 2) to curve x + ye^{3 - 4} = 0 is: \]

• (1) \( \sqrt{e^3 + 2} \)
• (2) \( \frac{e^3 + 2}{\sqrt{e^3 + 1}} \)
• (3) \( \frac{e^3 + 2}{\sqrt{e^6 + 1}} \)
• (4) \( e \)

If \( (\alpha, \beta, \gamma) \) is the mirror image of the point \( (2, 3, 4) \) with respect to the line \[ \frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4}, \quad then 2\alpha + 3\beta + 4\gamma is: \]

• (1) 29
• (2) 30
• (3) 31
• (4) 32

A parabola has vertex \( (2, 3) \), equation of the directrix is \( 2x - y = 1 \), and the equation of ellipse is \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \quad e = \frac{1}{\sqrt{2}}, \quad and ellipse passing through the focus of parabola. Then the square of the length of the latus rectum of the ellipse is: \]

• (1) \( \frac{6564}{25} \)
• (2) \( \frac{3288}{25} \)
• (3) \( \frac{6272}{25} \)
• (4) \( \frac{4352}{25} \)

The value of \[ \frac{120}{\pi^3} \int_0^{\pi} \frac{x^2 \sin x \cos x}{(\sin x)^4 + (\cos x)^4} \, dx is: \]

The number of ways to distribute the 21 identical apples to three children so that each child gets at least 2 apples is:

If \( A = \{1, 2, 3, \dots, 100\}, R = \{(x, y) \mid 2x = 3y, x, y \in A \} \) is a symmetric relation on \( A \), and the number of elements in \( R \) is \( n \), the smallest integer value of \( n \) is:

Matrix \( A \) of order \( 3 \times 3 \) is such that \( |A| = 2 \), if \[ n = adj(adj(adj(\dots (A))\dots )) \quad 2024 times, \]
\text{then the remainder when \( n \) \text{ is divided by 9 is: