JEE Main 2023 Mathematics Question Paper Jan 24 Shift 1- Download PDF

The distance of the point (7, -3, -4) from the plane passing through the points (2, -3, 1), (-1, 1, -2), and (3, -4, 2) is:

• (1) \( 4 \)
• (2) \( 5 \)
• (3) \( 5\sqrt{2} \)
• (4) \( 4\sqrt{2} \)

The value of the following limit is: \[ \lim_{t \to 0} \left( \frac{1}{1^{2n^2 t} + 2^{\frac{1}{2n^2 t}} + \ldots + n^{\frac{1}{2n^2 t}}} \right)^{2n^2 t} \]

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

Let \( \mathbf{u} = \hat{i} - \hat{j} - 2\hat{k}, \mathbf{v} = 2\hat{i} + \hat{j} - \hat{k}, \mathbf{w} = 2 \) and \( \mathbf{v} \times \mathbf{w} = \mathbf{u} + \lambda \mathbf{v}. \) Then \( \mathbf{u} \cdot \mathbf{w} \) is equal to:

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

The value of \( \sum_{r=0}^{22} 22C_r 23C_{r} \) is:

• (1) \( 45 C_{23} \)
• (2) \( 44 C_{23} \)
• (3) \( 45 C_{24} \)
• (4) \( 44 C_{22} \)

Let a tangent to the curve \( y^2 = 24x \) meet the curve \( xy = 2 \) at the points A and B. Then the midpoints of such line segments AB lie on a parabola with the:

• (1) Directrix \( 4x = 3 \)
• (2) Directrix \( 4x = -3 \)
• (3) Length of latus rectum \( \frac{3}{2} \)
• (4) Length of latus rectum 2

Let \( N \) denote the number that turns up when a fair die is rolled. If the probability that the system of equations \[ x + y + z = 1 \] \[ 2x + Ny + 2z = 2 \] \[ 3x + 3y + Nz = 3 \]
has unique solution is \( \frac{k}{6} \), then the sum of value of \( k \) and all possible values of \( N \) is:

• (1) 18
• (2) 19
• (3) 20
• (4) 21

tan^{-1 \left( \frac{1 + \sqrt{3{3 + \sqrt{3 \right) + \sec^{-1 \sqrt{\left( \frac{8 + 4 \sqrt{3{6 + 3 \sqrt{3 \right) is equal to:

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

Let PQR be a triangle. The points A, B and C are on the sides QR, RP and PQ respectively such that \[ \frac{QA}{AR} = \frac{RB}{BP} = \frac{PC}{CQ} = \frac{1}{2} \]
Then \[ \frac{Area of \, \Delta PQR}{Area of \, \Delta ABC} is equal to: \]

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

If A and B are two non-zero n \(\times\) n matrices such that \(A^2 + B = A^2 B\), then:

• (1) \( AB = I \)
• (2) \( A^2 B = I \)
• (3) \( A^2 = I \) or \( B = I \)
• (4) \( A^2 B = BA^2 \)

Let \( y = y(x) \) be the solution of the differential equation \( x^3 \frac{dy}{dx} + (xy - 1) dx = 0, x > 0, y(1) = e^{-3} \). Then \( y(1) \) is equal to:

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

The area enclosed by the curves \( y^2 + 4x = 4 \) and \( y - 2x = 2 \) is:

• (1) \( \frac{25}{3} \)
• (2) \( \frac{22}{3} \)
• (3) \( 9 \)
• (4) \( \frac{23}{3} \)

Let \( \alpha \) be a root of the equation \( (a - c) x^2 + (b - a) x + (c - b) = 0 \), where \( a, b, c \) are distinct real numbers such that the matrix \[ \begin{pmatrix} \alpha^2 & \alpha & 1
1 & 1 & 1
a & b & c \end{pmatrix} \]
is singular. Then the value of \( (a - c)^2 \) is:

• (1) 6
• (2) 9
• (3) 12
• (4) 5

The distance of the point \( (-1, 9, -16) \) from the plane \( 2x + 3y - z = 5 \) measured parallel to the line \( x+4 = 2y - z = 3 \) is:

• (1) \( 13 \sqrt{2} \)
• (2) \( 31 \)
• (3) \( 26 \)
• (4) \( 20 \sqrt{2} \)

For three positive integers \( p, q, r \), such that \( x^n = y^m = z^p \) and \( r = pq + 1 \), and \( 3, 3 \log x, 3 \log y, 7 \log z \) are in A.P. with common difference \( \frac{1}{2} \). Then \( r - p - q \) is equal to:

• (1) 2
• (2) 6
• (3) 12
• (4) -6

Let \( p, q \in \mathbb{R} \) and \( (1 - \sqrt{3}i)^{200} = 2^{199} (p + iq) \), then \( p \) and \( q \) are equal to:

• (1) \( \sqrt{3} \) and \( -\sqrt{3} \)
• (2) \( 2 \) and \( -3 \)
• (3) \( 4 \) and \( 5 \)
• (4) \( 3 \) and \( 2 \)

The relation \( R = \{(a, b): gcd(a, b) = 1, 2a = b, a, b \in \mathbb{Z}\} \) is:

• (1) transitive but not reflexive
• (2) symmetric but not transitive
• (3) reflexive but not symmetric
• (4) neither symmetric nor transitive

The compound statement \( (\neg (P \land Q)) \lor ((\neg P) \land Q) \) is equivalent to:

• (1) \( (\neg P) \lor Q \)
• (2) \( \neg Q \lor P \)
• (3) \( (\neg Q) \land (\neg P) \)
• (4) \( (\neg P) \lor Q \)

If \( f(x) = \frac{\sin(\frac{1}{x})}{x} \), then at \( x = 0 \):

• (1) \( f(x) \) is continuous but not differentiable
• (2) \( f(x) \) is continuous but \( f' \) is not continuous
• (3) \( f' \) is continuous but not differentiable
• (4) \( f' \) is continuous and differentiable

The equation \( x^2 - 4x + [x] + 3 = [x] \), where \( [x] \) denotes the greatest integer function, has:

• (1) exactly two solutions in \( (-\infty, \infty) \)
• (2) no solution
• (3) a unique solution in \( (-\infty, 1) \)
• (4) a unique solution in \( (-\infty, 0) \)

Let \( \Omega \) be the sample space and \( A \subset \Omega \) be an event. Given below are two statements:
(S1): If \( P(A) = 0 \), then \( A = \Omega \)
(S2): If \( P(A) = 1 \), then \( A = \Omega \).
Which of the following is correct?

• (1) only (S1) is true
• (2) only (S2) is true
• (3) both (S1) and (S2) are true
• (4) both (S1) and (S2) are false

Let C be the largest circle centered at (2, 0) and inscribed in the ellipse \( \frac{x^2}{36} + \frac{y^2}{16} = 1 \). If the point \( (1, \alpha) \) lies on C, then \( 10 \alpha^2 \) is equal to:

Suppose \( \sum_{r=0}^{2023} r^2 \binom{2023}{r} = 2023 \times \alpha \times 2022^{2022} \). Then the value of \( \alpha \) is:

The value of \( \int_0^{12} |x^2 - 3x + 2| dx \) is:

The number of 9-digit numbers that can be formed using all the digits of the number 123412341 such that the even digits occupy only even places, is:

Let \( \lambda \in \mathbb{R} \) and let the equation \( E = |x^2 - 2|x| + |x - 3| = 0 \). Then the largest element in the set \( S = \{ x + \lambda : x is an integer solution of E \} \) is:

A boy needs to select five courses from 12 available courses, out of which 5 courses are language courses. If he can choose at most two language courses, then the number of ways he can choose five courses is:

Let a tangent to the curve \( 9x^2 + 16y^2 = 144 \) intersect the coordinate axes at the points A and B. Then, the minimum length of the line segment AB is:

The value of \( \int_0^{\frac{\pi}{2}} \frac{( \cos x )^{2023}}{( \sin x )^{2023} + ( \cos x )^{2023}} dx \) is:

The shortest distance between the lines \( \frac{x-2}{3} = \frac{y+1}{2} = \frac{z-6}{2} \) and \( \frac{x-6}{1} = \frac{y-1}{2} = \frac{z+8}{3} \) is equal to:

The 4^th term of a GP is 500 and its common ratio is \( \frac{1}{m} \), where \( m \in \mathbb{N} \). Let \( S_n \) denote the sum of the first \( n \) terms of this GP. If \( S_6 > S_5 + 1 \) and \( S_7 < S_6 + \frac{1}{2} \), then the number of possible values of \( m \) is: