JEE Main 2023 Mathematics Question Paper Jan 24 Shift 2- Download PDF

Let the six numbers \( a_1, a_2, a_3, a_4, a_5, a_6 \) be in A.P., and \( a_1 + a_3 = 10 \). If the mean of these six numbers is \( \frac{19}{2} \) and their variance is \( \sigma^2 \), then \( 8\sigma^2 \) is equal to:

\begin{flushleft

• (1) 220
• (2) 210
• (3) 200
• (4) 105

Let \( f(x) \) be a function such that \( f(x + y) = f(x)f(y) \) for all \( x, y \in \mathbb{N} \). If \( f(1) = 3 \) and \( \sum_{k=1}^{n} f(k) = 3279 \), then the value of \( n \) is:

\begin{flushleft

• (1) 6
• (2) 8
• (3) 7
• (4) 9

The number of real solutions of the equation \( 3(x^2 + \frac{1}{x^2}) - 2(x+\frac{1}{x}) \), is:

\begin{flushleft

• (1) 4
• (2) 0
• (3) 3
• (4) 2

If \( f(x) = \frac{2^{2x}}{2^{2x} + 2} \), \( x \in \mathbb{R} \), then \( f\left(\frac{1}{2023}\right) + f\left(\frac{2}{2023}\right) + \ldots + f\left(\frac{2022}{2023}\right) \) is equal to:

\begin{flushleft

• (1) 2011
• (2) 1010
• (3) 2010
• (4) 1011

If \( f(x) = x^3 - x^2 f'(1) + x f''(2) - f'''(3) \), \( x \in \mathbb{R} \), then:

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• (1) \( 3f(1) + f(2) = f(3) \)
• (2) \( f(3) - f(2) = f(1) \)
• (3) \( 2f(0) - f(1) + f(3) = f(2) \)
• (4) \( f(1) + f(2) + f(3) = f(0) \)

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

\begin{flushleft

• (1) 120
• (2) 168
• (3) 220
• (4) 48

If the system of equations \[ x + 2y + 3z = 3 \quad \cdots (i) \] \[ 4x + 3y - 4z = 4 \quad \cdots (ii) \] \[ 8x + 4y - z = 9 + \mu \quad \cdots (iii) \]
has infinitely many solutions, then the ordered pair \( (\lambda, \mu) \) is equal to:

\begin{flushleft

• (1) \( \left(\frac{72}{5}, \frac{21}{5}\right) \)
• (2) \( \left(\frac{-72}{5}, \frac{-21}{5}\right) \)
• (3) \( \left(\frac{72}{5}, \frac{-21}{5}\right) \)
• (4) \( \left(\frac{-72}{5}, \frac{21}{5}\right) \)

The value of \[ \left( \frac{1 + \sin \frac{2\pi}{9} + i \cos \frac{2\pi}{9}}{1 + \sin \frac{2\pi}{9} - i \cos \frac{2\pi}{9}} \right)^3 \]
is:

\begin{flushleft

• (1) \( -\frac{1}{2}(1 - i\sqrt{3}) \)
• (2) \( \frac{1}{2}(1 - i\sqrt{3}) \)
• (3) \( -\frac{1}{2}(\sqrt{3} - i) \)
• (4) \( \frac{1}{2}(\sqrt{3} + i) \)

The equations of the sides AB and AC of a triangle ABC are: \[ (\lambda + 1)x + \lambda y = 4 \quad and \quad \lambda x + (1 - \lambda)y + \lambda = 0, \]
respectively. Its vertex A is on the y-axis and its orthocentre is \( (1, 2) \). The length of the tangent from the point C to the part of the parabola \( y^2 = 6x \) in the first quadrant is:

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• (1) \( \sqrt{6} \)
• (2) \( 2\sqrt{2} \)
• (3) 2
• (4) 4

The set of all values of \( a \) for which \[ \lim_{x \to a} \left([\![x - 5]\!] - [\![2x + 2]\!]\right) = 0, \]
where \( [\![x]\!] \) denotes the greatest integer less than or equal to \( x \), is equal to:

\begin{flushleft

• (1) \( (-7.5, -6.5) \)
• (2) \( (-7.5, -6.5] \)
• (3) \( [-7.5, -6.5] \)
• (4) \( [-7.5, -6.5) \)

If \[ {}({}^{30}C_1)^2+ 2({}^{30}C_2)^2 + 3({}^{30}C_2)^2 + \ldots + 30({}^{30}C_{30})^2 = \frac{\alpha \cdot 60!}{(30!)^2}, \]
then \( \alpha \) is equal to:

\begin{flushleft

• (1) 30
• (2) 60
• (3) 15
• (4) 10

Let the plane containing the line of intersection of the planes \[ P_1: x + (\lambda + 4)y + z = 1 \quad and \quad P_2: 2x + y + z = 2 \]
pass through the points \( (0, 1, 0) \) and \( (1, 0, 1) \). Then the distance of the point \( (2\lambda, \lambda, -\lambda) \) from the plane \( P_2 \) is:

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• (1) \( 5\sqrt{6} \)
• (2) \( 4\sqrt{6} \)
• (3) \( 2\sqrt{6} \)
• (4) \( 3\sqrt{6} \)

Let \( \vec{\alpha} = 4\hat{i} + 3\hat{j} + 5\hat{k} \) and \( \vec{\beta} = \hat{i} + 2\hat{j} - 4\hat{k} \). Let \( \vec{\beta}_1 \) be parallel to \( \vec{\alpha} \) and \( \vec{\beta}_2 \) be perpendicular to \( \vec{\alpha} \). If \( \vec{\beta} = \vec{\beta}_1 + \vec{\beta}_2 \), then the value of \( 5\vec{\beta}_2 \cdot (\hat{i} + \hat{j} + \hat{k}) \) is:

\begin{flushleft

• (1) 6
• (2) 11
• (3) 7
• (4) 9

The locus of the midpoints of the chords of the circle \( C_1: (x - 4)^2 + (y - 5)^2 = 4 \), which subtend an angle \( \theta_1 \) at the centre of the circle \( C_1 \), is a circle of radius \( r_1 \). If \( \theta_1 = \frac{\pi}{3} \), \( \theta_3 = \frac{2\pi}{3} \), and \( r_1^2 = r_2^2 + r_3^2 \), then \( \theta_2 \) is equal to:

\begin{flushleft

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

If the foot of the perpendicular drawn from \( (1, 9, 7) \) to the line passing through the point \( (3, 2, 1) \) and parallel to the planes \( x + 2y + z = 0 \) and \( 3y - z = 3 \) is \( (\alpha, \beta, \gamma) \), then \( \alpha + \beta + \gamma \) is equal to:

\begin{flushleft

• (1) \( -1 \)
• (2) \( 3 \)
• (3) \( 1 \)
• (4) \( 5 \)

Let \( y = y(x) \) be the solution of the differential equation \[ (x^2 - 3y^2)dx + 3xy \, dy = 0, \quad y(1) = 1. \]
Then \( 6y^2(e) \) is equal to:

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

Let \( p \) and \( q \) be two statements. Then \( \sim(p \land (p \to \sim q)) \) is equivalent to:

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• (1) \( p \lor (p \land \sim q) \)
• (2) \( p \lor (\sim p \land q) \)
• (3) \( \sim p \lor q \)
• (4) \( p \lor (p \land q) \)

The number of square matrices of order 5 with entries from the set \( \{0, 1\} \), such that the sum of all the elements in each row is 1 and the sum of all the elements in each column is also 1, is:

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• (1) \( 225 \)
• (2) \( 120 \)
• (3) \( 150 \)
• (4) \( 125 \)

The value of the integral: \[ \int_{\frac{3\sqrt{2}}{4}}^{\frac{3\sqrt{3}}{4}} \frac{48}{\sqrt{9 - 4x^2}} \, dx \quad is equal to: \]

\begin{flushleft

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

Let \( A \) be a \( 3 \times 3 \) matrix such that \( |adj(adj(A))| = 12^4 \). Then \( |A^{-1}adj(A)| \) is equal to:

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• (1) \( 2\sqrt{3} \)
• (2) \( \sqrt{6} \)
• (3) \( 12 \)
• (4) \( 1 \)

The urns \( A \), \( B \), and \( C \) contain \( 4 \) red, \( 6 \) black; \( 5 \) red, \( 5 \) black, and \( \lambda \) red; \( 4 \) black balls respectively. One of the urns is selected at random, and a ball is drawn. If the ball drawn is red and the probability that it is drawn from urn \( C \) is \( 0.4 \), then the square of the length of the side of the largest equilateral triangle, inscribed in the parabola \( y^2 = \lambda x \) with one vertex at the vertex of the parabola, is:

If the area of the region bounded by the curves \( y^2 - 2y = -x \) and \( x + y = 0 \) is \( A \), then \( 8A \) is equal to:

If \[ \frac{1^3 + 2^3 + 3^3 + \ldots (up to \( n \) terms)}{1 \cdot 3 + 2 \cdot 5 + 3 \cdot 7 + \ldots (up to \( n \) terms)} = \frac{9}{5}, \]
then the value of \( n \) is:

Let \( f \) be a differentiable function defined on \( \left(0, \frac{\pi}{2}\right) \) such that \( f(x) > 0 \) and \[ f(x) + \int_0^x f(t)\sqrt{1 - (\log_e f(t))^2} \, dt = e, \quad \forall x \in \left[0, \frac{\pi}{2}\right]. \]
Then \( \left(6 \log_e f\left(\frac{\pi}{6}\right)\right)^2 \) is equal to:

The minimum number of elements that must be added to the relation \( R = \{(a, b), (b, c), (b, d)\} \) on the set \( \{a, b, c, d\} \) so that it is an equivalence relation, is:

Let \( \mathbf{a} = \mathbf{i} + 2\mathbf{j} + \lambda \mathbf{k}, \, \mathbf{b} = 3\mathbf{i} - 5\mathbf{j} - \lambda \mathbf{k}, \, \mathbf{a} \cdot \mathbf{c} = 7, \, 2\mathbf{b} \cdot \mathbf{c} + 43 = 0, \, \mathbf{a} \times \mathbf{c} = \mathbf{b} \times \mathbf{c} \). Then \( |\mathbf{a} \cdot \mathbf{b}| \) is equal to:

Let the sum of the coefficients of the first three terms in the expansion of \[ \left(x - \frac{3}{x^2}\right)^n, \quad x \neq 0, \, n \in \mathbb{N}, \]
be \( 376 \). Then the coefficient of \( x^4 \) is:

If the shortest distance between the lines \[ \frac{x + \sqrt{6}}{2} = \frac{y - \sqrt{6}}{4} = \frac{z}{5}, \quad \frac{x - \lambda}{3} = \frac{y - 2\sqrt{6}}{4} = \frac{z + 2\sqrt{6}}{5} \]
is \( 6 \), then the square of the sum of all possible values of \( \lambda \) is:

Let \( S = \{ \theta \in [0, 2\pi) : \tan(\cos \theta) + \tan(\sin \theta) = 0 \} \). Then \( \sum_{\theta \in S} \sin^2 \left(\theta + \frac{\pi}{4}\right) \) is equal to:

The equations of the sides \( AB \), \( BC \), and \( CA \) of a triangle \( \Delta ABC \) are: \[ 2x + y = 0, \quad x + py = 21a \, (a \neq 0), \quad x - y = 3, \]
and \( P(2, a) \) is the centroid of \( \Delta ABC \). Then \( (BC)^2 \) is equal to: