JEE Main 2023 Mathematics Question Paper Jan 31 Shift 2- Download Paper with Solution PDF Here

JEE Main 2023 Mathematics Questions with Solutions

SECTION A

Question 1:

If \( \phi(x) = \frac{1}{\sqrt{x}} \int_{\frac{x}{4}}^{x} \left( 4\sqrt{2} \sin t - 3 \phi(t) \right) \, dt, \, x > 0, \)

then \( \phi \left( \frac{\pi}{4} \right) \) is equal to:

• (1) \( \frac{8}{\sqrt{\pi}} \)
• (2) \( \frac{6}{6 + \sqrt{\pi}} \)
• (3) \( \frac{8}{6 + \sqrt{\pi}} \)
• (4) \( \frac{4}{6 - \sqrt{\pi}} \)

If a point \( P(\alpha, \beta, \gamma) \) satisfying the equation \[ \begin{pmatrix} 2 & 10 & 8
9 & 3 & 8
8 & 4 & 8 \end{pmatrix} \begin{pmatrix} \alpha
\beta
\gamma \end{pmatrix} = \begin{pmatrix} 0
0
0 \end{pmatrix} \]
lies on the plane \( 2x + 4y + 3z = 5 \), then \( 6\alpha + 9\beta + 7\gamma \) is equal to:

• (1) \( 1 \)
• (2) \( \frac{11}{5} \)
• (3) \( \frac{5}{4} \)
• (4) \( 11 \)

Let \( a_1, a_2, a_3, \dots \) be an A.P. If \( a_4 = 3 \), the product \( a_1 a_4 \) is minimum and the sum of its first \( n \) terms is zero, then \( n! - 4a_n(a_{n+2}) \) is equal to:

• (1) \( 24 \)
• (2) \( \frac{33}{4} \)
• (3) \( \frac{381}{4} \)
• (4) \( 9 \)

Let \( (a, b) \subset (0, 2\pi) \) be the largest interval for which \[ \sin^{-1}(\sin \theta) - \cos^{-1}(\sin \theta) > 0, \quad \theta \in (0, 2\pi) \]
holds. If \[ \alpha x^2 + \beta x + \sin^{-1}\left( (x^2 - 6x + 10) \right) + \cos^{-1}\left( (x^2 - 3)^2 + 1 \right) = 0 \]
and \( \alpha - \beta = b - a \), then \( \alpha \) is equal to:

• (1) \( \frac{\pi}{48} \)
• (2) \( \frac{\pi}{16} \)
• (3) \( \frac{\pi}{12} \)
• (4) \( \frac{\pi}{8} \)

Let \( y = y(x) \) be the solution of the differential equation \[ (3y^2 - 5x^2) y \, dx + 2x(x^2 - y^2) \, dy = 0, \]
such that \( y(1) = 1 \). Then \[ \left( y(2) \right)^3 - 12y(2) \, is equal to: \]

• (1) \( 32\sqrt{2} \)
• (2) \( 64 \)
• (3) \( 16\sqrt{2} \)
• (4) \( 32 \)

The set of all values of \( a^2 \) for which the line \( x + y = 0 \) bisects two distinct chords drawn from a point \( P\left( \frac{1 + a}{2}, \frac{1 - a}{2} \right) \) on the circle \[ 2x^2 + 2y^2 - (1 + a)x - (1 - a)y = 0 \]
is equal to:

• (1) \( (8, \infty) \)
• (2) \( (4, \infty) \)
• (3) \( (0, 4) \)
• (4) \( (2, 12) \)

Among the relations \[ S = \left\{ (a, b) : a, b \in \mathbb{R} \setminus \{ 0 \}, a^2 + b^2 > 0 \right\} \]
And \[ T = \left\{ (a, b) : a, b \in \mathbb{R}, a^2 - b^2 \in \mathbb{Z} \right\} \]
which of the following is true?

• (1) \( S \) is transitive but \( T \) is not.
• (2) \( T \) is symmetric but \( S \) is not.
• (3) Neither \( S \) nor \( T \) is transitive.
• (4) Both \( S \) and \( T \) are symmetric.

The equation \[ e^x + 8e^{2x} + 13e^x - 8e^x + 1 = 0, \quad x \in \mathbb{R} \]
has:

• (1) two solutions and both are negative
• (2) no solution
• (3) four solutions, two of which are negative
• (4) two solutions and only one of them is negative

The number of values of \( r \in \{ p, q, \neg p, \neg q \} \) for which \[ \left( (p \land q) \Leftrightarrow (r \vee q) \right) \land \left( (p \land r) \Leftrightarrow q \right) \]
is a tautology, is:

• (1) 3
• (2) 2
• (3) 1
• (4) 4

Let \( f: \mathbb{R} \setminus \{ 2, 6 \} \to \mathbb{R} \) be the real-valued function defined as \[ f(x) = \frac{x^2 + 2x + 1}{x^2 - 8x + 12}. \]
Then the range of \( f \) is:

• (1) \( \left( -\infty, \frac{-21}{4} \right] \cup [0, \infty) \)
• (2) \( \left( -\infty, \frac{-21}{4} \right] \cup (0, \infty) \)
• (3) \( \left( -\infty, \frac{-21}{4} \right] \cup \left[ \frac{21}{4}, \infty \right) \)
• (4) \( \left[ \frac{-21}{4}, \infty \right) \cup [0, \infty) \)

Evaluate the limit: \[ \lim_{x \to 1} \frac{\left( \sqrt{3x+1} + \sqrt{3x-1} \right)^6}{(x + \sqrt{x^2 - 1})^3 + \left( \sqrt{3x+1} - \sqrt{3x-1} \right)^6} \]

• (1) is equal to 9
• (2) is equal to 27
• (3) does not exist
• (4) is equal to \( \frac{27}{2} \)

Let P be the plane, passing through the point \( (1, -1, -5) \) and perpendicular to the line joining the points \( (4, 1, -3) \) and \( (2, 4, 3) \). Then the distance of P from the point \( (3, -2, 2) \) is:

• (1) 6
• (2) 4
• (3) 5
• (4) 7

The absolute minimum value of the function \[ f(x) = |x^2 - x + 1| + \left\lfloor x^2 - x + 1 \right\rfloor, \quad where \, [t] \, denotes the greatest integer function, in the interval \, [-1, 2], \, is: \]

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

Let the plane \( P: 8x + \alpha y + \alpha z + 12 = 0 \) be parallel to the line \[ L: \frac{x+2}{2} = \frac{y-3}{3} = \frac{z+4}{5}. \]
If the intercept of P on the y-axis is 1, then the distance between P and L is:

• (1) \( \sqrt{14} \)
• (2) \( \frac{6}{\sqrt{14}} \)
• (3) \( \frac{\sqrt{2}}{7} \)
• (4) \( \frac{\sqrt{7}}{2} \)

The foot of perpendicular from the origin \( O \) to a plane \( P \) which meets the coordinate axes at the points A, B, C is \( (2, 4, 4) \). If the volume of the tetrahedron \( OABC \) is 144 unit\(^3\), then which of the following points is NOT on \( P \)?

• (1) \( (2, 2, 4) \)
• (2) \( (0, 4, 4) \)
• (3) \( (3, 0, 4) \)
• (4) \( (0, 6, 6) \)

Let the mean and standard deviation of marks of class A of 100 students be respectively 40 and \( \alpha > 0 \), and the mean and standard deviation of marks of class B of \( n \) students be respectively 55 and \( 30 - \alpha \). If the mean and variance of the marks of the combined class of \( 100 + n \) students are respectively 50 and 350, then the sum of variances of classes A and B is:

• (1) 500
• (2) 650
• (3) 450
• (4) 900

Let \[ \mathbf{a} = \hat{i} + 2\hat{j} + 3\hat{k}, \quad \mathbf{b} = \hat{i} - \hat{j} + 2\hat{k}, \quad \mathbf{c} = 5\hat{i} - 3\hat{j} + 3\hat{k} \]
be three vectors. If \( \mathbf{r} \) is a vector such that \( \mathbf{r} \times \mathbf{b} = \mathbf{c} \times \mathbf{b} \) and \( \mathbf{r} \cdot \mathbf{a} = 0 \), then \( 25|\mathbf{r}|^2 \) is equal to:

• (1) 449
• (2) 336
• (3) 339
• (4) 560

Let \( H \) be the hyperbola, whose foci are \( (1 \pm \sqrt{2}, 0) \) and eccentricity is \( \sqrt{2} \). Then the length of its latus rectum is:

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

Let \( \alpha > 0 \). If \[ \int_{\alpha}^{x} \frac{x}{\sqrt{x + \alpha - \sqrt{x}}} \, dx = \frac{16 + 20 \sqrt{2}}{15}, \]
then \( \alpha \) is equal to:

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

The complex number \[ z = \frac{i-1}{\cos \frac{\pi}{3} + i \sin \frac{\pi}{3}} \]
is equal to:

• (1) \( \sqrt{2} \left( \cos \frac{5\pi}{12} + i \sin \frac{5\pi}{12} \right) \)
• (2) \( \cos \frac{\pi}{12} - i \sin \frac{\pi}{12} \)
• (3) \( \sqrt{2} \left( \cos \frac{\pi}{12} + i \sin \frac{\pi}{12} \right) \)
• (4) \( \sqrt{2} \left( \cos \frac{5\pi}{12} - i \sin \frac{5\pi}{12} \right) \)

SECTION B

Question 21:

The coefficient of \( x^{-6} \), in the expansion of \[ \left( \frac{4x}{5} + \frac{5}{2x^2} \right)^9 , is: \]

Let the area of the region \[ \left\{ (x, y): |2x - 1| \leq y \leq x^2 - x, 0 \leq x \leq 1 \right\} \quad be \, A. \]
Then \( (6A + 11)^2 \) is equal to:

If \[ \frac{(2n+1)P_{n-1}}{2nP_n} = \frac{11}{21}, \quad then \quad n^2 + n + 15 \, is equal to: \]

If the constant term in the binomial expansion of \[ \left( \frac{x^{5/2}}{2} - \frac{4}{x} \right)^9 is -84 and the coefficient of x^{-3} is 2\alpha\beta, \] \[ where \beta < 0 is an odd number, then |\alpha - \beta| is equal to: \]

Let \( \vec{a}, \vec{b}, \vec{c} \) be three vectors such that \[ |\vec{a}| = \sqrt{31}, \quad |\vec{b}| = 4, \quad |\vec{c}| = 2, \quad 2(\vec{a} \times \vec{b}) = 3(\vec{c} \times \vec{a}). \]
If the angle between \( \vec{b} \) and \( \vec{c} \) is \( \frac{2\pi}{3} \), then \( \left( \frac{\vec{a} \times \vec{c}}{\vec{a} \cdot \vec{b}} \right)^2 \) is equal to:

Let \( S \) be the set of all \( a \in \mathbb{N} \) such that the area of the triangle formed by the tangent at the point \( P(b, c), b, c \in \mathbb{N} \) on the parabola \[ y^2 = 2ax \quad and the lines \quad x = b, \, y = 0 \quad is \, 16 \, unit^2, then \quad \sum_{a \in S} a \, is equal to: \]

The sum \[ 1^2 - 2 \cdot 3^2 + 3.5^2 - 4.7^2 + 5.9^2 - \dots + 15.29^2 \, is: \]

Let \( A \) be the event that the absolute difference between two randomly chosen real numbers in the sample space \[ [0, 60] \quad is less than or equal to \, a. \, If \, P(A) = \frac{11}{36}, \, then \, a \, is equal to: \]

Let \( A = [a_{ij}] \), where \( a_{ij} \in \mathbb{Z} \cap [0, 4], 1 \leq i, j \leq 2 \). The number of matrices \( A \) such that the sum of all entries is a prime number \( p \in \{2, 13\} \) is:

Let \( A \) be an \( n \times n \) matrix such that \( |A| = 2 \). If the determinant of the matrix \[ Adj (2 \cdot Adj (2A^{-1})) is 2^{84}, then n is equal to: \]