JEE Main 2023 Mathematics Question Paper Jan 25 Shift 1- Download PDF

Let \(M\) be the maximum value of the product of two positive integers when their sum is \(66\). Let the sample space \(S = \{x \in \mathbb{Z} : (66 - x)x \geq \frac{5}{9}M\}\) and the event \(A = \{x \in S : x is a multiple of 3\}\). Then \(P(A)\) is equal to:

• (A) \(\frac{15}{44}\)
• (B) \(\frac{1}{3}\)
• (C) \(\frac{7}{22}\)
• (D) \(\frac{1}{2}\)

Let \(\vec{a}, \vec{b}, \vec{c}\) be three non-zero vectors such that \(\vec{b} \cdot \vec{c} = 0\) and \(\vec{a} \times \vec{b} = \frac{\vec{b} - \vec{c}}{2}\). If \(\vec{d}\) is a vector such that \(\vec{b} \cdot \vec{d} = \vec{a} \cdot \vec{b}\), then \((\vec{a} \times \vec{b}) \cdot (\vec{c} \times \vec{d})\) is equal to:

• (A) \(1\)
• (B) \(\frac{1}{4}\)
• (C) \(2\)
• (D) \(\frac{1}{2}\)

Let \(y = y(x)\) be the solution curve of the differential equation \[ \frac{dy}{dx} = \frac{y}{x}(1 + xy^2(1 + \log x)), \quad x > 0, \, y(1) = 3. \]
Then \(\frac{y^2(x){9}\) is equal to:

• (A) \(\frac{x^2}{5 - 2x^3(2 + \log x^3)\)
• (B) \(\frac{x^2}{2x^3(2 + \log x^3) - 3}\)
• (C) \(\frac{x^2}{3x^3(1 + \log x^2) - 2}\)
• (D) \(\frac{x^2}{7 - 3x^3(2 + \log x^2)\)

The value of \[ \lim_{n \to \infty} \frac{1 + 2 - 3 + 4 + 5 - 6 + \ldots + (3n - 2) + (3n - 1) - 3n}{\sqrt{2n^4 + 4n + 3} - \sqrt{n^4 + 5n + 4}} \]
is:

• (A) \(\frac{\sqrt{2} + 1}{2}\)
• (B) \(3(\sqrt{2} + 1)\)
• (C) \(\frac{3}{2}(\sqrt{2} + 1)\)
• (D) \(\frac{3}{2}\sqrt{2}\)

The points of intersection of the line \(ax + by = 0\), \((a \neq b)\) and the circle \(x^2 + y^2 - 2x = 0\) are \(A(\alpha, 0)\) and \(B(1, \beta)\). The image of the circle with \(AB\) as a diameter in the line \(x + y + 2 = 0\) is:

• (A) \(x^2 + y^2 + 5x + 5y + 12 = 0\)
• (B) \(x^2 + y^2 + 3x + 5y + 8 = 0\)
• (C) \(x^2 + y^2 + 3x + 3y + 4 = 0\)
• (D) \(x^2 + y^2 - 5x - 5y + 12 = 0\)

The mean and variance of the marks obtained by the students in a test are 10 and 4 respectively. Later, the marks of one of the students is increased from 8 to 12. If the new mean of the marks is 10.2, then their new variance is equal to:

• (A) \(4.04\)
• (B) \(4.08\)
• (C) \(3.96\)
• (D) \(3.92\)

Let \[ y(x) = (1 + x)(1 + x^2)(1 + x^4)(1 + x^8)(1 + x^{16}). \]
Then \(y' - y''\) at \(x = -1\) is equal to:

• (A) \(976\)
• (B) \(464\)
• (C) \(496\)
• (D) \(944\)

The vector \(\vec{a} = -\hat{i} + 2\hat{j} + \hat{k}\) is rotated through a right angle, passing through the y-axis in its way, and the resulting vector is \(\vec{b}\). Then the projection of \(3\vec{a} + \sqrt{2}\vec{b}\) on \(\vec{c} = 5\hat{i} + 4\hat{j} + 3\hat{k}\) is:

• (A) \(3\sqrt{2}\)
• (B) 1
• (C) \(\sqrt{6}\)
• (D) \(2\sqrt{3}\)

The minimum value of the function \[ f(x) = \int_{0}^{2} e^{|k-t|} dt \]
is:

• (A) \(2(e - 1)\)
• (B) \(2e - 1\)
• (C) 2
• (D) \(e(e - 1)\)

Consider the lines \(L_1\) and \(L_2\) given by \[ L_1: \frac{x-1}{2} = \frac{y-3}{2} = \frac{z-2}{2}, \quad L_2: \frac{x-2}{1} = \frac{y-2}{2} = \frac{z-3}{3}. \]
A line \(L_3\) having direction ratios \(1, -1, -2\) intersects \(L_1\) and \(L_2\) at the points \(P\) and \(Q\) respectively. Then the length of line segment \(PQ\) is:

• (A) \(2\sqrt{6}\)
• (B) \(3\sqrt{2}\)
• (C) \(4\sqrt{3}\)
• (D) 4

Let \(x = 2\) be a local minima of the function \[ f(x) = 2x^4 - 18x^2 + 8x + 12, \quad x \in (-4, 4). \]
If \(M\) is the local maximum value of the function \(f(x)\) in \((-4, 4)\), then \(M\) is:

• (A) \(12\sqrt{6} - \frac{33}{2}\)
• (B) \(12\sqrt{6} - \frac{31}{2}\)
• (C) \(18\sqrt{6} - \frac{33}{2}\)
• (D) \(18\sqrt{6} - \frac{31}{2}\)

Let \(z_1 = 2 + 3i\) and \(z_2 = 3 + 4i\). The set \[ S = \{ z \in \mathbb{C} : |z - z_1|^2 - |z - z_2|^2 = |z_1 - z_2|^2 \} \]
 represents a:

• (A) straight line with sum of its intercepts on the coordinate axes 14
• (B) hyperbola with the length of the transverse axis 7
• (C) straight line with the sum of its intercepts on the coordinate axes equals \(-18\)
• (D) hyperbola with eccentricity 2

The distance of the point \((6, -2\sqrt{2})\) from the common tangent \(y = mx + c, \, m > 0\), of the curves \(x = 2y^2\) and \(x = 1 + y^2\) is:

• (A) \(\frac{1}{3}\)
• (B) \(5\)
• (C) \(\frac{14}{3}\)
• (D) \(5\sqrt{3}\)

Let \(S_1\) and \(S_2\) be respectively the sets of all \(a \in \mathbb{R} - \{0\}\) for which the system of linear equations: \[ \begin{aligned} ax + 2ay - 3az &= 1,
(2a + 1)x + (2a + 3)y + (a + 1)z &= 2,
(3a + 5)x + (a + 5)y + (a + 2)z &= 3, \end{aligned} \]
has unique solution and infinitely many solutions. Then:

• (A) \(n(S_1) = 2\) and \(S_2\) is an infinite set
• (B) \(S_1\) is an infinite set and \(n(S_2) = 2\)
• (C) \(S_1 = \emptyset\) and \(S_2 = \mathbb{R} - \{0\}\)
• (D) \(S_1 = \mathbb{R} - \{0\}\) and \(S_2 = \emptyset\)

Let \(f(x) = \int \frac{2x}{x^2 + 1}(x^2 + 3) \, dx\). If \(f(3) = \frac{1}{2}(\log_e 5 - \log_e 6)\), then \(f(4)\) is equal to:

• (A) \(\frac{1}{2}(\log_e 17 - \log_e 19)\)
• (B) \(\log_e 17 - \log_e 18\)
• (C) \(\frac{1}{2}(\log_e 19 - \log_e 17)\)
• (D) \(\log_e 19 - \log_e 17\)

The statement \( (p \land (\sim q)) \Rightarrow (p \Rightarrow (\sim q)) \) is:

• (A) Equivalent to \( (\sim p) \lor (\sim q) \)
• (B) A tautology
• (C) Equivalent to \( p \lor q \)
• (D) A contradiction

Let \(f : (0, 1) \to \mathbb{R}\) be a function defined by \[ f(x) = \frac{1}{1 - e^{-x}}, \]
and \[ g(x) = (f(-x) - f(x)). \]
Consider two statements:

[(I)] \(g\) is an increasing function in \((0, 1)\),
[(II)] \(g\) is one-one in \((0, 1)\).

Then:

• (A) Only (I) is true
• (B) Only (II) is true
• (C) Neither (I) nor (II) is true
• (D) Both (I) and (II) are true

The distance of the point P(4, 6, -2) from the line passing through the point \((-3, 2, 3)\) and parallel to a line with direction ratios 3, 3, -1 is equal to:

• (A) 3
• (B) \(\sqrt{6}\)
• (C) \(2\sqrt{3}\)
• (D) \(\sqrt{14}\)

Let \(x, y, z > 1\) and \[ A = \begin{bmatrix} 1 & \log_x y & \log_x z
\log_y x & 2 & \log_y z
\log_z x & \log_z y & 3 \end{bmatrix}. \]
Then \(adj(adj A^2)\) is equal to:

• (A) \(6^4\)
• (B) \(2^8\)
• (C) \(4^8\)
• (D) \(2^4\)

If \(a_r\) is the coefficient of \(x^{10-r}\) in the binomial expansion of \((1 + x)^{10}\), then \[ \sum_{r=1}^{10} r^3 \left( \frac{a_r}{a_{r-1}} \right)^2 is equal to: \]

• (A) 4895
• (B) 1210
• (C) 5445
• (D) 3025

Number of Non-Empty Subsets with Sum Divisible by 3

Problem: Let \( S = \{1, 2, 3, 5, 7, 10, 11\} \). The number of non-empty subsets of \( S \) such that the sum of their elements is divisible by 3 is __.

For some \( a, b, c \in \mathbb{N} \), let \( f(x) = ax - 3 \) and \( g(x) = x^b + c \), \( x \in \mathbb{R} \). If \( (f \circ g)^{-1}(x) = \left(\frac{x - 7}{2}\right)^{1/3} \), then \( (f \circ g)(ac) + (g \circ f)(b) \) is equal to __.

The vertices of a hyperbola \( H \) are \( (\pm 6, 0) \) and its eccentricity is \( \frac{\sqrt{5}}{2} \). Let \( N \) be the normal to \( H \) at a point in the first quadrant and parallel to the line \( \sqrt{2}x + y = 2\sqrt{2} \). If \( d \) is the length of the line segment of \( N \) between \( H \) and the y-axis, then \( d^2 \) is equal to __.

Let \( S = \left\{\alpha : \log_2 \left(9^{2\alpha-4} + 13\right) - \log_2 \left(\frac{5}{2} \cdot 3^{2\alpha-4} + 1\right) = 2 \right\}. \)
Then the maximum value of \( \beta \) for which the equation \[ x^2 - 2\left(\sum_{\alpha \in S} \alpha\right)x + \sum_{\alpha \in S} (\alpha + 1)^2 \beta = 0 \]
has real roots, is __.

The constant term in the expansion of \( \left(2x + \frac{1}{x^7} + 3x^2\right)^5 \) is __.

Let \( A_1, A_2, A_3 \) be the three A.P. with the same common difference \( d \) and having their first terms as \( A, A+1, A+2 \), respectively. Let \( a, b, c \) be the 7th, 9th, and 17th terms of \( A_1, A_2, A_3 \), respectively, such that \[ \begin{vmatrix} a & 7 & 1
2b & 17 & 1
c & 17 & 1 \end{vmatrix} + 70 = 0. \]
If \( a = 29 \), then the sum of the first 20 terms of an AP whose first term is \( c - a - b \) and common difference is \( \frac{d}{12} \), is equal to __.

If the sum of all the solutions of \[ \tan^{-1}\left(\frac{2x}{1-x^2}\right) + \cot^{-1}\left(\frac{1-x^2}{2x}\right) = \frac{\pi}{3}, \]
where \( -1 < x < 1, x \neq 0 \), is \( \alpha - \frac{4}{\sqrt{3}} \), then \( \alpha \) is equal to __.

Let the equation of the plane passing through the line \[ x - 2y - z - 5 = 0 \quad and \quad x + y + 3z - 5 = 0, \]
and parallel to the line \[ x + y + 2z - 7 = 0 \quad and \quad 2x + 3y + z - 2 = 0, \]
be \( ax + by + cz = 65 \). Then the distance of the point \( (a, b, c) \) from the plane \( 2x + 2y - z + 16 = 0 \) is __.

Let \( x \) and \( y \) be distinct integers where \( 1 \leq x \leq 25 \) and \( 1 \leq y \leq 25 \). Then, the number of ways of choosing \( x \) and \( y \), such that \( x + y \) is divisible by 5, is __.

It the area enclosed by the parabolas \( P_1: 2y = 5x^2 \) and \( P_2: x^2 - y + 6 = 0 \) is equal to the area enclosed by \( P_1 \) and \( y = \alpha x, \alpha > 0 \), then \( \alpha^3 \) is equal to __.