JEE Main 2023 Mathematics April 8 Shift 2 Question Paper with Solution PDF- Download Here

Let \( A = \left\{ \theta \in (0, 2\pi) : \frac{1 + 2i \sin \theta}{1 - i \sin \theta} \, is purely imaginary \right\}. \) Then the sum of the elements in \( A \) is:

• (1) \(\pi\)
• (2) \(3\pi\)
• (3) \(4\pi\)
• (4) \(2\pi\)

Let \( P \) be the plane passing through the line \[ \frac{x - 1}{1} = \frac{y - 2}{-3} = \frac{z + 5}{7} \]
and the point \( (2, 4, -3) \). If the image of the point \( (-1, 3, 4) \) in the plane \( P \) is \( (\alpha, \beta, \gamma) \), then \( \alpha + \beta + \gamma \) is equal to:

• (A) \( 12 \)
• (B) \( 9 \)
• (C) \( 10 \)
• (D) \( 11 \)

If \( A = \begin{bmatrix} 1 & 5
\lambda & 10 \end{bmatrix} \), \( A^{-1} = \alpha A + \beta I \), and \( \alpha + \beta = -2 \), then \( 4\alpha^2 + \beta^2 + \lambda^2 \) is equal to:
 

• (1) 14
• (2) 12
• (3) 19
• (4) 10

The area of the quadrilateral \(ABCD\) with vertices \(A(2, 1, 1)\), \(B(1, 2, 5)\), \(C(-2, -3, 5)\), and \(D(1, -6, -7)\) is equal to:
 

• (1) 54
• (2) \(9\sqrt{38}\)
• (3) 48
• (4) \(8\sqrt{38}\)

\( 25^{190} - 19^{190} - 8^{190} + 2^{190} \) is divisible by:
 

• (1) 34 but not by 14
• (2) 14 but not by 34
• (3) Both 14 and 34
• (4) Neither 14 nor 34

Let \( O \) be the origin and \( OP \) and \( OQ \) be the tangents to the circle \[ x^2 + y^2 - 6x + 4y + 8 = 0 \]
at the points \( P \) and \( Q \). If the circumcircle of the triangle \( OPQ \) passes through the point \[ \left( \alpha, \frac{1}{2} \right), \]
then the value of \( \alpha \) is:
 

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

Let \( a_n \) be the \( n^{th} \) term of the series \[ 5 + 8 + 14 + 23 + 35 + 50 + \ldots \]
and \( S_n = \sum_{k=1}^n a_k \). Then \( S_{30} - a_{40} \) is equal to:

• (1) 11260
• (2) 11280
• (3) 11290
• (4) 11310

If \(\alpha > \beta > 0\) are the roots of the equation \(ax^2 + bx + 1 = 0\), and \[ \lim_{x \to \frac{1}{\alpha}} \left[\frac{1 - \cos(x^2 + bx + a)}{2(1 - ax)^2}\right]^{\frac{1}{2}} = \frac{1}{k}\left(\frac{1}{\beta}- \frac{1}{\alpha}\right), \]
then \(k\) is equal to:
 

• (1) \(\beta\)
• (2) \(2\alpha\)
• (3) \(2\beta\)
• (4) \(\alpha\)

If the number of words, with or without meaning, which can be made using all the letters of the word \textbf{MATHEMATICS} in which C and S do not come together, is \((6! \cdot k)\), \(k\) is equal to:

• (1) \( 1890 \)
• (2) \( 945 \)
• (3) \( 2835 \)
• (4) \( 5670 \)

Let \( S \) be the set of all values of \( \theta \in [-\pi, \pi] \) for which the system of linear equations \[ x + y + \sqrt{3}z = 0, \] \[ -x + (\tan\theta)y + \sqrt{7}z = 0, \] \[ x + y + (\tan\theta)z = 0 \]
has a non-trivial solution. Then \( \frac{120}{\pi} \sum_{\theta \in S} \theta \) is equal to:
 

• (1) \(20\)
• (2) \(40\)
• (3) \(30\)
• (4) \(10\)

For \(a, b \in \mathbb{Z}\) and \(|a - b| \leq 10\), let the angle between the plane \(P: ax + y - z = b\) and the line \(L: x - 1 = a - y = z + 1\) be \(\cos^{-1}\left(\frac{1}{3}\right)\). If the distance of the point \((6, -6, 4)\) from the plane \(P\) is \(3\sqrt{6}\), then \(a^4 + b^2\) is equal to:
 

• (1) \(85\)
• (2) \(48\)
• (3) \(25\)
• (4) \(32\)

Let the vectors \(\mathbf{u}_1 = \hat{i} + \hat{j} + a\hat{k}, \mathbf{u}_2 = \hat{i} + b\hat{j} + \hat{k}\), and \(\mathbf{u}_3 = c\hat{i} + \hat{j} + \hat{k}\) be coplanar. If the vectors \(\mathbf{v}_1 = (a + b)\hat{i} + c\hat{j} + c\hat{k}, \mathbf{v}_2 = a\hat{i} + (b + c)\hat{j} + a\hat{k}, \mathbf{v}_3 = b\hat{i} + b\hat{j} + (c + a)\hat{k}\) are also coplanar, then \(6(a + b + c)\) is equal to:
 

• (1) \(4\)
• (2) \(12\)
• (3) \(6\)
• (4) \(0\)

The absolute difference of the coefficients of \(x^{10}\) and \(x^7\) in the expansion of \(\left(2x^2 + \frac{1}{2x}\right)^{11}\) is equal to:

• (1) \(10^3 - 10\)
• (2) \(11^3 - 11\)
• (3) \(12^3 - 12\)
• (4) \(13^3 - 13\)

Let \( A = \{1, 2, 3, 4, 5, 6, 7\} \). Then the relation \( R = \{(x, y) \in A \times A : x + y = 7\} \) is:
 

• (1) Symmetric but neither reflexive nor transitive
• (2) Transitive but neither symmetric nor reflexive
• (3) An equivalence relation
• (4) Reflexive but neither symmetric nor transitive

If the probability that the random variable \( X \) takes values \( x \) is given by \( P(X = x) = k(x + 1) 3^{-x}, x = 0, 1, 2, \dots \), where \( k \) is a constant, then \( P(X \geq 2) \) is equal to:
 

• (1) \(\frac{7}{27}\)
• (2) \(\frac{11}{18}\)
• (3) \(\frac{7}{18}\)
• (4) \(\frac{20}{27}\)

The integral \(\int \left(\frac{x}{2}\right)^x + \left(\frac{2}{x}\right)^x \log x \, dx\) is equal to:
 

• (1) \(\left(\frac{x}{2}\right)^x \log_2 \left(\frac{2}{x}\right) + C\)
• (2) \(\left(\frac{x}{2}\right)^x - \left(\frac{2}{x}\right)^x + C\)
• (3) \(\left(\frac{x}{2}\right)^x \log_2 \left(\frac{x}{2}\right) + C\)
• (4) \(\left(\frac{x}{2}\right)^x + \left(\frac{2}{x}\right)^x + C\)

The value of \( 36 \big(4 \cos^2 9^\circ - 1\big) \big(4 \cos^2 27^\circ - 1\big) \big(4 \cos^2 81^\circ - 1\big) \big(4 \cos^2 243^\circ - 1\big) \) is:
 

• (1) 27
• (2) 54
• (3) 18
• (4) 36

Let \( A(0, 1) \), \( B(1, 1) \), and \( C(1, 0) \) be the midpoints of the sides of a triangle with incentre at the point \( D \). If the focus of the parabola \( y^2 = 4ax \) passing through \( D \) is \( (\alpha + \beta \sqrt{3}, 0) \), where \( \alpha \) and \( \beta \) are rational numbers, then \( \frac{\alpha}{\beta^2} \) is equal to:
 

• (1) 6
• (2) 8
• (3) \(\frac{9}{2}\)
• (4) 12

The negation of \( (p \land (\sim q)) \lor (\sim p) \) is equivalent to:
 

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

Let the mean and variance of 12 observations be \( \frac{9}{2} \) and 4, respectively. Later on, it was observed that two observations were considered as 9 and 10 instead of 7 and 14 respectively. If the correct variance is \( \frac{m}{n} \), where \( m \) and \( n \) are coprime, then \( m + n \) is equal to:
 

• (1) 316
• (2) 317
• (3) 315
• (4) 314

Let \( R = \{a, b, c, d, e\} \) and \( S = \{1, 2, 3, 4\} \). Total number of onto functions \( f: R \to S \) such that \( f(a) \neq 1 \), is equal to:

Let \( m \) and \( n \) be the numbers of real roots of the quadratic equations \( x^2 - 12x + [x] + 31 = 0 \) and \( x^2 - 5|x+2| - 4 = 0 \), respectively, where \( [x] \) denotes the greatest integer less than or equal to \( x \). Then \( m^2 + mn + n^2 \) is equal to ______.

Let \( P_1 \) be the plane \( 3x - y - 7z = 11 \) and \( P_2 \) be the plane passing through the points \( (2, -1, 0) \), \( (2, 0, -1) \), and \( (5, 1, 1) \). If the foot of the perpendicular drawn from the point \( (7, 4, -1) \) on the line of intersection of the planes \( P_1 \) and \( P_2 \) is \( (\alpha, \beta, \gamma) \), then \( \alpha + \beta + \gamma \) is equal to _____.

If the domain of the function \[ f(x) = \log_e \left(\frac{6x^2 + 5x + 1}{2x - 1}\right) + \cos^{-1}\left(\frac{2x^2 - 3x + 4}{3x - 5}\right) \]
is \( (\alpha, \beta) \cup (\gamma, \delta) \), then \( 18(\alpha^2 + \beta^2 + \gamma^2 + \delta^2) \) is equal to _____.

Let the area enclosed by the lines \( x + y = 2 \), \( y = 0 \), \( x = 0 \), and the curve \( f(x) = \min \left\{ x^2 + \frac{3}{4}, 1 + [x] \right\} \), where \( [x] \) denotes the greatest integer less than or equal to \( x \), be \( A \). Then the value of \( 12A \) is _____.

Let \( 0 < z < y < x \) be three real numbers such that \( \frac{1}{x}, \frac{1}{y}, \frac{1}{z} \) are in an arithmetic progression and \( x, \sqrt{2}y, z \) are in a geometric progression. If \( xy + yz + zx = \frac{3}{\sqrt{2}} xyz \), then \( 3(x + y + z)^2 \) is equal to ______.

Let the solution curve \( x = x(y) \), \( 0 < y \leq \frac{\pi}{2} \), of the differential equation \[ (\log_e (\cos y))^2 \cos y \, dx - (1 + 3x \log_e (\cos y)) \sin y \, dy = 0 \]
satisfy \( x \left( \frac{\pi}{3} \right) = \frac{1}{2 \log_e 2} \). If \( x \left( \frac{\pi}{6} \right) = \frac{1}{\log_e m - \log_e n} \), where \( m \) and \( n \) are co-prime integers, then \( mn \) is equal to ______.

Let \([t]\) denote the greatest integer function. If \[ \int_0^{2.4} [x^2] dx = \alpha + \beta \sqrt{2} + \gamma \sqrt{3} + \delta \sqrt{5}, \]
then \( \alpha + \beta + \gamma + \delta \) is equal to ______.

The ordinates of the points \( P \) and \( Q \) on the parabola \( y^2 = 12x \) are in the ratio \( 3:1 \). If \( R(\alpha, \beta) \) is the point of intersection of the tangents to the parabola at \( P \) and \( Q \), then \( \frac{\beta^2}{\alpha} \) is equal to _____.

Let \( k \) and \( m \) be positive real numbers such that the function \[ f(x) = \begin{cases} 3x^2 + \frac{k}{\sqrt{x} + 1}, & 0 < x < 1,
mx^2 + k^2, & x \geq 1 \end{cases} \]
is differentiable for all \( x > 0 \). Then \( 8f'(8) \left(\frac{1}{f(8)}\right) \) is equal to ______.