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, \( A^{-1} = \alpha A + \beta I \) and \( \alpha + \beta = -2 \), then \( 4\alpha^{2} + \beta^{2} + \lambda^{2} \) is equal to:

Let the vectors \( \vec{u}_{1} = \hat{i} + \hat{j} + a\hat{k} \), \( \vec{u}_{2} = \hat{i} + b\hat{j} + \hat{k} \) and \( \vec{u}_{3} = c\hat{i} + \hat{j} + \hat{k} \) be coplanar. If the vectors \( \vec{v}_{1} = (a+b)\hat{i} + c\hat{j} + c\hat{k} \), \( \vec{v}_{2} = a\hat{i} + (b+c)\hat{j} + a\hat{k} \) and \( \vec{v}_{3} = b\hat{i} + b\hat{j} + (c+a){k} \) are also coplanar, then \( 6(a+b+c) \) is equal to:

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 \( \cos^{-1}\!\left(\dfrac{1}{3}\right) \). 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:

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

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 non-trivial solution. Then \( \dfrac{120}{\pi} \displaystyle\sum_{\theta \in S} \theta \) is equal to:

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 \( Q \) on it. If the circumcircle of the triangle \( OPQ \) passes through the point \( \left(\alpha, \dfrac{1}{2}\right) \), then a value of \( \alpha \) is:

Let \( P \) be the plane passing through the line \( \dfrac{x-1}{1}=\dfrac{y-2}{-3}=\dfrac{z+5}{7} \) 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:

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

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

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

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

Let \( A(0,1), B(1,1) \) and \( C(1,0) \) be the mid-points 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 \( (\alpha+\beta\sqrt{2},0) \), where \( \alpha \) and \( \beta \) are rational numbers, then \( \dfrac{\alpha}{\beta^{2}} \) is equal to:

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

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

Let the mean and variance of \( 12 \) observations be \( \dfrac{9}{2} \) 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 \( \dfrac{m}{n} \), where \( m \) and \( n \) are coprime, then \( m+n \) is equal to:

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

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

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

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

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:

Let the solution curve \( x=x(y) \), \( 0<y<\dfrac{\pi}{2}\), of the differential equation \((\log_e(\cos y))^2 \cos y \, dx - (1 + 3x \log_e(\cos y)) \sin y \, dy = 0\) satisfy \(x\left(\dfrac{\pi}{3}\right) = \dfrac{1}{2\log_e 2}\).If \(\displaystyle x\left(\dfrac{\pi}{6}\right) = \dfrac{1}{\log_e m - \log_e n}\),where \(m\) and \(n\) are coprime, then \(mn\) 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 _______.

Let the area enclosed by the lines \( x+y=2 \), \( y=0 \), \( x=0 \) and the curve \( f(x)=\min\!\left\{x^{2}+\dfrac{3}{4},\,1+[ x ] \right\} \) where \( [ x ] \) denotes the greatest integer \( \le x \), be \( A \). Then the value of \( 12A \) is ________.

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 \( \le x \). Then \( m^{2}+mn+n^{2} \) is equal to __________.

The ordinates of the points \( P \) and \( Q \) on the parabola with focus \( (3,0) \) and directrix \( x=-3 \) are in the ratio \( 3:1 \). If \( R(\alpha,\beta) \) is the point of intersection of the tangents to the parabola of \( P \) and \( Q \), then \( \dfrac{\beta^2}{\alpha} \) is equal to ________.

Let \( k \) and \( m \) be positive real numbers such that the function is differentiable for all \(x>0\).Then\(\displaystyle \frac{8 f'(8)}{f'\left(\dfrac{1}{8}\right)}\) is equal to ______.

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

Let \( [t] \) denote the greatest integer function. If \( \displaystyle \int_{0}^{2.4}\!\!\left[\!x^{2}\!\right] \,dx=\alpha+\beta\sqrt{2}+\gamma\sqrt{3}+\delta\sqrt{5} \), then \( \alpha+\beta+\gamma+\delta \) is equal to ________.

If domain of the function \( \log_{e}\!\left(\dfrac{6x^{2}+5x+1}{2x-1}\right)+\cos^{-1}\!\left(\dfrac{2x^{2}-3x+4}{3x-5}\right) \) is \( (\alpha,\beta)\cup(\gamma,\delta] \), then \( 18(\alpha^{2}+\beta^{2}+\gamma^{2}+\delta^{2}) \) is equal to ________.

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

JEE Main 2023 8th April Evening Shift

Physics Chemistry Mathematics

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