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Let \( S=\left\{x \in [-6,3]-\{-2,2\}:\; \dfrac{|x+3|-1}{|x|-2}\ge 0\right\} \) and \( T=\left\{x\in \mathbb{Z}: x^2-7|x|+9\le 0\right\}. \) Then the number of elements in \( S\cap T \) is:

Let \( \alpha,\beta \) be the roots of the equation \( x^{2}-\sqrt{2}\,x+\sqrt{6}=0 \) and \( \dfrac{1}{\alpha^{2}}+1,\, \dfrac{1}{\beta^{2}}+1 \) be the roots of the equation \( x^{2}+ax+b=0 \). Then the roots of the equation \( x^{2}-(a+b-2)x+(a+b+2)=0 \) are:

Let \( A \) and \( B \) be any two \( 3\times 3 \) symmetric and skew symmetric matrices respectively. Then which of the following is NOT true?

Let \( f(x)=ax^{2}+bx+c \) be such that \( f(1)=3 \), \( f(-2)=\lambda \) and \( f(3)=4 \). If \( f(0)+f(1)+f(-2)+f(3)=14 \), then \( \lambda \) is equal to:

The function \( f:\mathbb{R}\to \mathbb{R} \) defined by \( f(x)=\displaystyle\lim_{n\to\infty}\frac{\cos(2\pi x)-x^{2n}\sin(x-1)}{1+x^{2n+1}-x^{2n}} \) is continuous for all \( x \) in:

The function \( f(x)=x e^{x(1-x)},\; x\in \mathbb{R}, \) is:

The sum of the absolute maximum and absolute minimum values of the function \( f(x) = \tan^{-1}(\sin x - \cos x) \) in the interval \( [0, \pi] \) is:

Let \( x(t)=2\sqrt{2}\cos t\,\sqrt{\sin 2t} \) and \( y(t)=2\sqrt{2}\sin t\,\sqrt{\sin 2t} \), \( t\in\left(0,\dfrac{\pi}{2}\right) \). Then \(\displaystyle \dfrac{1 + \left(\dfrac{dy}{dx}\right)^2}{\dfrac{d^2y}{dx^2}}\) at \( t=\dfrac{\pi}{4} \) is equal to:

Let \( I_n(x)=\displaystyle\int\limits_{0}^{x}\dfrac{1}{(t^2+5)^n}\,dt,\ n=1,2,3,\ldots \). Then:

The area enclosed by the curves \( y=\log_e(x+e^2) \), \( x=\log_e\!\left(\dfrac{2}{y}\right) \) and \( x=\log_e 2 \), above the line \( y=1 \) is:

Let \( y=y(x) \) be the solution curve of the differential equation \( \dfrac{dy}{dx}+\dfrac{1}{x^2-1}y=\left(\dfrac{x-1}{x+1}\right)^{1/2},\ x>1 \) passing through the point \( \left(2,\sqrt{\dfrac{1}{3}}\right) \). Then \( \sqrt{7}\,y(8) \) is equal to:

The differential equation of the family of circles passing through the points \( (0,2) \) and \( (0,-2) \) is:

Let the tangents at two points A and B on the circle \( x^2 + y^2 - 4x + 3 = 0 \) meet at origin \( O(0, 0) \). Then the area of the triangle OAB is:

Let the hyperbola \( H: \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1 \) pass through the point \( (2\sqrt{2}, -2\sqrt{2}) \). A parabola is drawn whose focus is same as the focus of H with positive abscissa and the directrix of the parabola passes through the other focus of H. If the length of the latus rectum of the parabola is \( e \) times the length of the latus rectum of H, where \( e \) is the eccentricity of H, then which of the following points lies on the parabola

Let the lines \(\displaystyle \frac{x-1}{\lambda} = \frac{y-2}{1} = \frac{z-3}{2} \) and \(\displaystyle \frac{x+26}{-2} = \frac{y+18}{3} = \frac{z+28}{\lambda} \) be coplanar and \( P \) be the plane containing these two lines. Then which of the following points does NOT lie on \( P \)?

A plane \( P \) is parallel to two lines whose direction ratios are \( -2, 1, -3 \) and \( -1, 2, -2 \) and it contains the point \( (2, 2, -2) \). Let \( P \) intersect the co-ordinate axes at the points \( A, B, C \) making the intercepts \( \alpha, \beta, \gamma \). If \( V \) is the volume of the tetrahedron \( OABC \), where \( O \) is the origin, and \( p = \alpha + \beta + \gamma \), then the ordered pair \( (V, \ p) \) is equal to:

Let \( S \) be the set of all \( a \in \mathbb{R} \) for which the angle between the vectors \( \vec{u} = a(\log_{e} b)\hat{\imath} - 6\hat{\jmath} + 3\hat{k} \) and \( \vec{v} = (\log_{e} b)\hat{\imath} + 2\hat{\jmath} + 2a(\log_{e} b)\hat{k}, \ (b>1) \) is acute. Then \( S \) is equal to:

A horizontal park is in the shape of a triangle \( OAB \) with \( AB = 16 \). A vertical lamps post \( OP \) is erected at the point \( O \) such that \( \angle PAO = \angle PBO = 15^\circ \) and \( \angle PCO = 45^\circ \), where \( C \) is the midpoint of \( AB \). Then \( (OP)^2 \) is equal to:

Let A and B two events such that \( P(B \mid A) = \dfrac{2}{5} \), \( P(A \mid B) = \dfrac{1}{7} \) and \( P(A \cap B) = \dfrac{1}{9} \). Consider:
(S1) \( P(A \cup B) = \dfrac{5}{6} \), (S2) \( P(A' \cup B') = \dfrac{1}{18} \).
Then:

Let
\( p \): Ramesh listens to music
\( q \): Ramesh is out of his village
\( r \): It is Sunday
\( s \): It is Saturday
Then the statement “Ramesh listens to music only if he is in his village and it is Sunday or Saturday” can be expressed as:

Let the coefficients of the middle terms in the expansion of \( \left(\dfrac{1}{\sqrt{6}} + \beta x\right)^4 \), \( (1 - 3 \beta x)^2 \) and \( \left(1 - \dfrac{\beta}{2} x\right)^6 \), \( \beta > 0 \) , respectively from the first three terms of an A.P. If \( d \) is the common difference of this A.P., then \( 50 -\dfrac{ 2d}{\beta^2} \) is equal to ____________.

A class contains \( b \) boys and \( g \) girls. If the number of ways of selecting \( 3 \) boys and \( 2 \) girls from the class is \( 168 \), then \( b + 3g \) is equal to ______________.

Let the tangents at the points \( P \) and \( Q \) on the ellipse \( \dfrac{x^{2}}{2} + \dfrac{y^{2}}{4} = 1 \) meet at the point \( R(\sqrt{2},\, 2\sqrt{2} - 2) \). If \( S \) is the focus of the ellipse on its negative major axis, then \( SP^{2} + SQ^{2} \) is equal to __________________.

If \( 1 + (2 + {}^{49}C_{1} + {}^{49}C_{2} + \ldots + {}^{49}C_{49})({}^{50}C_{2} + {}^{50}C_{4} + \ldots + {}^{50}C_{50}) \) is equal to \( 2^{n} \cdot m \), where \( m \) is odd, then \( n + m \) is equal to ______________.

Two tangent lines \(l_1\) and \(l_2\) are drawn from the point \( (2, 0) \) to the parabola \( 2y^2 = -x \). If the lines \(l_1\) and \(l_2\) are also tangent to the circle \( (x-5)^2 + y^2 = r \), then \( 17r \) is equal to _______________.

If \(\displaystyle \frac{6}{3^{12}} + \frac{10}{3^{11}} + \frac{20}{3^{10}} + \frac{40}{3^{9}} + \ldots + \frac{10240}{3} = 2^{n} \cdot m \) , where \( m \) is odd, then \( m \cdot n \) is equal to _______________.

Let \(\displaystyle S=\left[-\pi,\dfrac{\pi}{2}\right)-\left\{-\frac{\pi}{2},-\frac{\pi}{4},\frac{3\pi}{4},\frac{\pi}{4}\right\} \). Then the number of elements in the set \( A=\left\{\theta\in S:\tan\theta\left(1+\sqrt{5}\tan(2\theta)\right)=\sqrt{5}-\tan(2\theta)\right\} \) is _______________.

Let \( z=a+ib ,\; b\neq 0 \) be complex numbers satisfying \( z^{2}=\overline{z}\cdot 2^{\,1-|z|} \). Then the least value of \( n\in\mathbb{N} \), such that \( z^{n}=(z+1)^{n} \), is equal to _______________.

A bag contains \( 4 \) white and \( 6 \) black balls. Three balls are drawn at random from the bag. Let \( X \) be the number of white balls, among the drawn balls. If \( \sigma^{2} \) is the variance of \( X \), then \( 100\sigma^{2} \) is equal to _______________.

The value of integral \(\displaystyle \int\limits_{0}^{\pi/2} 60\,\frac{\sin(6x)}{\sin x}\,dx \) is equal to __________________.

JEE Main 2022 28th July Evening shift

Physics Chemistry Mathematics

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The differential equation of the family of circles passing through the points \( (0,2) \) and \( (0,-2) \) is: