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Let \( f : R \to R \) be defined as \( f(x) = x - 1 \) and \( g : R - \{ 1, -1 \} \to R \) be defined as \( g(x) = \frac{x^2}{x^2 - 1} \). Then the function \( fog \) is:

If the system of equations \( ax + y + z = 5 \), \( x + 2y + 3z = 4 \), \( x + 3y + 5z = \beta \) has infinitely many solutions, then the ordered pair \( (\alpha, \beta) \) is equal to:

If \( A = \displaystyle \sum_{n=1}^{\infty} \frac{1}{\left(3 + (-1)^n\right)^n} \) and \( B = \displaystyle \sum_{n=1}^{\infty} \frac{(-1)^n}{\left(3 + (-1)^n\right)^n} \) then \( \frac{A}{B} \) is equal to:

\( \displaystyle \lim_{x \to 0} \frac{\cos(\sin x) - \cos x}{x^4} \) is equal to:

Let \( f(x) = \min \{ 1, 1 + x \sin x \} \), \( 0 \leq x \leq 2\pi \). If \( m \) is the number of points, where \( f \) is not differential and \( n \) is the number of points, where \( f \) is not continuous, then the ordered pair \( (m, n) \) is equal to:

Consider a cuboid of sides \( 2x \), \( 4x \) and \( 5x \) and a closed hemisphere of radius \( r \). If the sum of their surface areas is a constant \( k \), then the ratio \( x : r \), for which the sum of their volumes is maximum, is:

The area of the region bounded by \( y^2 = 8x \) and \( y^2 = 16(3 - x) \) is equal to:

If \( \int{}^{} \frac{1}{x}\sqrt{\frac{1-x}{1+x}} \, dx = g(x) + c, \, g(1) = 0 \), then \( g\!\left(\tfrac{1}{2}\right) \) is equal to:

If \( y = y(x) \) is the solution of the differential equation \( x \dfrac{dy}{dx} + 2y = x e^{x} , \, y(1) = 0 \) then the local maximum value of the function \( z(x) = x^{2} y(x) - e^{x} , \, x \in \mathbb{R} \) is:

If the solution of the differential equation \( \dfrac{dy}{dx} + e^{x}(x^{2} - 2)y = (x^{2} - 2x)(x^{2} - 2)e^{2x} \) satisfies \( y(0) = 0 \), then the value of \( y(2) \) is ________.

If \( m \) is the slope of a common tangent to the curves \( \dfrac{x^{2}}{16} + \dfrac{y^{2}}{9} = 1 \) and \( x^{2} + y^{2} = 12 \), then \( 12 m^{2} \) is equal to:

The locus of the mid point of the line segment joining the point \( (4, 3) \) and the points on the ellipse \( x^2 + 2y^2 = 4 \) is an ellipse with eccentricity:

The normal to the hyperbola \( \frac{x^2}{a^2}-\frac{y^2}{9}=1 \) at the point \( (8,\,3\sqrt{3}) \) on it passes through the point:

If the plane \( 2x+y-5z=0 \) is rotated about its line of intersection with the plane \( 3x-y+4z-7=0 \) by an angle of \( \frac{\pi}{2} \), then the plane after the rotation passes through the point:

If the lines \( \vec{r}=(\hat{i}-\hat{j}+\hat{k})+\lambda(3\hat{j}-\hat{k}) \) and \( \vec{r}=(\alpha\hat{i}-\hat{j})+\mu(2\hat{i}-3\hat{k}) \) are co-planar, then the distance of the plane containing these two lines from the point \( (\alpha,0,0) \) is:

Let $\vec{a} = \hat{i} + \hat{j} + 2\hat{k}$, $\vec{b} = 2\hat{i} - 3\hat{j} + \hat{k}$ and $\vec{c} = \hat{i} - \hat{j} + \hat{k}$ be three given vectors. Let $\vec{v}$ be a vector in the plane of $\vec{a}$ and $\vec{b}$ whose projection on $\vec{c}$ is $\frac{2}{\sqrt{3}}$. If $\vec{v} \cdot \hat{j} = 7$, $\vec{v} \cdot (\hat{i} + \hat{k})$ is equal to:

The mean and standard deviation of \( 50 \) observations are \( 15 \) and \( 2 \) respectively. It was found that one incorrect observation was taken such that the sum of correct and incorrect observations is \( 70 \). If the correct mean is \( 16 \), then the correct variance is equal to:

\( 16\sin(20^\circ)\sin(40^\circ)\sin(80^\circ) \) is equal to:

If the inverse trigonometric functions take principal values, then \( \cos^{-1}\!\left(\frac{3}{10}\cos\!\left(\tan^{-1}\!\left(\frac{4}{3}\right)\right)+\frac{2}{5}\sin\!\left(\tan^{-1}\!\left(\frac{4}{3}\right)\right)\right) \) is equal to:

Let $r \in \{p, q, \sim p, \sim q\}$ be such that the logical statement $r \vee (\sim p) \Rightarrow (p \wedge q) \vee r$ is a tautology. Then $r$ is equal to:

Let \( f:\mathbb{R}\to\mathbb{R} \) satisfies \( f(x+y)=2^{x}f(y)+4^{y}f(x),\ \forall x,y\in\mathbb{R}. \) If \( f(2)=3 \), then \( 14\cdot \frac{f'(4)}{f'(2)} \) is equal to ______.

Let \( p \) and \( q \) be two real numbers such that \( p+q=3 \) and \( p^{4}+q^{4}=369 \). Then \( \left(\frac{1}{p}+\frac{1}{q}\right)^{-2} \) is equal to ______.

If \( z^{2}+z+1=0, \; z\in \mathbb{C} \), then \( \left| \displaystyle \sum_{n=1}^{15}\left(z^{n}+(-1)^{n}\frac{1}{z^{n}}\right)^{2}\right|^{} \) is equal to ______.

\( Y = \alpha I + \beta X + \gamma X^2 \) and \( Z = \alpha^2 I - \alpha \beta X + (\beta^2 - \alpha \gamma) X^2, \ \alpha, \beta, \gamma \in \mathbb{R}. \)
Then \( (\alpha - \beta + \gamma)^2 \) is equal to _______.

The total number of \( 3 \)-digit numbers, whose greatest common divisor with \( 36 \) is \( 2 \), is _______.

If $\left(^{40}C_{0}\right) + \left(^{41}C_{1}\right) + \left(^{42}C_{2}\right) + \dots + \left(^{60}C_{20}\right) = \frac{m}{n} {}^{60}C_{20}$, $m$ and $n$ are coprime, then $m+n$ is equal to _____________.

If \( a_1(>0), a_2, a_3, a_4, a_5 \) are in a G.P., \( a_2 + a_4 = 2a_3 + 1 \) and \( 3a_2 + a_3 = 2a_4 \), then \( a_2 + a_4 + 2a_5 \) is equal to _______.

The integral \( \dfrac{24}{\pi} \displaystyle \int_{0}^{\sqrt{2}} \dfrac{(2 - x^2)\, dx}{(2 + x^2)\sqrt{4 + x^4}} \) is equal to _______.

Let a line \( L_1 \) be tangent to the hyperbola \( \dfrac{x^2}{16} - \dfrac{y^2}{4} = 1 \) and let \( L_2 \) be the line passing through the origin and perpendicular to \( L_1 \). If the locus of the point of intersection of \( L_1 \) and \( L_2 \) is \( (x^2 + y^2)^2 = \alpha x^2 + \beta y^2 \), then \( \alpha + \beta \) is equal to _______.

If the probability that a randomly chosen \( 6 \)-digit number formed by using digits \( 1 \) and \( 8 \) only is a multiple of \( 21 \) is \( p \), then \( 96p \) is equal to _________.

JEE Main 2022 26th June Evening Shift

Physics Chemistry Mathematics

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Let \( p \) and \( q \) be two real numbers such that \( p+q=3 \) and \( p^{4}+q^{4}=369 \). Then \( \left(\frac{1}{p}+\frac{1}{q}\right)^{-2} \) is equal to ______.