JEE Main 2022 (Online) 25th July Morning Shift

Let \(\mathrm{P}\) be the plane containing the straight line \(\frac{x-3}{9}=\frac{y+4}{-1}=\frac{z-7}{-5}\) and perpendicular to the plane containing the straight lines \(\frac{x}{2}=\frac{y}{3}=\frac{z}{5}\) and $$\frac{x}{3}=\frac{y}{7}=...

The line of shortest distance between the lines \(\frac{x-2}{0}=\frac{y-1}{1}=\frac{z}{1}\) and \(\frac{x-3}{2}=\frac{y-5}{2}=\frac{z-1}{1}\) makes an angle of \(\cos ^{-1}\left(\sqrt{\frac{2}{27}}\right)\) with the plane $$\mathrm{P}: \mat...

If the absolute maximum value of the function \(f(x)=\left(x^{2}-2 x+7\right) \mathrm{e}^{\left(4 x^{3}-12 x^{2}-180 x+31\right)}\) in the interval \([-3,0]\) is \(f(\alpha)\), then :

The curve \(y(x)=a x^{3}+b x^{2}+c x+5\) touches the \(x\)-axis at the point \(\mathrm{P}(-2,0)\) and cuts the \(y\)-axis at the point \(Q\), where \(y^{\prime}\) is equal to 3 . Then the local maximum value of \(y(x)\) is:

The area of the region given by
\(A=\left\{(x, y): x^{2} \leq y \leq \min \{x+2,4-3 x\}\right\}\) is :

Let the locus of the centre \((\alpha, \beta), \beta>0\), of the circle which touches the circle \(x^{2}+(y-1)^{2}=1\) externally and also touches the \(x\)-axis be \(\mathrm{L}\). Then the area bounded by \(\mathrm{L}\) and the line $$y=4$...

If the maximum value of the term independent of \(t\) in the expansion of \(\left(\mathrm{t}^{2} x^{\frac{1}{5}}+\frac{(1-x)^{\frac{1}{10}}}{\mathrm{t}}\right)^{15}, x \geqslant 0\), is \(\mathrm{K}\), then \(8 \mathrm{~K}\) is equal to ___...

For \(\mathrm{n} \in \mathbf{N}\), let \(\mathrm{S}_{\mathrm{n}}=\left\{z \in \mathbf{C}:|z-3+2 i|=\frac{\mathrm{n}}{4}\right\}\) and \(\mathrm{T}_{\mathrm{n}}=\left\{z \in \mathbf{C}:|z-2+3 i|=\frac{1}{\mathrm{n}}\right\}\). Then the numbe...

For any real number \(x\), let \([x]\) denote the largest integer less than equal to \(x\). Let \(f\) be a real valued function defined on the interval \([-10,10]\) by $$f(x)=\left\{\begin{array}{l}x-[x], \text { if }[x] \text { is odd } \\...

$$
\begin{aligned}
&\text { If } \lim _{n \rightarrow \infty} \frac{(n+1)^{k-1}}{n^{k+1}}[(n k+1)+(n k+2)+\ldots+(n k+n)] \\
&=33 \cdot \lim _{n \rightarrow \infty} \frac{1}{n^{k+1}} \cdot\left[1^{k}+2^{k}+3^{k}+\ldots+n^{k}\right]
\end{ali...

The slope of the tangent to a curve \(C: y=y(x)\) at any point \((x, y)\) on it is \(\frac{2 \mathrm{e}^{2 x}-6 \mathrm{e}^{-x}+9}{2+9 \mathrm{e}^{-2 x}}\).

If \(C\) passes through the points $$\left(0, \frac{1}{2}+\frac{\pi}{2 \sqrt{2}}\r...

The general solution of the differential equation \(\left(x-y^{2}\right) \mathrm{d} x+y\left(5 x+y^{2}\right) \mathrm{d} y=0\) is :

The total number of functions,
\(f:\{1,2,3,4\} \rightarrow\{1,2,3,4,5,6\}\)
such that \(f(1)+f(2)=f(3)\), is equal to :

A tower PQ stands on a horizontal ground with base \(Q\) on the ground. The point \(R\) divides the tower in two parts such that \(Q R=15 \mathrm{~m}\). If from a point \(A\) on the ground the angle of elevation of \(R\) is \(60^{\circ}\) a...

Let the equation of two diameters of a circle \(x^{2}+y^{2}-2 x+2 f y+1=0\) be \(2 p x-y=1\) and \(2 x+p y=4 p\). Then the slope m \(\in\) \((0, \infty)\) of the tangent to the hyperbola \(3 x^{2}-y^{2}=3\) passing through the centre of t...

If \(\mathop {\lim }\limits_{n \to \infty } \left( {\sqrt {{n^2} - n - 1} + n\alpha + \beta } \right) = 0\), then \(8(\alpha+\beta)\) is equal to :

Let $$f(x)=\left\{\begin{array}{l}\left|4 x^{2}-8 x+5\right|, \text { if } 8 x^{2}-6 x+1 \geqslant 0 \\ {\left[4 x^{2}-8 x+5\right], \text { if } 8 x^{2}-6 x+1<0,}\end{array}\right.$$ where \([\alpha]\) denotes the greatest integer less tha...

Which of the following statements is a tautology ?

The number of \(\theta \in(0,4 \pi)\) for which the system of linear equations

$$ \begin{aligned} &3(\sin 3 \theta) x-y+z=2 \\\\ &3(\cos 2 \theta) x+4 y+3 z=3 \\\\ &6 x+7 y+7 z=9 \end{aligned} $$
has no solution, is :

Let $$A=\left(\begin{array}{rrr}2 & -1 & -1 \\ 1 & 0 & -1 \\ 1 & -1 & 0\end{array}\right)$$ and \(B=A-I\). If \(\omega=\frac{\sqrt{3} i-1}{2}\), then the number of elements in the $$\operatorname{set}\left\{n \in\{1,2, \ldots, 100\}: A^{n}+...

The sum of diameters of the circles that touch (i) the parabola \(75 x^{2}=64(5 y-3)\) at the point \(\left(\frac{8}{5}, \frac{6}{5}\right)\) and (ii) the \(y\)-axis, is equal to ______________.

The letters of the word 'MANKIND' are written in all possible orders and arranged in serial order as in an English dictionary. Then the serial number of the word 'MANKIND' is _____________.

If the sum and the product of mean and variance of a binomial distribution are 24 and 128 respectively, then the probability of one or two successes is :

If the numbers appeared on the two throws of a fair six faced die are \(\alpha\) and \(\beta\), then the probability that \(x^{2}+\alpha x+\beta>0\), for all \(x \in \mathbf{R}\), is :

If \(\alpha, \beta, \gamma, \delta\) are the roots of the equation \(x^{4}+x^{3}+x^{2}+x+1=0\), then \(\alpha^{2021}+\beta^{2021}+\gamma^{2021}+\delta^{2021}\) is equal to :

Let \(a, b\) be two non-zero real numbers. If \(p\) and \(r\) are the roots of the equation \(x^{2}-8 \mathrm{a} x+2 \mathrm{a}=0\) and \(\mathrm{q}\) and s are the roots of the equation \(x^{2}+12 \mathrm{~b} x+6 \mathrm{~b}=0\), such that...

Let \(a_{1}=b_{1}=1, a_{n}=a_{n-1}+2\) and \(b_{n}=a_{n}+b_{n-1}\) for every natural number \(n \geqslant 2\). Then \(\sum\limits_{n = 1}^{15} {{a_n}.{b_n}}\) is equal to ___________.

A line, with the slope greater than one, passes through the point \(A(4,3)\) and intersects the line \(x-y-2=0\) at the point B. If the length of the line segment \(A B\) is \(\frac{\sqrt{29}}{3}\), then \(B\) also lies on the line :

The number of solutions of \(|\cos x|=\sin x\), such that \(-4 \pi \leq x \leq 4 \pi\) is :

Let \(\mathrm{ABC}\) be a triangle such that $$\overrightarrow{\mathrm{BC}}=\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{CA}}=\overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{AB}}=\overrightarrow{\mathrm{c}},|\overrightarrow{\ma...