JEE Main 2022 (Online) 28th July Evening Shift

Let the lines \(\frac{x-1}{\lambda}=\frac{y-2}{1}=\frac{z-3}{2}\) and \(\frac{x+26}{-2}=\frac{y+18}{3}=\frac{z+28}{\lambda}\) be coplanar and \(\mathrm{P}\) be the plane containing these two lines. Then which of the following points does NO...

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 \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) making the intercept...

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

The area enclosed by the curves \(y=\log _{e}\left(x+\mathrm{e}^{2}\right), x=\log _{e}\left(\frac{2}{y}\right)\) and \(x=\log _{\mathrm{e}} 2\), above the line \(y=1\) is:

Let the coefficients of the middle terms in the expansion of \(\left(\frac{1}{\sqrt{6}}+\beta x\right)^{4},(1-3 \beta x)^{2}\) and \(\left(1-\frac{\beta}{2} x\right)^{6}, \beta>0\), respectively form the first three terms of an A.P. If d is...

If \(1 + (2 + {}^{49}{C_1} + {}^{49}{C_2} + \,\,...\,\, + \,\,{}^{49}{C_{49}})({}^{50}{C_2} + {}^{50}{C_4} + \,\,...\,\, + \,\,{}^{50}{C_{50}})\) is equal to \(2^{\mathrm{n}} \cdot \mathrm{m}\), where \(\mathrm{m}\) is odd, then $$\mathrm{n...

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

Let \(\mathrm{z}=a+i b, b \neq 0\) be complex numbers satisfying \(z^{2}=\bar{z} \cdot 2^{1-z}\). Then the least value of \(n \in N\), such that \(z^{n}=(z+1)^{n}\), is equal to __________.

Let \(I_{n}(x)=\int_{0}^{x} \frac{1}{\left(t^{2}+5\right)^{n}} d t, n=1,2,3, \ldots .\) Then :

The value of the integral \(\int\limits_{0}^{\frac{\pi}{2}} 60 \frac{\sin (6 x)}{\sin x} d x\) is equal to _________.

Let \(y=y(x)\) be the solution curve of the differential equation \(\frac{d y}{d x}+\frac{1}{x^{2}-1} y=\left(\frac{x-1}{x+1}\right)^{1 / 2}\), \(x >1\) passing through the point \(\left(2, \sqrt{\frac{1}{3}}\right)\). Then $$\sqrt{7}\, y...

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

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

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

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

A horizontal park is in the shape of a triangle \(\mathrm{OAB}\) with \(\mathrm{AB}=16\). A vertical lamp post \(\mathrm{OP}\) is erected at the point \(\mathrm{O}\) such that \(\angle \mathrm{PAO}=\angle \mathrm{PBO}=15^{\circ}\) and $$\an...

Let the hyperbola \(H: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) pass through the point \((2 \sqrt{2},-2 \sqrt{2})\). A parabola is drawn whose focus is same as the focus of \(\mathrm{H}\) with positive abscissa and the directrix of the p...

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 :

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

Let
\(\mathrm{p}\) : Ramesh listens to music.
\(\mathrm{q}\) : Ramesh is out of his village.
\(\mathrm{r}\) : It is Sunday.
\(\mathrm{s}\) : It is Saturday.
Then the statement "Ramesh listens to music only if he is in his village and it is ...

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

Two tangent lines \(l_{1}\) and \(l_{2}\) are drawn from the point \((2,0)\) to the parabola \(2 \mathrm{y}^{2}=-x\). 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 __________...

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 \(\mathrm{b}+3 \mathrm{~g}\) is equal to ____________.

Let \(\mathrm{A}\) and \(\mathrm{B}\) be two events such that \(P(B \mid A)=\frac{2}{5}, P(A \mid B)=\frac{1}{7}\) and \(P(A \cap B)=\frac{1}{9} \cdot\) Consider
(S1) \(P\left(A^{\prime} \cup B\right)=\frac{5}{6}\),
(S2) $$P\left(A^{\prime}...

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

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

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

\({6 \over {{3^{12}}}} + {{10} \over {{3^{11}}}} + {{20} \over {{3^{10}}}} + {{40} \over {{3^9}}} + \,\,...\,\, + \,\,{{10240} \over 3} = {2^n}\,.\,m\), where m is odd, then m . n is equal to ____________.

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

Let S be the set of all a \(\in R\) for which the angle between the vectors \(\vec{u}=a\left(\log _{e} b\right) \hat{i}-6 \hat{j}+3 \hat{k}\) and \(\vec{v}=\left(\log _{e} b\right) \hat{i}+2 \hat{j}+2 a\left(\log _{e} b\right) \hat{k}\), $...