JEE Main 2022 (Online) 27th July Evening Shift

If the length of the perpendicular drawn from the point \(P(a, 4,2)\), a \(>0\) on the line \(\frac{x+1}{2}=\frac{y-3}{3}=\frac{z-1}{-1}\) is \(2 \sqrt{6}\) units and \(Q\left(\alpha_{1}, \alpha_{2}, \alpha_{3}\right)\) is the image of the ...

If the line of intersection of the planes \(a x+b y=3\) and \(a x+b y+c z=0\), a \(>0\) makes an angle \(30^{\circ}\) with the plane \(y-z+2=0\), then the direction cosines of the line are :

A water tank has the shape of a right circular cone with axis vertical and vertex downwards. Its semi-vertical angle is \(\tan ^{-1} \frac{3}{4}\). Water is poured in it at a constant rate of 6 cubic meter per hour. The rate (in square mete...

The area of the region enclosed by \(y \leq 4 x^{2}, x^{2} \leq 9 y\) and \(y \leq 4\), is equal to :

Consider a curve \(y=y(x)\) in the first quadrant as shown in the figure. Let the area \(\mathrm{A}_{1}\) is twice the area \(\mathrm{A}_{2}\). Then the normal to the curve perpendicular to the line \(2 x-12 y=15\) does NOT pass through the...

Let for the \(9^{\text {th }}\) term in the binomial expansion of \((3+6 x)^{\mathrm{n}}\), in the increasing powers of \(6 x\), to be the greatest for \(x=\frac{3}{2}\), the least value of \(\mathrm{n}\) is \(\mathrm{n}_{0}\). If $$\mathrm...

A circle \(C_{1}\) passes through the origin \(\mathrm{O}\) and has diameter 4 on the positive \(x\)-axis. The line \(y=2 x\) gives a chord \(\mathrm{OA}\) of circle \(\mathrm{C}_{1}\). Let \(\mathrm{C}_{2}\) be the circle with $$\mathrm{OA...

Let S be the set of all \((\alpha, \beta), \pi<\alpha, \beta<2 \pi\), for which the complex number \(\frac{1-i \sin \alpha}{1+2 i \sin \alpha}\) is purely imaginary and \(\frac{1+i \cos \beta}{1-2 i \cos \beta}\) is purely real. Let $$Z_{\a...

Let \(f(x)=2+|x|-|x-1|+|x+1|, x \in \mathbf{R}\).
Consider
\((\mathrm{S} 1): f^{\prime}\left(-\frac{3}{2}\right)+f^{\prime}\left(-\frac{1}{2}\right)+f^{\prime}\left(\frac{1}{2}\right)+f^{\prime}\left(\frac{3}{2}\right)=2\)
$$(\mathrm{S} 2):...

\(\int\limits_{0}^{2}\left(\left|2 x^{2}-3 x\right|+\left[x-\frac{1}{2}\right]\right) \mathrm{d} x\), where [t] is the greatest integer function, is equal to :

Let \(f(x)=\min \{[x-1],[x-2], \ldots,[x-10]\}\) where [t] denotes the greatest integer \(\leq \mathrm{t}\). Then $$\int\limits_{0}^{10} f(x) \mathrm{d} x+\int\limits_{0}^{10}(f(x))^{2} \mathrm{~d} x+\int\limits_{0}^{10}|f(x)| \mathrm{d} x$...

Let f be a differentiable function satisfying \(f(x)=\frac{2}{\sqrt{3}} \int\limits_{0}^{\sqrt{3}} f\left(\frac{\lambda^{2} x}{3}\right) \mathrm{d} \lambda, x>0\) and \(f(1)=\sqrt{3}\). If \(y=f(x)\) passes through the point \((\alpha, 6)\)...

For the curve \(C:\left(x^{2}+y^{2}-3\right)+\left(x^{2}-y^{2}-1\right)^{5}=0\), the value of \(3 y^{\prime}-y^{3} y^{\prime \prime}\), at the point \((\alpha, \alpha)\), \(\alpha>0\), on C, is equal to ____________.

The number of functions \(f\), from the set \(\mathrm{A}=\left\{x \in \mathbf{N}: x^{2}-10 x+9 \leq 0\right\}\) to the set \(\mathrm{B}=\left\{\mathrm{n}^{2}: \mathrm{n} \in \mathbf{N}\right\}\) such that \(f(x) \leq(x-3)^{2}+1\), for every...

The angle of elevation of the top P of a vertical tower PQ of height 10 from a point A on the horizontal ground is \(45^{\circ}\). Let R be a point on AQ and from a point B, vertically above \(\mathrm{R}\), the angle of elevation of $$\math...

A common tangent \(\mathrm{T}\) to the curves \(\mathrm{C}_{1}: \frac{x^{2}}{4}+\frac{y^{2}}{9}=1\) and \(C_{2}: \frac{x^{2}}{42}-\frac{y^{2}}{143}=1\) does not pass through the fourth quadrant. If \(\mathrm{T}\) touches \(\mathrm{C}_{1}\) ...

The domain of the function \(f(x)=\sin ^{-1}\left[2 x^{2}-3\right]+\log _{2}\left(\log _{\frac{1}{2}}\left(x^{2}-5 x+5\right)\right)\), where [t] is the greatest integer function, is :

If for \(\mathrm{p} \neq \mathrm{q} \neq 0\), the function \(f(x)=\frac{\sqrt[7]{\mathrm{p}(729+x)}-3}{\sqrt[3]{729+\mathrm{q} x}-9}\) is continuous at \(x=0\), then :

If the truth value of the statement \((P \wedge(\sim R)) \rightarrow((\sim R) \wedge Q)\) is F, then the truth value of which of the following is \(\mathrm{F}\) ?

Let $$A=\left(\begin{array}{rr}4 & -2 \\ \alpha & \beta\end{array}\right)$$.
If \(\mathrm{A}^{2}+\gamma \mathrm{A}+18 \mathrm{I}=\mathrm{O}\), then \(\operatorname{det}(\mathrm{A})\) is equal to _____________.

Consider a matrix $$A=\left[\begin{array}{ccc}\alpha & \beta & \gamma \\ \alpha^{2} & \beta^{2} & \gamma^{2} \\ \beta+\gamma & \gamma+\alpha & \alpha+\beta\end{array}\right]$$, where \(\alpha, \beta, \gamma\) are three distinct natural numb...

If the length of the latus rectum of a parabola, whose focus is \((a, a)\) and the tangent at its vertex is \(x+y=a\), is 16, then \(|a|\) is equal to :

Let X have a binomial distribution B(n, p) such that the sum and the product of the mean and variance of X are 24 and 128 respectively. If \(P(X>n-3)=\frac{k}{2^{n}}\), then k is equal to :

A six faced die is biased such that
\(3 \times \mathrm{P}(\)a prime number\()\,=6 \times \mathrm{P}(\)a composite number\()\,=2 \times \mathrm{P}(1)\).
Let X be a random variable that counts the number of times one gets a perfect square on ...

If \(\alpha, \beta\) are the roots of the equation
\(x^{2}-\left(5+3^{\sqrt{\log _{3} 5}}-5^{\sqrt{\log _{5} 3}}\right)x+3\left(3^{\left(\log _{3} 5\right)^{\frac{1}{3}}}-5^{\left(\log _{5} 3\right)^{\frac{2}{3}}}-1\right)=0\),
then the e...

Let the sum of an infinite G.P., whose first term is a and the common ratio is r, be 5 . Let the sum of its first five terms be \(\frac{98}{25}\). Then the sum of the first 21 terms of an AP, whose first term is $$10\mathrm{a r}, \mathrm{n}...

\(\frac{2^{3}-1^{3}}{1 \times 7}+\frac{4^{3}-3^{3}+2^{3}-1^{3}}{2 \times 11}+\frac{6^{3}-5^{3}+4^{3}-3^{3}+2^{3}-1^{3}}{3 \times 15}+\cdots+ \frac{30^{3}-29^{3}+28^{3}-27^{3}+\ldots+2^{3}-1^{3}}{15 \times 63}\) is equal to _____________.

The equations of the sides \(\mathrm{AB}, \mathrm{BC}\) and CA of a triangle ABC are \(2 x+y=0, x+\mathrm{p} y=39\) and \(x-y=3\) respectively and \(\mathrm{P}(2,3)\) is its circumcentre. Then which of the following is NOT true?

Let \(S=\left\{\theta \in\left(0, \frac{\pi}{2}\right): \sum\limits_{m=1}^{9} \sec \left(\theta+(m-1) \frac{\pi}{6}\right) \sec \left(\theta+\frac{m \pi}{6}\right)=-\frac{8}{\sqrt{3}}\right\}\). Then

Let \(\overrightarrow a\), \(\overrightarrow b\), \(\overrightarrow c\) be three non-coplanar vectors such that \(\overrightarrow a\) \(\times\) \(\overrightarrow b\) = 4\(\overrightarrow c\), \(\overrightarrow b\) \(\times\) $$\over...