JEE Main 2022 (Online) 29th July Evening Shift

Let \(Q\) be the foot of perpendicular drawn from the point \(P(1,2,3)\) to the plane \(x+2 y+z=14\). If \(R\) is a point on the plane such that \(\angle P R Q=60^{\circ}\), then the area of \(\triangle P Q R\) is equal to :

If \((2,3,9),(5,2,1),(1, \lambda, 8)\) and \((\lambda, 2,3)\) are coplanar, then the product of all possible values of \(\lambda\) is:

If the tangent to the curve \(y=x^{3}-x^{2}+x\) at the point \((a, b)\) is also tangent to the curve \(y = 5{x^2} + 2x - 25\) at the point (2, \(-\)1), then \(|2a + 9b|\) is equal to __________.

\(\sum\limits_{r=1}^{20}\left(r^{2}+1\right)(r !)\) is equal to

\(\text { If } \sum\limits_{k=1}^{10} K^{2}\left(10_{C_{K}}\right)^{2}=22000 L \text {, then } L \text { is equal to }\) ________.

Let \(A B\) be a chord of length 12 of the circle \((x-2)^{2}+(y+1)^{2}=\frac{169}{4}\). If tangents drawn to the circle at points \(A\) and \(B\) intersect at the point \(P\), then five times the distance of point \(P\) from chord \(A B\) ...

\(\text { Let } S=\left\{(x, y) \in \mathbb{N} \times \mathbb{N}: 9(x-3)^{2}+16(y-4)^{2} \leq 144\right\}\) and
\(T=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}:(x-7)^{2}+(y-4)^{2} \leq 36\right\}\). Then \(n(S \cap T)\) is equal to _____...

If \(z \neq 0\) be a complex number such that \(\left|z-\frac{1}{z}\right|=2\), then the maximum value of \(|z|\) is :

Let \(\mathrm{S}=\{z=x+i y:|z-1+i| \geq|z|,|z|<2,|z+i|=|z-1|\}\). Then the set of all values of \(x\), for which \(w=2 x+i y \in \mathrm{S}\) for some \(y \in \mathbb{R}\), is :

If \([t]\) denotes the greatest integer \(\leq t\), then the value of \(\int_{0}^{1}\left[2 x-\left|3 x^{2}-5 x+2\right|+1\right] \mathrm{d} x\) is :

If the solution curve of the differential equation \(\frac{d y}{d x}=\frac{x+y-2}{x-y}\) passes through the points \((2,1)\) and \((\mathrm{k}+1,2), \mathrm{k}>0\), then

Let \(y=y(x)\) be the solution curve of the differential equation \(\frac{d y}{d x}+\left(\frac{2 x^{2}+11 x+13}{x^{3}+6 x^{2}+11 x+6}\right) y=\frac{(x+3)}{x+1}, x>-1\), which passes through the point \((0,1)\). Then \(y(1)\) is equal to ...

For \(I(x)=\int \frac{\sec ^{2} x-2022}{\sin ^{2022} x} d x\), if \(I\left(\frac{\pi}{4}\right)=2^{1011}\), then

The domain of the function \(f(x)=\sin ^{-1}\left(\frac{x^{2}-3 x+2}{x^{2}+2 x+7}\right)\) is :

$$ \text { Let the function } f(x)=\left\{\begin{array}{cl} \frac{\log _{e}(1+5 x)-\log _{e}(1+\alpha x)}{x} & ;\text { if } x \neq 0 \\ 10 & ; \text { if } x=0 \end{array} \text { be continuous at } x=0 .\right. $$
Then \(\alpha\) is equal...

If \([t]\) denotes the greatest integer \(\leq t\), then the number of points, at which the function \(f(x)=4|2 x+3|+9\left[x+\frac{1}{2}\right]-12[x+20]\) is not differentiable in the open interval \((-20,20)\), is __________.

The statement \((p \Rightarrow q) \vee(p \Rightarrow r)\) is NOT equivalent to

Which of the following matrices can NOT be obtained from the matrix $$\left[\begin{array}{cc}-1 & 2 \\ 1 & -1\end{array}\right]$$ by a single elementary row operation ?

If the system of equations
$$ \begin{aligned} &x+y+z=6 \\ &2 x+5 y+\alpha z=\beta \\ &x+2 y+3 z=14 \end{aligned} $$
has infinitely many solutions, then \(\alpha+\beta\) is equal to

Let $$X=\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]$$ and $$A=\left[\begin{array}{ccc}-1 & 2 & 3 \\ 0 & 1 & 6 \\ 0 & 0 & -1\end{array}\right]$$. For \(\mathrm{k} \in N\), if \(X^{\prime} A^{k} X=33\), then \(\mathrm{k}\) is equal to...

The number of natural numbers lying between 1012 and 23421 that can be formed using the digits \(2,3,4,5,6\) (repetition of digits is not allowed) and divisible by 55 is _________.

Bag I contains 3 red, 4 black and 3 white balls and Bag II contains 2 red, 5 black and 2 white balls. One ball is transferred from Bag I to Bag II and then a ball is drawn from Bag II. The ball so drawn is found to be black in colour. Then ...

The sum and product of the mean and variance of a binomial distribution are 82.5 and 1350 respectively. Then the number of trials in the binomial distribution is ____________.

Let \(\alpha, \beta(\alpha>\beta)\) be the roots of the quadratic equation \(x^{2}-x-4=0 .\) If \(P_{n}=\alpha^{n}-\beta^{n}\), \(n \in \mathrm{N}\), then \(\frac{P_{15} P_{16}-P_{14} P_{16}-P_{15}^{2}+P_{14} P_{15}}{P_{13} P_{14}}\) is equ...

$$ \begin{aligned} &\text { Let }\left\{a_{n}\right\}_{n=0}^{\infty} \text { be a sequence such that } a_{0}=a_{1}=0 \text { and } \\\\ &a_{n+2}=3 a_{n+1}-2 a_{n}+1, \forall n \geq 0 . \end{aligned} $$
Then $$a_{25} a_{23}-2 a_{25} a_{22}-2...

Let \(m_{1}, m_{2}\) be the slopes of two adjacent sides of a square of side a such that \(a^{2}+11 a+3\left(m_{1}^{2}+m_{2}^{2}\right)=220\). If one vertex of the square is \((10(\cos \alpha-\sin \alpha), 10(\sin \alpha+\cos \alpha))\), wh...

Let \(\mathrm{A}(\alpha,-2), \mathrm{B}(\alpha, 6)\) and \(\mathrm{C}\left(\frac{\alpha}{4},-2\right)\) be vertices of a \(\triangle \mathrm{ABC}\). If \(\left(5, \frac{\alpha}{4}\right)\) is the circumcentre of \(\triangle \mathrm{ABC}\), ...

The number of elements in the set \(S=\left\{x \in \mathbb{R}: 2 \cos \left(\frac{x^{2}+x}{6}\right)=4^{x}+4^{-x}\right\}\) is :

Let \(\vec{a}, \vec{b}, \vec{c}\) be three coplanar concurrent vectors such that angles between any two of them is same. If the product of their magnitudes is 14 and
$$(\vec{a} \times \vec{b}) \cdot(\vec{b} \times \vec{c})+(\vec{b} \times \...

Let \(\vec{a}\) and \(\vec{b}\) be two vectors such that \(|\vec{a}+\vec{b}|^{2}=|\vec{a}|^{2}+2|\vec{b}|^{2}, \vec{a} \cdot \vec{b}=3\) and \(|\vec{a} \times \vec{b}|^{2}=75\). Then \(|\vec{a}|^{2}\) is equal to __________.