JEE Main 2022 (Online) 26th July Evening Shift

A vector \(\vec{a}\) is parallel to the line of intersection of the plane determined by the vectors \(\hat{i}, \hat{i}+\hat{j}\) and the plane determined by the vectors \(\hat{i}-\hat{j}, \hat{i}+\hat{k}\). The obtuse angle between $$\vec{a...

The largest value of \(a\), for which the perpendicular distance of the plane containing the lines \(\vec{r}=(\hat{i}+\hat{j})+\lambda(\hat{i}+a \hat{j}-\hat{k})\) and \(\vec{r}=(\hat{i}+\hat{j})+\mu(-\hat{i}+\hat{j}-a \hat{k})\) from the ...

The plane passing through the line \(L: l x-y+3(1-l) z=1, x+2 y-z=2\) and perpendicular to the plane \(3 x+2 y+z=6\) is \(3 x-8 y+7 z=4\). If \(\theta\) is the acute angle between the line \(L\) and the \(y\)-axis, then $$415 \cos ^{2} \th...

If the maximum value of \(a\), for which the function \(f_{a}(x)=\tan ^{-1} 2 x-3 a x+7\) is non-decreasing in \(\left(-\frac{\pi}{6}, \frac{\pi}{6}\right)\), is \(\bar{a}\), then \(f_{\bar{a}}\left(\frac{\pi}{8}\right)\) is equal to :

The area bounded by the curves \(y=\left|x^{2}-1\right|\) and \(y=1\) is

\(\sum\limits_{\matrix{ {i,j = 0} \cr {i \ne j} \cr } }^n {{}^n{C_i}\,{}^n{C_j}}\) is equal to

Let the abscissae of the two points \(P\) and \(Q\) on a circle be the roots of \(x^{2}-4 x-6=0\) and the ordinates of \(\mathrm{P}\) and \(\mathrm{Q}\) be the roots of \(y^{2}+2 y-7=0\). If \(\mathrm{PQ}\) is a diameter of the circle $$x^{...

If \(z=x+i y\) satisfies \(|z|-2=0\) and \(|z-i|-|z+5 i|=0\), then :

\(\int\limits_{0}^{20 \pi}(|\sin x|+|\cos x|)^{2} d x \text { is equal to }\)

Let the solution curve \(y=f(x)\) of the differential equation \(\frac{d y}{d x}+\frac{x y}{x^{2}-1}=\frac{x^{4}+2 x}{\sqrt{1-x^{2}}}\), \(x\in(-1,1)\) pass through the origin. Then $$\int\limits_{-\frac{\sqrt{3}}{2}}^{\frac{\sqrt{3}}{2}}...

Suppose \(y=y(x)\) be the solution curve to the differential equation \(\frac{d y}{d x}-y=2-e^{-x}\) such that \(\lim\limits_{x \rightarrow \infty} y(x)\) is finite. If \(a\) and \(b\) are respectively the \(x\) - and \(y\)-intercepts of t...

The value of \(\log _{e} 2 \frac{d}{d x}\left(\log _{\cos x} \operatorname{cosec} x\right)\) at \(x=\frac{\pi}{4}\) is

The acute angle between the pair of tangents drawn to the ellipse \(2 x^{2}+3 y^{2}=5\) from the point \((1,3)\) is :

If the line \(x-1=0\) is a directrix of the hyperbola \(k x^{2}-y^{2}=6\), then the hyperbola passes through the point :

\(\text { The integral } \int \frac{\left(1-\frac{1}{\sqrt{3}}\right)(\cos x-\sin x)}{\left(1+\frac{2}{\sqrt{3}} \sin 2 x\right)} d x \text { is equal to }\)

If \(0 < x < {1 \over {\sqrt 2 }}\) and \({{{{\sin }^{ - 1}}x} \over \alpha } = {{{{\cos }^{ - 1}}x} \over \beta }\), then the value of \(\sin \left( {{{2\pi \alpha } \over {\alpha + \beta }}} \right)\) is :

Let \(\beta=\mathop {\lim }\limits_{x \to 0} \frac{\alpha x-\left(e^{3 x}-1\right)}{\alpha x\left(e^{3 x}-1\right)}\) for some \(\alpha \in \mathbb{R}\). Then the value of \(\alpha+\beta\) is :

Negation of the Boolean expression \(p \Leftrightarrow(q \Rightarrow p)\) is

$$
\text { Let } A=\left[\begin{array}{l}
1 \\
1 \\
1
\end{array}\right] \text { and } B=\left[\begin{array}{ccc}
9^{2} & -10^{2} & 11^{2} \\
12^{2} & 13^{2} & -14^{2} \\
-15^{2} & 16^{2} & 17^{2}
\end{array}\right] \text {, then the value ...

The number of matrices $$A=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)$$, where \(a, b, c, d \in\{-1,0,1,2,3, \ldots \ldots, 10\}\), such that \(A=A^{-1}\), is ___________.

Let \(\mathrm{P}\) and \(\mathrm{Q}\) be any points on the curves \((x-1)^{2}+(y+1)^{2}=1\) and \(y=x^{2}\), respectively. The distance between \(P\) and \(Q\) is minimum for some value of the abscissa of \(P\) in the interval :

The equation of a common tangent to the parabolas \(y=x^{2}\) and \(y=-(x-2)^{2}\) is

Numbers are to be formed between 1000 and 3000 , which are divisible by 4 , using the digits \(1,2,3,4,5\) and 6 without repetition of digits. Then the total number of such numbers is ____________.

Let \(X\) be a binomially distributed random variable with mean 4 and variance \(\frac{4}{3}\). Then, \(54 \,P(X \leq 2)\) is equal to :

The minimum value of the sum of the squares of the roots of \(x^{2}+(3-a) x+1=2 a\) is:

If \(\sum\limits_{k=1}^{10} \frac{k}{k^{4}+k^{2}+1}=\frac{m}{n}\), where m and n are co-prime, then \(m+n\) is equal to _____________.

Different A.P.'s are constructed with the first term 100, the last term 199, and integral common differences. The sum of the common differences of all such A.P.'s having at least 3 terms and at most 33 terms is ___________.

Let \(A=\{1,2,3,4,5,6,7\}\) and \(B=\{3,6,7,9\}\). Then the number of elements in the set \(\{C \subseteq A: C \cap B \neq \phi\}\) is ___________.

The mean and standard deviation of 40 observations are 30 and 5 respectively. It was noticed that two of these observations 12 and 10 were wrongly recorded. If \(\sigma\) is the standard deviation of the data after omitting the two wrong ob...

If the sum of solutions of the system of equations \(2 \sin ^{2} \theta-\cos 2 \theta=0\) and \(2 \cos ^{2} \theta+3 \sin \theta=0\) in the interval \([0,2 \pi]\) is \(k \pi\), then \(k\) is equal to __________.