JEE MAIN 2022 Previous Year Questions (PYQs) – Page 1 of 31
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Let $\vec{a} = \alpha \hat{i} + \hat{j} - \hat{k}$ and $\vec{b} = 2\hat{i} + \hat{j} - \alpha \hat{k}$, $\alpha > 0$.
If the projection of $\vec{a} \times \vec{b}$ on the vector $-\hat{i} + 2\hat{j} - 2\hat{k}$ is $30$, then $\alpha$ is equal to:
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Solution
Let $E_1, E_2, E_3$ be three mutually exclusive events such that
$P(E_1)=\dfrac{2+3p}{6}$, $P(E_2)=\dfrac{2-p}{8}$ and $P(E_3)=\dfrac{1-p}{2}$.
If the maximum and minimum values of $p$ are $p_1$ and $p_2$, then $(p_1+p_2)$ is equal to:
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Solution
$\tan\!\left(2\tan^{-1}\!\tfrac{1}{5}+\sec^{-1}\!\tfrac{\sqrt{5}}{2}+2\tan^{-1}\!\tfrac{1}{8}\right)$ is equal to:
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Solution
The minimum value of the sum of the squares of the roots of
$x^{2}+(3-a)x+1=2a$ is:
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Solution
If $z = x+iy$ satisfies $|z|-2=0$ and $|z-i|-|z+5i|=0$, then :
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Solution
Let $A=\begin{bmatrix}1\\1\\1\end{bmatrix}$ and
$B=\begin{bmatrix}
9^{2} & -10^{2} & 11^{2}\\
12^{2} & 13^{2} & -14^{2}\\
-15^{2} & 16^{2} & 17^{2}
\end{bmatrix}$,
then the value of $A'BA$ is:
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Solution
Let $P$ and $Q$ 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:
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Solution
If the maximum value of $a$, for which the function $f_a(x)=\tan^{-1}(2x)-3ax+7$ is non-decreasing in $\left(-\tfrac{\pi}{6},\,\tfrac{\pi}{6}\right)$, is $\bar a$, then $f_{\bar a}\!\left(\tfrac{\pi}{8}\right)$ is equal to :
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Solution
Let $\beta=\lim_{x\to 0}\dfrac{\alpha x-(e^{3x}-1)}{\alpha x(e^{3x}-1)}$ for some $\alpha\in\mathbb{R}$.
Then the value of $\alpha+\beta$ is:
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Solution
The value of $\log_{e}2 \, \dfrac{d}{dx}\!\big(\log_{\cos x} \csc x\big)$ at $x=\tfrac{\pi}{4}$ is:
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Solution
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