JEE MAIN 2023 Previous Year Questions (PYQs) – Page 12 of 34
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$\displaystyle \lim_{t\to 0}\left(1^{\frac{1}{\sin^2 t}}+2^{\frac{1}{\sin^2 t}}+\cdots+n^{\frac{1}{\sin^2 t}}\right)^{\sin^2 t}$ is equal to
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Solution
Let $w_1$ be the point obtained by the rotation of $z_1 = 5 + 4i$ about the origin through a right angle in the anticlockwise direction,
and $w_2$ be the point obtained by the rotation of $z_2 = 3 + 5i$ about the origin through a right angle in the clockwise direction.
Then the principal argument of $w_1 - w_2$ is equal to:
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Solution
$\tan^{-1}\!\left(\dfrac{1+\sqrt{3}}{3+\sqrt{3}}\right) + \sec^{-1}\!\left(\sqrt{\dfrac{8+4\sqrt{3}}{6+3\sqrt{3}}}\right)$ is equal to:
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Solution
Let $y = y(x)$ be a solution curve of the differential equation
\[
(1 - x^2 y^2)\,dx = y\,dx + x\,dy.
\]
If the line $x = 1$ intersects the curve $y = y(x)$ at $y = 2$ and the line $x = 2$ intersects the curve $y = y(x)$ at $y = \alpha$,
then a value of $\alpha$ is:
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Solution
Let $y=y(x)$ be the solution of the differential equation
$x^{3}\,dy+(xy-1)\,dx=0,\quad x>0,$ with $y\!\left(\dfrac{1}{2}\right)=3-e.$
Then $y(1)$ is equal to:
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Solution
The value of the integral
\[
\int_{-\log_e 2}^{\log_e 2} e^x \left( \log_e\!\left( e^x + \sqrt{1 + e^{2x}} \right) \right) dx
\]
is equal to:
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Solution
Let $\Omega$ be the sample space and $A \subseteq \Omega$ be an event.
Given below are two statements:
(S1): If $P(A)=0$, then $A=\varnothing$
(S2): If $P(A)=1$, then $A=\Omega$
Then:
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Solution
Let sets A and B have 5 elements each. Let the mean of the elements in sets A and B be 5 and 8 respectively and the variance of the elements in sets A and B be 12 and 20 respectively. A new set C of 10 elements is formed by subtracting 3 from each element of
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Solution
The area enclosed by the curves $y^2 + 4x = 4$ and $y - 2x = 2$ is:
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Solution
For any vector $\vec{a} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}$, with $10|a_i| < 1, \; i = 1, 2, 3$,
consider the following statements:
(A): $\max \{|a_1|, |a_2|, |a_3|\} \le |\vec{a}|$
(B): $|\vec{a}| \le 3 \max \{|a_1|, |a_2|, |a_3|\}$
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Solution
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