JEE MAIN 2023 Previous Year Questions (PYQs) – Page 22 of 34
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The number of numbers, strictly between 5000 and 10000 can be formed using the digits 1, 3, 5, 7, 9 without repetition, is :
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Solution
$ \text{For } x\in\mathbb{R}, \text{ two real valued functions } f(x) \text{ and } g(x) \text{ are such that } g(x)=\sqrt{x}+1 \text{ and } (f\circ g)(x)=x+3-\sqrt{x}. \text{ Then } f(0) \text{ is equal to: } $
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Let $y=y(t)$ be a solution of the differential equation $\dfrac{dy}{dt}+\alpha y=\gamma e^{-\beta t}$ where $\alpha>0$, $\beta>0$ and $\gamma>0$.
Then $\displaystyle \lim_{t\to\infty} y(t)$
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For the differentiable function $f:\mathbb{R}\setminus{0}\to\mathbb{R}$, let $3f(x)+2f!\left(\dfrac{1}{x}\right)=\dfrac{1}{x}-10$. Then $\left|,f(3)+f'!\left(\dfrac{1}{4}\right)\right|$ is equal to:
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Let
$A=\begin{bmatrix}\dfrac{1}{\sqrt{10}} & \dfrac{3}{\sqrt{10}}\\[4pt]-\dfrac{3}{\sqrt{10}} & \dfrac{1}{\sqrt{10}}\end{bmatrix}$
and
$B=\begin{bmatrix}1 & -i\\[2pt] 0 & 1\end{bmatrix}$, where $i=\sqrt{-1}$.
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$\displaystyle \max_{0\le x\le \pi}\left\{x-2\sin x\cos x+\frac{1}{3}\sin(3x)\right\}=$
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Solution
Evaluate the sum:
$\displaystyle \sum_{k=0}^{6} \binom{51-k}{3}$
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$\text{The number of symmetric matrices of order }3\text{, with all entries from the set }{0,1,2,3,4,5,6,7,8,9}\text{ is:}$
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Let $f(x)=2x^{n}+\lambda$, $\lambda\in \mathbb{R}$, $n\in \mathbb{N}$, and $f(4)=133$, $f(5)=255$.
Then the sum of all the positive integer divisors of $\bigl(f(3)-f(2)\bigr)$ is:
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Let $a_1, a_2, a_3, \dots$ be a G.P. of increasing positive numbers. Let the sum of its $6^{th}$ and $8^{th}$ terms be $2$ and the product of its $3^{rd}$ and $5^{th}$ terms be $\dfrac{1}{9}$. Then $6(a_2 + a_4)(a_4 + a_6)$ is equal to
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Solution
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