JEE MAIN 2023 Previous Year Questions (PYQs) – Page 17 of 34
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The vector $\vec{a}=-\hat{i}+2\hat{j}+\hat{k}$ is rotated through a right angle, passing through the $y$-axis in its way and the resulting vector is $\vec{b}$.
Then the projection of $3\vec{a}+\sqrt{2}\,\vec{b}$ on $\vec{c}=5\hat{i}+4\hat{j}+3\hat{k}$ is:
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Solution
Let the function $f:[0,2]\to\mathbb{R}$ be defined as
\[
f(x)=
\begin{cases}
e^{\min\{x^2,\; x-[x]\}}, & x\in[0,1),\\[4pt]
e^{[\,x-\log_e x\,]}, & x\in[1,2],
\end{cases}
\]
where $[t]$ denotes the greatest integer less than or equal to $t$.
Then the value of the integral $\displaystyle \int_{0}^{2} x f(x)\,dx$ is:
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Solution
Let $x=2$ be a local minima of the function $f(x)=2x^{4}-18x^{2}+8x+12,\ x\in(-4,4)$.
If $M$ is the local maximum value of the function $f$ in $(-4,4)$, then $M=$
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Solution
The domain of the function
\[
f(x)=\frac{1}{\sqrt{[x]^2-3[x]-10}}
\]
(where $[x]$ denotes the greatest integer less than or equal to $x$) is:
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Solution
The mean and variance of the marks obtained by the students in a test are 10 and 4 respectively. Later, the marks of one of the students is increased from 8 to 12. If the new mean of the marks is 10.2, then their new variance is equal to :
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Solution
Two dice $A$ and $B$ are rolled. Let the numbers obtained on $A$ and $B$ be $\alpha$ and $\beta$ respectively.
If the variance of $\alpha-\beta$ is $\dfrac{p}{q}$, where $p$ and $q$ are co-prime, then the sum of the positive divisors of $p$ is equal to:
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Solution
The value of $\displaystyle \lim_{n\to\infty}
\frac{1+2-3+4+5-6+\cdots+(3n-2)+(3n-1)-3n}
{\sqrt{2n^{4}+4n+3}-\sqrt{n^{4}+5n+4}}$ is:
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Solution
Let $\mathbf{A}=\begin{bmatrix} 1 & \tfrac{1}{51} \\[2pt] 0 & 1 \end{bmatrix}$.
If $\mathbf{B}=\begin{bmatrix} 1 & 2 \\ -1 & -1 \end{bmatrix}\mathbf{A}\begin{bmatrix} -1 & -2 \\ 1 & 1 \end{bmatrix}$,
then the sum of all the elements of the matrix $\displaystyle \sum_{n=1}^{50} \mathbf{B}^n$ is equal to:
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Solution
Let $M$ be the maximum value of the product of two positive integers when their sum is $66$.
Let the sample space $S=\{\,x\in\mathbb{Z}: x(66-x)\ge \tfrac{5}{9}M\,\}$ and the event $A=\{\,x\in S:\ x\ \text{is a multiple of }3\,\}$.
Then $P(A)$ is equal to:
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Solution
Let $y=y(x),\ y>0$, be a solution curve of the differential equation
\[
(1+x^2)\,dy = y(x-y)\,dx.
\]
If $y(0)=1$ and $y(2\sqrt{2})=\beta$, then
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Solution
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