JEE MAIN 2023 Previous Year Questions (PYQs) – Page 26 of 34
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Consider the following system of equations
\[
\begin{cases}
\alpha x+2y+z=1,\\
2\alpha x+3y+z=1,\\
3x+\alpha y+2z=\beta
\end{cases}
\]
for some $\alpha,\beta\in\mathbb{R}$. Then which of the following is NOT correct?
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Solution
The total number of three-digit numbers, divisible by 3, which can be formed using the digits
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Solution
If the vectors $\vec a=\lambda\,\hat i+\mu\,\hat j+4\,\hat k$, $\vec b=-2\,\hat i+4\,\hat j-2\,\hat k$ and
$\vec c=2\,\hat i+3\,\hat j+\hat k$ are coplanar and the projection of $\vec a$ on the vector $\vec b$ is
$\sqrt{54}$ units, then the sum of all possible values of $\lambda+\mu$ is equal to:
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Solution
Let ABCD be a quadrilateral. If E and F are the mid points of the diagonals AC and BD respectively and
$\overrightarrow{(AB-BC)}+\overrightarrow{(AD-DC)}=k\,\overrightarrow{FE}$, then $k$ is equal to:
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Solution
Let $x=2$ be a root of the equation $x^{2}+px+q=0$ and define
\[
f(x)=
\begin{cases}
\dfrac{1-\cos\!\big(x^{2}-4px+q^{2}+8q+16\big)}{(x-2p)^{4}}, & x\ne 2p,\\[6pt]
0, & x=2p.
\end{cases}
\]
Then $\displaystyle \lim_{x\to 2p^{+}} \big[\,f(x)\,\big]$, where $[\cdot]$ denotes the greatest integer function, is:
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Solution
If the domain of the function $f(x)=\log_e(4x^2+11x+6)+\sin^{-1}(4x+3)+\cos^{-1}\!\left(\dfrac{10x+6}{3}\right)$ is $(\alpha,\beta]$, then $36|\alpha+\beta|$ is equal to:
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Solution
Let $y=y(x)$ be the solution of the differential equation
$$x\log_e x \,\frac{dy}{dx}+y=x^2\log_e x,\quad (x>1).$$
If $y(2)=2$, then $y(e)$ is equal to:
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Solution
The mean and standard deviation of 10 observations are 20 and 8 respectively. Later on, it was observed that one observation was recorded as 50 instead of 40. Then the correct variance is :
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Solution
Let \(R\) be a relation defined on \(\mathbb{N}\) as \(aRb\) iff \(2a+3b\) is a multiple of \(5\), \(a,b\in\mathbb{N}\).
Then \(R\) is:
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Solution
Let $[x]$ denote the greatest integer function and
$f(x)=\max\{\,1+x+[x],\ 2+x,\ x+2[x]\,\},\ 0\le x\le 2.$
Let $m$ be the number of points in $[0,2]$, where $f$ is not continuous and $n$ be the number of points in $(0,2)$, where $f$ is not differentiable. Then $(m+n)^2+2$ is equal to:
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Solution
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