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:
Previous 10 Questions — JEE Main 2023 (29 January Morning Shift)
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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\…
Topic: JEE Main 2023 (29 January Morning Shift)
Consider the following system of equations
\[
\begin{cases}
\alpha x+2y+z=1,\\
2\alpha x+3y+z=1,\\
3x+\alpha y+2z=\beta…
Topic: JEE Main 2023 (29 January Morning Shift)
Let
$$A=\{(x,y)\in\mathbb{R}^{2}:\ y\ge 0,\ 2x\le y\le \sqrt{4-(x-1)^{2}}\}$$
and
$$B=\{(x,y)\in\mathbb{R}\times\math…
Topic: JEE Main 2023 (29 January Morning Shift)
Let $[x]$ denote the greatest integer $\le x$. Consider the function
$$f(x)=\max\{x^{2},\,1+[x]\}.$$
Then the value o…
Topic: JEE Main 2023 (29 January Morning Shift)
Let $f:\mathbb{R}\to\mathbb{R}$ be a function such that
$$f(x)=\frac{x^{2}+2x+1}{x^{2}+1}.$$
Then:
Topic: JEE Main 2023 (29 January Morning Shift)
The domain of
$$f(x)=\frac{\log_{(x+1)}(x-2)}{e^{2\log_e x}-(2x+3)},\quad x\in\mathbb{R}$$
is:
Topic: JEE Main 2023 (29 January Morning Shift)
Let $\Delta$ be the area of the region $\{(x,y)\in\mathbb{R}^{2}:\ x^{2}+y^{2}\le 21,\ y^{2}\le 4x,\ x\ge 1\}$.
Then …
Topic: JEE Main 2023 (29 January Morning Shift)
For two non-zero complex numbers $z_{1}$ and $z_{2}$, if $\operatorname{Re}(z_{1}z_{2})=0$ and $\operatorname{Re}(z_{1}…
Topic: JEE Main 2023 (29 January Morning Shift)
A light ray emits from the origin making an angle $30^\circ$ with the positive $x$-axis.
After getting reflected by t…
Topic: JEE Main 2023 (29 January Morning Shift)
Let $\alpha$ and $\beta$ be real numbers. Consider a $3\times 3$ matrix $A$ such that $A^{2}=3A+\alpha I$.
If $A^{4}=…
Topic: JEE Main 2023 (29 January Morning Shift)
