Download App

JEE Main 7-April-2025 Paper - Shift 2 — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEEJEE Main 7-April-2025 Paper - Shift 24 Marks
MCQ
If the range of the function $f(x)=\frac{5-x}{x^{2}-3 x+2}$, $x \neq 1,2$, is $(-\infty, \alpha] \cup[\beta, \infty)$, then $\alpha^{2}+\beta^{2}$ is equal to :
  • A
    190
  • B
    192
  • C
    188
  • D
    194
✓

Answer

D. 194
$y=\frac{5-x}{x^{2}-3 x+2}$
$\mathrm{yx}^{2}-3 \mathrm{xy}+2 \mathrm{y}+\mathrm{x}-5=0$
$y z^{2}+(-3 y+1) x+(2 y-5)=0$
Case I : If $\mathrm{y}=0$ (Accepted)
\begin{equation*}
\Rightarrow x=5
\end{equation*}
Case II : If $y \neq 0$
\begin{equation*}
\mathrm{D} \geq 0
\end{equation*}
$(-3 y+1)^{2}-4(y)(2 y-5) \geq 0$
$9 y^{2}+1-6 y-8 y^{2}+20 y \geq 0$
$y^{2}+14 y+1 \geq 0$
$(y+7)^{2}-48 \geq 0$
$|y+7| \geq 4 \sqrt{3}$
$\Rightarrow y+7 \geq 4 \sqrt{3}$ or $y+7 \leq-4 \sqrt{3}$
$\Rightarrow \mathrm{y} \geq 4 \sqrt{3}-7$ or $\mathrm{y} \leq-4 \sqrt{3}-7$
From Case I and Case II
$\mathrm{y} \in(-\infty,-4 \sqrt{3}-7] \cup[4 \sqrt{3}-7, \infty)$
So $\alpha=-4 \sqrt{3}-7$
\begin{equation*}
\begin{aligned}
\beta= & 4 \sqrt{3}-7 \\
\Rightarrow \mathrm{a}^{2}+\mathrm{b}^{2} & =(-4 \sqrt{3}-7)^{2}+(4 \sqrt{3}-7)^{2} \\
& =2(48+49) \\
= & 194
\end{aligned}
\end{equation*}

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "SECTION - A [MATHS - MCQ]" from JEE Main 7-April-2025 Paper - Shift 2→JEE Main 7-April-2025 Paper - Shift 2 — all question types→JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

The product of a matrix and its transpose is an identity matrix. the value of determinant of this matrix is
→
The area of the region $\left\{ {\left( {x,y} \right):{x^2} + {y^2} \leqslant 1 \leqslant x + y} \right\}$ is
→
Let $r_k=\frac{\int_0^1\left(1-x^7\right)^k d x}{\int_0^1\left(1-x^7\right)^{k+1} d x}, k \in N$. Then the value of $\sum_{\mathrm{k}=1}^{10} \frac{1}{7\left(\mathrm{r}_{\mathrm{k}}-1\right)}$ is equal to ...........
→
The value of the integral $\int_{-\pi / 2}^{\pi / 2}\left(x^2+\ln \frac{\pi+x}{\pi-x}\right) \cos x d x$ is
→
The values of $A$ and $B$ such that the function $f(x) = \left\{ {\begin{array}{*{20}{c}}{ - 2\sin x,}&{x \le - \frac{\pi }{2}}\\{A\sin x + B,}&{ - \frac{\pi }{2} < x < \frac{\pi }{2}}\\{\cos x,}&{x \ge \frac{\pi }{2}}\end{array}} \right.$, is continuous everywhere are
→
If the given lines $y = {m_1}x + {c_1},y = {m_2}x + {c_2}$ and $y = {m_3}x + {c_3}$ be concurrent, then
→
Let $\mathrm{f}:[0,3] \rightarrow \mathrm{R}$ be defined by

$f(x)=\min \{x-[x], 1+[x]-x\}$

where $[\mathrm{x}]$ is the greatest integer less than or equal to $\mathrm{x}$. Let $\mathrm{P}$ denote the set containing all $x \in[0,3]$ where $f$ is discontinuous, and $Q$ denote the set containing all $x \in(0,3)$ where $f$ is not differentiable. Then the sum of number of elements in $\mathrm{P}$ and $\mathrm{Q}$ is equal to $......$

→
The value of $\int_0^1 {{{\tan }^{ - 1}}\left( {\frac{{2x - 1}}{{1 + x - {x^2}}}} \right)} \,dx$ is
→
Let $f: R \rightarrow R$ be a differentiable function with $f(0)=0$. If $y=f(x)$ satisfies the differential equation $\frac{ dy }{ dx }=(2+5 y )(5 y -2)$ then the value of $\lim _{x \rightarrow-\infty} f(x)$ is. . . . . .
→
The set of all those points, where the function $f(x) = \frac{x}{{1 + |x|}}$ is differentiable, is
→