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JEE Main 22-Jan-2025 Paper - Shift 1 — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEEJEE Main 22-Jan-2025 Paper - Shift 14 Marks
Question
Let the function,$
f(x)=\left\{\begin{array}{ll}
-3 a x^2-2, & x<1 \\
a^2+b x, & x \geq 1
\end{array}\right.
$
Be differentiable for all $x \in R$, where $a >1, b \in R$. If the area of the region enclosed by $y=f(x)$ and the line $y=-20$ is $\alpha+\beta \sqrt{3}, \alpha, \beta, \in Z$, then the value of $\alpha+\beta$ is $\qquad$
✓

Answer

34
Sol. $f(x)$ is continuous and differentiable<br>$
\begin{array}{l}
\text { at } x=1 ; \quad LHL=RHL, LHD=RHD \\
-3 a-2=a^2+b,-6 a=b \\
a=2,1 ; b=-12 \\
f(x)=\left\{\begin{array}{cc}
-6 x^2-2 & ; 
4<1 \\
4-12 x & ; x \geq 1
\end{array}\right.
\end{array}
$
Image

$\begin{array}{l}\text { Area }=\int_{-\sqrt{3}}^1\left(-6 x^2-2+20\right) d x+\int_1^2(4-12 x+20) d x \\ 16+12 \sqrt{3}+6=22+12 \sqrt{3}\end{array}$

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