JEE MAIN - Mathematics (2023 - 31st January Morning Shift - No. 22)
Let for $$x \in \mathbb{R}$$,
$$ f(x)=\frac{x+|x|}{2} \text { and } g(x)=\left\{\begin{array}{cc} x, & x<0 \\ x^{2}, & x \geq 0 \end{array}\right. \text {. } $$
Then area bounded by the curve $$y=(f \circ g)(x)$$ and the lines $$y=0,2 y-x=15$$ is equal to __________.
Answer
72
Explanation
$f(x)=\frac{x+|x|}{2}=\left[\begin{array}{ll}x & x \geq 0 \\ 0 & x<0\end{array}\right.$
$g(x)=\left[\begin{array}{cc}x^{2} & x \geq 0 \\ x & x<0\end{array}\right.$
$f(x)=f[g(x)]=\left[\begin{array}{cl}g(x), & g(x) \geq 0 \\ 0, & g(x)<0\end{array}\right.$
$\operatorname{fog}(x)=\left[\begin{array}{cc}x^{2}, & x \geq 0 \\ 0, & x<0\end{array}\right.$
$2 y-x=15$
_31st_January_Morning_Shift_en_22_1.png)
$\mathrm{A}=\int_{0}^{3}\left(\frac{\mathrm{x}+15}{2}-\mathrm{x}^{2}\right) \mathrm{dx}+\frac{1}{2} \times \frac{15}{2} \times 15$
= $\frac{x^{2}}{4}+\frac{15 x}{2}-\left.\frac{x^{3}}{3}\right|_{0} ^{3}+\frac{225}{4}$
$=\frac{9}{4}+\frac{45}{2}-9+\frac{225}{4}=\frac{99-36+225}{4}$
$=\frac{288}{4}=72$
$g(x)=\left[\begin{array}{cc}x^{2} & x \geq 0 \\ x & x<0\end{array}\right.$
$f(x)=f[g(x)]=\left[\begin{array}{cl}g(x), & g(x) \geq 0 \\ 0, & g(x)<0\end{array}\right.$
$\operatorname{fog}(x)=\left[\begin{array}{cc}x^{2}, & x \geq 0 \\ 0, & x<0\end{array}\right.$
$2 y-x=15$
$\mathrm{A}=\int_{0}^{3}\left(\frac{\mathrm{x}+15}{2}-\mathrm{x}^{2}\right) \mathrm{dx}+\frac{1}{2} \times \frac{15}{2} \times 15$
= $\frac{x^{2}}{4}+\frac{15 x}{2}-\left.\frac{x^{3}}{3}\right|_{0} ^{3}+\frac{225}{4}$
$=\frac{9}{4}+\frac{45}{2}-9+\frac{225}{4}=\frac{99-36+225}{4}$
$=\frac{288}{4}=72$
Comments (0)
