Q113 / 160 · AP EAMCET 2023 (15 May Shift 1) · Mathematics

Assertion (A): $\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{(\sin x)^{\sqrt{2}} d x}{(\sin x)^{\sqrt{2}}+(\cos x)^{\sqrt{2}}}=\frac{\pi}{12}$ Reason (R): $\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{f(x) d x}{f(x)+f\left(\frac{\pi}{2}-x\right)}=\frac{\pi}{12}$

  • Correct answer
    A
    A is true, $\mathrm{R}$ is true and $\mathrm{R}$ is the correct explanation of A
  • B
    A is true, $\mathrm{R}$ is true but $\mathrm{R}$ is not the correct explanation of $\mathrm{A}$
  • C
    $\mathrm{A}$ is true, $\mathrm{R}$ is false
  • D
    A is false, $\mathrm{R}$ is true

Solution

Let $\int_{\pi / 6}^{\pi / 3} \frac{(\sin x)^{\sqrt{2}} d x}{(\sin x)^{\sqrt{2}}+(\cos x)^{\sqrt{2}}}...(1)$ Since $\int_a^b f(x) d x=\int_a^b f(a+b-x) d x$ Hence $I=\int_{\pi / 6}^{\pi / 3} \frac{(\cos x)^{\sqrt{2}} \cdot d x}{(\sin x)^{\sqrt{2}}+(\cos x)^{\sqrt{2}}}...(2)$ $$ \begin{aligned} & \text { Equation (1)+(2), } \\ & \Rightarrow I \int_{1 / 3}^{/ 3} \frac{(\sin x)^{\sqrt{2}}(\cos x)^{\sqrt{2}}}{(\sin x)^{\sqrt{2}}(\cos x)^{\sqrt{2}}} d x \\ & \Rightarrow I=\frac{\pi}{12} \end{aligned} $$ Hence Assertion ' $A$ ' is correct. $$ \begin{aligned} & \text { Let } P=\int_{\pi / 6}^{\pi / 3} \frac{f(x) d x}{f(x)+f\left(\frac{\pi}{2}-x\right)}...(3) \\ & \Rightarrow P=\int_{\pi / 6}^{\pi / 3} \frac{f\left(\frac{\pi}{2}-x\right) d x}{f(x)+f\left(\frac{\pi}{2}-x\right)}...(4) \end{aligned} $$ Equation (3) + (4), $$ 2 P=\int_{\pi / 6}^{\pi / 3} d x \Rightarrow P=\frac{\pi}{12} $$ Hence Reason $R$ is also true putting $f(x)=(\sin x)^{\sqrt{2}}$ in Reason ' $R$ ' we get$\int_{\pi / 6}^{\pi / 3} \frac{(\sin x)^{\sqrt{2}} d x}{(\sin x)^{\sqrt{2}}+(\cos x)^{\sqrt{2}}}=\frac{\pi}{12}$ Hence $R$ is correct explanation of ' $A$ '

Hence, the correct answer is option: 'A'.

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