JEE Advanced 2015 · Paper 2 · Mathematics Q53

← JEE Advanced 2015 · Paper 2

MathematicsMultiple correct+4 / -2Question 53 of 60

The option(s) with the values of $a$ and $L$ that satisfy the following equation is(are)
$\frac{\displaystyle\int_0^{4\pi}e^t(\sin^6at+\cos^4at)\,dt}{\displaystyle\int_0^{\pi}e^t(\sin^6at+\cos^4at)\,dt}=L?$

a=2,\ L=\dfrac{e^{4\pi}-1}{e^\pi-1}

a=2,\ L=\dfrac{e^{4\pi}+1}{e^\pi+1}

a=4,\ L=\dfrac{e^{4\pi}-1}{e^\pi-1}

a=4,\ L=\dfrac{e^{4\pi}+1}{e^\pi+1}

Answer and worked solution

Official answer: A, C

Period. $g(t)=\sin^6at+\cos^4at$ has period $\pi/a$ , which divides $\pi$ for both $a=2$ and $a=4$ .

Split the numerator. Break $[0,4\pi]$ into four intervals of length π and shift $t\to t+k\pi$ :
$\int_0^{4\pi}e^tg\,dt=\sum_{k=0}^3e^{k\pi}\int_0^\pi e^tg\,dt.$

Ratio.
$L=1+e^\pi+e^{2\pi}+e^{3\pi}=\frac{e^{4\pi}-1}{e^\pi-1}.$

(A), (C)

Topics:Properties of definite integrals

← Previous question Next question →

JEE Advanced 2015 · Paper 2, Mathematics question 53