Question

The areas $S_1,S_2,S_3$ bounded by $x-$axis and half waves of the curves $y=e^{-ax}\sin \beta x,x\ge 0$ from left to right then:

• A. $S_1,S_2,S_3$ are in A.P
• B. $S_1,S_2,S_3$ are in G.P
• C. $S_1^3+S_3^3>2S_2^3$
• D. $S_1=S_2=S_3$

Answer 1

Answer: (B), (C)

The correct options are
B $S_1,S_2,S_3$ are in G.P
C $S_1^3+S_3^3>2S_2^3$

$\because S_n=\frac{1}{\beta }\left|{\int }_{\frac{n\pi }{\beta }}^{\frac{(n+1)\pi }{\beta }}{e}^{-ax}\sin \beta xdx\right|$

$=\frac{1}{\beta }\left|{\int }_{n\pi }^{(n+1)\pi }{e}^{\frac{-alphax}{\beta }}\sin xdx\right|$

$=\frac{1}{\beta }\left|{\int }_0^{\pi }{e}^{-\frac{\alpha }{\beta }(x+n\pi )}sin(n\pi +x)dx\right|$

$=\frac{1}{\beta }\left|{\int }_0^{\pi }{e}^{-\frac{\alpha (x+2\pi )}{\beta }}{(-1)}^n\sin xdx\right|$

$S_n=e^{-\frac{n\pi \alpha }{\beta }I}$ .... (say)
$\therefore \frac{S_2}{S_1}=\frac{S_3}{S_2}=e^{-\frac{\pi \alpha }{\beta }}$
$\therefore S_1,S_2,S_3$ are in G.P
$\Rightarrow$A.M$>$G.M
$\Rightarrow \frac{S_1+S_3}{2}>S_2$
$\Rightarrow S_1+S_3>2S_2$
$\therefore S_1^n,S_2^n,S_3^n$ are also in G.P
$S_1^n+S_3^n>2S_2^n$