GATE Papers >> EEE >> 2019 >> Question No 38

Question No. 38

A periodic function $f(t)$, with a period of $2\pi$, is represented as its Fourier series,

$f(t)=a_0+{\textstyle\sum_{n=1}^\infty}\;a_n\;\cos nt\;+{\textstyle\sum_{n=1}^\infty}\;b_n\sin\;nt.$

If

$f(t)=\left\{\begin{array}{lc}A\;\sin\;t,&0\;\leq\;t\;\pi\\0,&\;\;\;\;\;\pi\;<\;t\;<\;2\pi\end{array},\right.$

the Fourier series coefficients $a_1$ and $b_1$ of $f(t)$ are

Answer : (D) $a_1=0;\;b_1=\frac A2$

Solution of Question No 38 of GATE 2019 EEE Paper

As per the given description of f(t), if we draw its waveform, if looks like

→ One way to obtain its C.T.T.S is by obtaining its odd and even part and then by obtaining their individual C.T.F.S and finally we can add them to get complete C.T.F.S of f(t). However in this case we can pick the correct option by eliminating others.

$\mathrm f(\mathrm t)={\mathrm f}_{\mathrm o}(\mathrm t)1+\mathrm{fe}(\mathrm t)=\left[\frac{\mathrm f(\mathrm t)-\mathrm f(-\mathrm t)}2\right]+\left[\frac{\mathrm f(\mathrm t)+\mathrm f(-\mathrm t)}2\right]$

${\mathrm f}_{\mathrm o}(\mathrm t)=\left[\frac{\mathrm A}2{\mathrm{sinω}}_{\mathrm o}\mathrm t\right]+\sum\limits_{\mathrm n=1}^{\mathrm N}{\mathrm a}_{\mathrm n}{\mathrm{cosω}}_{\mathrm o}\mathrm t$

From this ${\mathrm b}_1=\frac{\mathrm A}2,$  So only option D satisfy this.