Problem $2$ Find the Fourier transform for the following functions, a) $\right)f\left(t\right)= \begin{cases} 1 \\ 0 \end{cases} $ $0<t<2$ otherwise $\left(10$ $Man^{1}<s\right)$ b) $\right)f\left(t-5\right)$ if $F\left(f\left(t\right)\right)=\dfrac {1} {\sqrt{2\pi } }sinc\left(w\right)$

Problem 1/ Find the Fourir series xpansion for the following function, Marks) $f\left(x\right)= \begin{cases} -8 \\ 8 \end{cases} $ $\pi <x<0$ $0<x<\pi ,f\left(x+2\pi \right)=f\left(x\right)$ $\left(5Ma$ Problem 2/ Find the Fourier transform for the following functions, $a\right)f\left(t\right)= \begin{cases} 1 \\ 0 \end{cases} $ $0<t<2$ $other1vise$ (10 Marks) $b\right)f\left(t-5\right)$ if $F\left(f\left(t\right)\right)=\dfrac {1} {\sqrt{2\pi } }sinc\left(w\right)$ Problem 31 The first term of a geometric progression is 12 and the fifth term is 55. Determine the 8" term and the 11" term. $\left(4$ $Mark5\right)$ Problem 4/ By multiplying the series expansions for sinx and $co5x$ prove that $2sinx.cosx=sin2x$ $\left(6Mar^{|ks}\right)$ Problem 5/ Solve the following pair of simultaneous differential equations : $\dfrac {d^{2}y} {dt^{2}}-x=y,$ $\dfrac {d^{2}y} {dt^{2}}+y=-x$ Given that at t = 0, $x=2,$ $y=-1,$ $\dfrac {dx} {dt}=0,$ $\dfrac {dy} {dt}=0$ (lo $1Ma/\times s\right)$

Problem $2|$ Find the Fourier transform for the following functions, $\right)f\left(t\right)= \begin{cases} 1 \\ 0 \end{cases} $ $0<t<2$ otherwise $\left(10$ $Ma\times $ $<5\right)$ $b\right)f\left(t-5\right)$ if $F\left(f\left(t\right)\right)=\dfrac {1} {\sqrt{2\pi } }sinc\left(w\right)$