(Solved): Please help me fast... 1. Evaluate the following integral using Cauchy's Residue Theorem. \[ \oint ...

Please help me fast...

1. Evaluate the following integral using Cauchy's Residue Theorem. \[ \oint_{|z-5|=3} \frac{z+3}{(z-3)(z-6)(z-9)} d z \text {. } \] 2. Show that \( \mathcal{L}\{\cos a t\}=\frac{s}{s^{2}+a^{2}} \). Hence find \( \mathcal{L}\{\sin 8 t \sin 6 t\} \). 3. Solve the following IVP using Laplace Transformation: \( y^{\prime \prime}+2 y^{\prime}-48 y=3 e^{2 t}+4 e^{-2 t} ; y(0)=0, y^{\prime}(0)=1 \).