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Total time: --
INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

1(a) Express the $$\frac{3}{1+i}-\frac{1}{2-i}+\frac{1}{1-i}$$/ in the form of a+ib.
6 M
1(b) Find the cube roots of 1-i.
7 M
1(c) Prove that $\left ( \frac{1+\cos \theta +i\sin \theta }{1+\cos \theta -i\sin \theta } \right )^n = \cos n\theta + i\sin n\theta.$
7 M

2(a) Find the nth derivative of $e^{ax}\cos \left ( bx+c \right )$.
7 M
2(b) Find the nth derivative of $\frac{x}{\left ( x-1 \right )\left ( 2x+3 \right )}$.
6 M
2(c) If $$y= a\cos \left ( logx \right )+ b \sin \left ( \log x \right )$$/ prove that $$x^2y_{n+2}+\left ( 2n+1 \right )xy_{n+1}+\left ( n^2+1 \right )y_n=0.$$/
7 M

3(a) With usual notations P.T $\tan \theta =\frac{rd\theta }{dr}$.
6 M
3(b) Find the angle between the pairs of curves $r=a\log \theta \ \ r=\frac{a}{\log \theta }.$
7 M
3(c) Find the Podal equation to the curve $r =a\left ( 1+\sin \theta \right )$.
7 M

4(a) State and prove Euler's theorem of Homogenous functions.
6 M
4(b) If u=(x-y, y-z, z-x) P.T $\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}+\frac{\partial u}{\partial z}=0$.
7 M
4(c) If $$u = \tan ^{-1}x+\tan ^{-1}y,\ \ V=\frac{x+y}{1-xy}\ \ \text{S.T}\frac{\partial \left ( u,v \right )}{\partial \left ( x,y \right )}=0.$$/
7 M

5(a) Obtain the Reduction formula for $$\int \sin ^m x \ \ \cos ^n x \ \ dx.$$/ Where m, n are positive integers.
7 M
5(b) Evaluate $\int_{1}^{2}\int_{0}^{{2-y}} xy \ \ dx \ \ dy.$
6 M
5(c) Evaluate $\int_{0}^{3}\int_{0}^{2}\int_{0}^{3}\ \ \left ( x+y+z \right )dz \ \ dx \ \ dy.$
7 M

6(a) Prove that$\bigg\lceil \left ( \frac{1}{2} \right )=\sqrt{\pi }.$
6 M
6(b) Prove that $\int_{0}^{\infty }x^2e^{-n{^4}}\int_{0}^{\infty }e{-x{^5}}\ \ dx = \frac{\pi }{8\sqrt{2}}.$
7 M

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