STUPIDSID
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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