1(a) If cos α cosh $ \beta =\frac{x}{2},\sin \alpha sinh \beta =\frac{y}{2}, $/ Prove that $ sec\left ( \alpha -i\beta \right )+sec\left ( \alpha +i\beta \right )= \frac{4x}{x^2+y^2}$/

1(b) If $ z=\log \left ( e^x+e^y \right ) $/, show that rt-s²=0, where $ r = \frac{\partial^2 z}{\partial x^2}, t = \frac{\partial^2 z}{\partial y^2}, s =\frac{\partial^2 z}{\partial x \partial y} $/

1(c) If x = u v, $$y = \frac{u+v}{u-v}. $$ Find $$\frac{\partial \left ( u,v \right )}{\partial \left ( x,y \right )}$$.

1(d) If $ y = 2^{x}\sin ^2x\cos x $/ find y_n

1(e) Express the matrix $ A =\begin{bmatrix} 1 & 0& 5& 3\\\\ -2& 1& 6& 1\\\\ 3 & 2& 7& 1\\\\ 4& -4& 2& 0 \end{bmatrix} $ as the sum of symmetric and skew- symmetric matrices.

1(f) Evaluate $$\lim_{x\rightarrow 0}\frac{e^2-\left ( 1+x \right )^2}{x\log \left ( 1+x \right )}$$

2(a) Show that the roots of x⁵=1 can be written as 1,
α,
α²,
α³,
α⁴. Hence show that $$\left ( 1-\alpha \right )\left ( 1-\alpha ^2 \right )\left ( 1 -\alpha ^3 \right )\left ( 1 -\alpha ^4 \right ) = 5$$

2(b) Reduce the following matrix to its normal from and hence find its rank $$A = \begin{bmatrix} 3 & -2& 0& 1\\ 0& 2& 2& 7\\ 1& -2& -3& 2\\ 0& 1& 2& 1 \end{bmatrix}$$

2(c) Solve the following system of equations by Gauss-Seidel Iterative Method upto four interations.
4x-2y-z=40
x-6y+2z=-28
x-2y+12z = -86

3(a) Investigate for what values of 'λ' and 'μ' the system of equations $x+y+z = 6 x+2 y+3 z = 10 x + 2 y + λ z = μ $/ has i) no solution
ii) a unique solution
iii) an infinite no. of solutions.

3(b) If $u = x^2+y^2+z^2 $/, where $ x = e^t, y = e^t \sin t, z = e^t \cos t $/ Prove that $$\frac{du}{dt} = 4e^{2t}$$

3(c)(i) Show that $$\sin \left ( e^x -1 \right ) = x+\frac{x^2}{2}-\frac{5x^4}{24}+.........$$

3(c)(ii) Expand 2x³+7x²+x-6 in power of x-2

4(a) If x=u+v+w,
y = uv+vw+uw,
z=uvw and φ is a function of x,y and z. Prove that $$x\frac{\partial\phi }{\partial x}+2y\frac{\partial^\phi }{\partial y}+3z\frac{\partial \phi }{\partial z} = u\frac{\partial \phi }{\partial u}+ v\frac{\partial\phi }{\partial v} + \frac{\partial \phi }{\partial w}$$

4(b) if $ \tan \left ( \theta+i\phi \right )=\tan \alpha + i\sec \alpha $/, Prove that
i) $$e^{2\phi } = \cot \frac{\alpha }{2}$$
ii) $$2\theta =n\pi +\frac{\pi }{2} + \alpha $$

4(c) Find the root of the equation x⁴+x³+7x²-x+5=0 which lies between 2 and 2.1 correct to three places of decimals using Regula Falsi Method.

5(a) If $ y = \left ( x+\sqrt{x^2-1} \right )^m $/, Prove That $$\left ( x^2-1 \right )y_{n+2}+(2n+1)xy_{n+1}+\left ( n^2-m^2 \right ) y _n =0$$.

5(b) Using the encoding matrix $ \begin{bmatrix} 1 & 1\\\\ 0& 1 \end{bmatrix} $/, encode and decode the message I* LOVEMUMBAI

5(c)(i) Consulting only principal values separate into real and imaginary parts $$i^\log \left ( 1+i \right )$$

5(c)(ii) Show that $$i\log \left ( \frac{x-i}{x+i} \right ) = \pi -2\tan ^{-1}x$$

6(a) Using De Moivre's theorem prove that $$\cos ^6\theta -\sin ^6\theta =\frac{1}{16}\left ( \cos 6\theta +15\cos 2\theta \right )$$

6(b) If$ u = sin ^{-1}\left ( \frac{x^\frac{1}{3}+y^\frac{1}{3}}{x^\frac{1}{2}-y^\frac{1}{2}} \right )^\frac{1}{2} $/, Prove that $$x^2\frac{\partial^2 u}{\partial x^2}+2xy\frac{\partial^2 u}{\partial x\partial y}+y^2\frac{\partial^2 u}{\partial y^2}=\frac{\tan u}{144}\left ( \tan ^2u +13 \right )$$

6(c) Discuss the maxima and minima of $$f\left ( x,y \right )= x^3y^2\left ( 1-x-y \right )$$