1(a) Prove that $tanh^{-1}(sin \theta) = cosh^{-1} (sec \theta)$

1(b) Prove that the matrix $\frac{1}{\sqrt{3}}[\matrix{1 & 1+i \\ 1-i & -1}]$ is unitary

1(c) If $x = uv$ & $y = \frac{u}{v}$ prove that $JJ^/ = 1$

1(d) If $Z = \tan^{-1}(\frac{x}{y})$ where $x = 2t, y = 1-t^2$, prove that $\frac{dZ}{dt} = \frac{2}{1+t^2}$

1(e) Find the $n^{th}$ derivative of $(\cos5x.\cos3x.\cos x)$

1(f) Evaluate $\lim_{x \rightarrow 0}(x)^{\frac{1}{x+1}}$

2(a) Find all values of $(1+i)^{\frac{1}{3}}$ & show that their continued product is (1+i)

2(b) Find the non-singular matrices P&Q such that PAQ is normal from where $A = \matrix {2 & -2 & 3 \\ 3 & -1 & 2 \\ 1 & 2 & -1 }$

2(c) Find the maximum and minimum values of $f(x,y) = x^3 + 3xy^2 - 15x^2 - 15 y^2 + 72x$

3(a) If $u = f(\frac{y-x}{xy},\frac{z-x}{xz}) $, show that

x^2\frac{\delta u}{\delta z} + y^2\frac{\delta u}{\delta y} + z^2\frac{\delta u}{\delta z} = 0

3(b) using Encoding matrix $[\matrix{1&1\\0&1}]$ encode & decode the message "MUMBAI"

3(c) Prove that $\log[\tan(\frac{\pi}{4}+\frac{ix}{2})] = i\tan^{-1}(sinh x)$

4(a) Obtain $\tan5\theta$ in terms of $\tan\theta$ & show that $1-10\tan^2\frac{\pi}{10} + 5\tan^4 \frac{\pi}{10} = 10$

4(b) If $y = e^{\tan^{-1}x}$ . Prove that $(1+x^2)y_{n+2} + [2(n+1)x - 1] y_{n+1} + n(n+1)y_n = 0.$

4(c)

• Express $(2x^3+3x^2-8x+7)$ in terms of (x-2) using Taylors theorem
• prove that $\tan^{-1} x = x- \frac{x^3}{3}+\frac{x^5}{5}-----------$

5(a) If $z = x^2 \tan^{-1}\frac{y}{x} - y^2\tan^{-1}\frac{x}{y}$ , prove that $\frac{\delta^2 z}{\delta x \delta y} = \frac{x^2 - y^2}{x^2 + y^2}$

5(b) Investigate the values of $\lambda$ & $\mu$ the equation

$2x + 3y + 5z = 9$

$7x + 3y - 2z = 8$

$2x + 3y + $\lambda$z = $\mu$$

1) have no solution

2) a unique solution

3) an infinite no of solution

5(c) Using Newton Raphson method, find approximate root of $x^3 - 2x - 5 = 0$ (correct to three places of decimals).

6(a) Find tanhx if 5sinhx - coshx = 5

6(b) If $u = \sin^{-1}(\frac{x+y}{\sqrt x + \sqrt y})$ ,

• $xu_x + yu_y = \frac{1}{2} \tan u$
• prove that $x^2u_{xx} + 2xy u_{xy} + y^2 u_{yy} = \frac{-\sin u \cos 2u}{4 \cos^3u}$

6(c) Solve the following equations by Gauss Seidal method:

$20x + y + 2z = 17$

$3x + 20y - z = -18$

$2x + 3y + 20z = 25$