1(a) Separate into a real part and imaginary part of $\cos^{-1} (\frac{3i}{4})$

1(b) Show that the matrix A is unitary where A = $\matrix{\alpha+i\gamma & -\beta+i\delta\\ \beta+i\delta & \alpha-i\gamma}$ is unitary if $\alpha^2 + \beta^2 + \gamma^2 + \delta^2 = 1$

1(c) If $z = \tan (y+ax)+(y-ax)^{\frac{3}{2}}$ then show that $\frac{\delta^2z}{\delta x^2} = a^2 \frac{\delta^2z}{\delta y^2}$

1(d) If $x = uv , y=\frac{u}{v}$ Prove that $JJ^/ = 1$

1(e) Find the $n^{th}$ derivative of $\frac{x^3}{(x+1)(x-2)}$

1(e) Using the matrix $A = \matrix{-1 & 2\\-1 &1}$decode the message matrix $ C = \matrix{4 & 11 & 12 & -2 \\ -4 & 4 & 9 & -2 }$

2(a) If $\sin^4\theta \cos^3\theta = a\cos\theta + b\cos3\theta + c\cos 7\theta$ then find a, b, c, d.

2(b) Using Newtons Raphson method Solve 3x - $\cos x$ - 1 = 0 Correct to 3 decimal places.

2(c) Find the stationary points of the function$ x^3 + 3xy^2 - 3x^2 - 3y^2 + 4$ & also find maximum amd minimum values of the function.

3(a) Show that $ x \hspace{0.1cm}cosec x = 1 + \frac{x^2}{6} + \frac{7}{360}x^4 + ................................$

3(b) Reduce matrix to PAQ normal form and find 2 non-singular matrices P&Q

$\matrix{1&2&-1&2\\ 2&5&-2&3\\1&2&1&2}$

3(c) If $y = cos(m \sin^{-1}x)$ prove that $(1- x^2) y_{n+2} - (2n+1)xy_{n+1} + (m^2 - n^2)y_n = 0$

4(a) State and prove Euler's theorem for three Variables.

4(b) Show that all the roots of $(x+1)^6 + (x-1)^6 = 0$ are given by $-i\cot \frac{(2k+1)\pi}{12}$ where k = 0, 1, 2, 3, 4, 5

4(c) Show that the equations

-2x + y + z = a

x -2y + z = b

x + y -2z = c

have no solutions unless a+b+c = 0 in which case they have infinitely many solutions.

Find these solutions when a=1, b=1, c=-2

5(a) If $z = f(x,y) , x=r\cos\theta$ and $y = r\sin \theta$

$(\frac{\delta z}{\delta x})^2 + (\frac{\delta z}{\delta y})^2 = (\frac{\delta z}{\delta r})^2 + \frac{1}{r^2}(\frac{\delta z}{\delta \theta})^2$

5(b) If $\cos hx = sec\theta$ prove that

• $x = \log({sec\theta + \tan \theta})$
• $ \theta = \frac{\pi}{2} - 2tan^{-1} (e^{-x})$

5(c) Solve by Gauss Jacobi Iteration method

5x - y + z = 10

2x + 4y = 12

x + y + 5z = -1

6(a) prove that $\cos^{-1} [\tan h(\log x)] = \pi - 2(x- \frac{x^3}{3}+\frac{x^5}{5}-----------)$

6(b) If $y = e^{2x} \sin\frac{x}{2} \cos{x}{2} sin3x$ Find $ y_n$

6(c)

• Evaluate $\lim_{x\rightarrow 0} (\cot x)^{\sin x}$
• Prove that $log[\frac{\sin{x+iy}}{\sin{x-iy}}] = 2i \tan^{-1} (\cot x \tan hy)$