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MU First Year Engineering (Semester 1)
Applied Mathematics 1
December 2016
Total marks: --
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) If cos α cosh $$\beta =\frac{x}{2},\sin \alpha sinh \beta =\frac{y}{2},$$/ Prove that $$sec\left ( \alpha -i\beta \right )+sec\left ( \alpha +i\beta \right )= \frac{4x}{x^2+y^2}$$/
3 M
1(b) If $$z=\log \left ( e^x+e^y \right )$$/, show that rt-s2=0, where $$r = \frac{\partial^2 z}{\partial x^2}, t = \frac{\partial^2 z}{\partial y^2}, s =\frac{\partial^2 z}{\partial x \partial y}$$/
3 M
1(c) If x = u v, $y = \frac{u+v}{u-v}.$ Find $\frac{\partial \left ( u,v \right )}{\partial \left ( x,y \right )}$.
3 M
1(d) If $$y = 2^{x}\sin ^2x\cos x$$/ find yn
3 M
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.
4 M
1(f) Evaluate $\lim_{x\rightarrow 0}\frac{e^2-\left ( 1+x \right )^2}{x\log \left ( 1+x \right )}$
4 M

2(a) Show that the roots of x5=1 can be written as 1,
α,
α2,
α3,
α4. Hence show that $\left ( 1-\alpha \right )\left ( 1-\alpha ^2 \right )\left ( 1 -\alpha ^3 \right )\left ( 1 -\alpha ^4 \right ) = 5$
6 M
2(b) Reduce the following matrix to its normal from and hence find its rank $A = \begin{bmatrix} 3 & -2& 0& 1\\ 0& 2& 2& 7\\ 1& -2& -3& 2\\ 0& 1& 2& 1 \end{bmatrix}$
6 M
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
8 M

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.
6 M
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 $\frac{du}{dt} = 4e^{2t}$
6 M
3(c)(i) Show that $\sin \left ( e^x -1 \right ) = x+\frac{x^2}{2}-\frac{5x^4}{24}+.........$
4 M
3(c)(ii) Expand 2x3+7x2+x-6 in power of x-2
4 M

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 $x\frac{\partial\phi }{\partial x}+2y\frac{\partial^\phi }{\partial y}+3z\frac{\partial \phi }{\partial z} = u\frac{\partial \phi }{\partial u}+ v\frac{\partial\phi }{\partial v} + \frac{\partial \phi }{\partial w}$
6 M
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) $2\theta =n\pi +\frac{\pi }{2} + \alpha$
6 M
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.
8 M

5(a) If $$y = \left ( x+\sqrt{x^2-1} \right )^m$$/, Prove That $\left ( x^2-1 \right )y_{n+2}+(2n+1)xy_{n+1}+\left ( n^2-m^2 \right ) y _n =0$.
6 M
5(b) Using the encoding matrix $$\begin{bmatrix} 1 & 1\\ 0& 1 \end{bmatrix}$$/, encode and decode the message I* LOVE*MUMBAI*
6 M
5(c)(i) Consulting only principal values separate into real and imaginary parts $i^\log \left ( 1+i \right )$
4 M
5(c)(ii) Show that $i\log \left ( \frac{x-i}{x+i} \right ) = \pi -2\tan ^{-1}x$
4 M

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

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