Applied Mathematics-I-Mumbai-Summer-2015

Time: 3 hours
Maximum marks: 80

Q.1 (a) If $latex \tan\frac x2=\tan h\frac u2$ then S.T.

$latex u=\log\;\tan\left(\frac\pi4+\frac x2\right)$ [3M]

(b) If $latex u=x^y\;find\;\frac{\partial^3u}{\partial x\;\partial y\;\partial x}$ [3M]

(c) If $latex ux=yz,\;vy=zx,\;wz=xy$ find $latex j\left(\frac{u,\;v,\;w}{x,\;y,\;z}\right)$ [3M]

(d) If $latex y=\left(x-1\right)^n\;then\;P.T.\;y+\frac{y_1}{1!}+\frac{y_2}{2!}+\frac{y_3}{3!}+…..\frac{y_n}{n!}=X^n$ [3M]

(e) P.T. $latex \sin hx=X+\frac{x^3}{3!}+\frac{x^5}{5!}+\frac{x^7}{7!}+$ [4M]

(f) Express the matrix A as sum of Hermition and skew Hermition matrix where

$latex \begin{bmatrix}3i&-1+i&3-2i\\1+i&-i&1+2i\\-3-2i&-1+2i&0\end{bmatrix}$ [4M]

2.  (a) Solve $latex x^7+x^4+i\left(x^3+1\right)=0$ [6M]

(b) Reduce the matrix A to normal form and hence find its rank where

$latex A=\begin{bmatrix}0&1&-3&-1\\1&0&4&3\\3&1&0&2\\1&1&-2&0\end{bmatrix}$ [6M]

(c) State and prove that Euler’s theorem for three variables and hence find $latex x\frac{\partial u}{\partial x}+y\frac{\displaystyle\partial u}{\displaystyle\partial y}+z\frac{\displaystyle\partial u}{\displaystyle\partial x}$ where

$latex u=\frac{x^3\;y^3\;z^3}{x^3+y^3+z^3}$

Q.3 (a) Solve the following system of equations

$latex \begin{array}{l}2x-2y-5z=0\\\\4x-y+z=0\\\\3z-2y+3z=0\\\\x-3y+7z=0\end{array}$ [6M]

(b) Find the maximum and minimum values of $latex x^3+3xy^2-3x^2-3y^2+4$ [6M]

(c) Separate into real and imaginary parts of $latex \tan h^{-1}\left(x+iy\right)$ [8M]

4. (a) If $latex u=2xy,\;y=x^2-y^2$ and $latex x=r\;\cos\theta,\;y=r\;\sin\theta$ then find $latex \frac{\partial\left(u_1\;v\right)}{\partial\left(\partial_1\;\theta\right)}$ [6M]

(b) If $latex i^{i^{i^{.^{.^{.^\infty}}}}}=A+iB,$ prove that $latex \left(\frac{\pi A}2\right)=\frac BA$ and $latex A^2+B^2=e^{-\pi B}$. [6M]

(c) Solve by crouts method the system of equations

$latex \begin{array}{l}3x+2y+7z=-4\\\\2x+3y+z=5\\\\3x+4y+z=7\end{array}$ [8M]

5. (a) By using De Moiverse theorem Express $latex \frac{\sin7\theta}{\sin\theta}$ in powers of sinθ only. [6M]

(b) By using Taylor’s series expand tan^-1 x in positive powers of (x – 1) upto first four non-zero terms. [6M]

(c) If $latex y=\sin\left[\log\left(x^2+2x+1\right)\right]$ prove that $latex \left(x+1\right)^2y_{n+2}+\left(2n+1\right)\left(x+1\right)y_{n+1}+\left(n^2+4\right)y_n=0$ [8M]

Q.6 (a) Determine linear dependance or independance of vectors $latex X_1=\left[1,\;3,\;4,\;2\right],\;X_2=\left[3,\;-5,\;2,\;6\right],\;X_3=\left[2,\;-1,\;3,\;4\right]$ and if dependent find the relation between them. [6M]

(b) If $latex u=x^2-y^2,\;v=2xy\;and\;z=f\left(u,\;v\right)$ prove that $latex \left(\frac{\partial z}{\partial x}\right)^2+\left(\frac{\partial z}{\partial y}\right)^2=4\sqrt{u^2+v^2}\left[\left(\frac{\partial z}{\partial u}\right)^2+\left(\frac{\partial z}{\partial v}\right)^2\right]$ [6M]

(c) (i) Evaluate $latex \lim_{x\rightarrow0}\frac{\sin x.\sin^{-1}x-x^2}{x^6}$ [4M]

(ii) Fit straight line to the following data (x, y) = (-1, -5), (1, 1), (2, 4), (3, 7), (4, 10) Estimate y when x = 7 [4M]