Applied Mathematics-I-Mumbai-Winter-2015

Time: 3 hours
Maximum marks: 80

Q.1 a) If log tanx = y then prove that $latex \sin h\left(n+1\right)y+\sin h\left(n-1\right)y=2\sin h\;ny\cdot\cos ec2x$ [3M]

b) If $latex z=\log\left(\tan x+\tan y\right)$ then prove that $latex \sin\;2x\frac{\partial z}{\partial x}+\sin2y\frac{\partial z}{\partial y}=2$ [3M]

c) If $latex x=r\;\sin\theta\;\cos\phi,\;y=r\;\sin\theta\;\sin\phi,\;z=r\;\cos\theta$ then find $latex \frac{\partial\left(r,\;\theta,\;\phi\right)}{\partial\left(x,\;y,\;z\right)}$ [3M]

d) Prove that $latex \log\;secx=\frac{x^2}2+\frac{x^4}{12}+\frac{x^6}{45}+……….$ [3M]

e) Find the values of a, b, c and $latex A^{-1}\;When\;A=\frac19\begin{bmatrix}-8&4&a\\1&4&b\\4&7&c\end{bmatrix}$ is orthogonal [4M][4M]

f) If $latex y=\sin\theta+\cos\theta$ then prove that $latex y_n=r^n\sqrt{1+\left(-1\right)^n\sin2\theta}$ where $latex \theta=rx$ [4M]

Q.2 a) If $latex z=-1+i\sqrt3$ then prove that $latex \left(\frac z2\right)^n+\left(\frac2z\right)^n=\left.=\right|_{-1\;if\;n\;is\;not\;multiple\;of\;3}^{2\;if\;n\;is\;multiple\;of\;3}$ [6M]

b) If $latex A=\begin{bmatrix}1&2&-2\\-1&3&0\\0&-2&1\end{bmatrix}$ then find two non singular matrices P and Q such that PAQ is in normal form also find $latex \rho\left(A\right)\;and\;A^{-1}$ [6M]

c) State and prove thatt Euler’s theorem for functions of two independent variable hence prove that

$latex \left(x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}\right)\left(x\frac{\partial v}{\partial x}+y\frac{\partial v}{\partial y}\right)=0\;$ if x = e^u tan v, y = e^u , secv [8M]

Q.3 a) Determine the value of a and b such that system $latex \left\{\begin{array}{l}\begin{array}{l}3x-2y+z=b\\5x-8y+9z=3\end{array}\\\;\;2x+y+az=-1\end{array}\right.$

has i) no solution, ii) a unique solution, iii) infinite number of solutions [6M]

b) Discuss the maximum and minimum of $latex \left(x,\;y\right)=x^3+3xy^2-15\left(x^2+y^2\right)+72x$ [6M]

c) show that $latex \tan^{-1}\left(\frac{x-ty}{x-ty}\right)=\frac\pi4+\frac t2\log\left(\frac{x+y}{x-y}\right)$ [8M]

Q.4 a) If $latex u=xyz,\;v=x^2+y^2+z^2,\;w=x+y+z$ then prove that $latex \frac{\partial x}{\partial u}=\frac1{\left(x-y\right){\displaystyle\left(x-z\right)}}$ [6M]

b) If $latex \sqrt{-i}^{\sqrt i^{\sqrt i………\infty}}=\alpha+i\beta$ then prove that i) $latex \alpha^2+\beta^2=e^{-\frac{\pi\beta}2}$ ii) $latex \left(\frac\beta\alpha\right)=\frac{\pi\alpha}4$ [6M]

c) Apply Crout’s method to solve $latex \left\{\begin{array}{l}\begin{array}{l}x-y+2z=2\\3x+2y-3z=2\end{array}\\\;\;4x-4y+2z=2\end{array}\right.$ [8M]

Q.5 a) If $latex \cos^6\theta+\sin^6\theta=\alpha\cos4\theta+\beta$ then prove that $latex \alpha+\beta=1$ [6M]

b) Find the values of a, b and such that $latex \lim_{x\rightarrow0}\frac{ae^x-be^{-x}+cx}{x-\sin x}=4$ [6M]

c) If $latex x=\cos\left[\log\left(y^\frac1m\right)\right]$ then prove that $latex \left(1-x^2\right)y_{n+2}-\left(2n+1\right)xy_{n+1}-\left(m^2+n^2\right)y_n=0$ [8M]

Q.6 a) Define linear dependence and independence of vectors, Examine for linear dependence of following set of vectors and find the relation between them if dependent

$latex X_1=\begin{bmatrix}1\\-1\\1\end{bmatrix},\;X_2=\begin{bmatrix}2\\1\\1\end{bmatrix},\;X_3=\begin{bmatrix}3\\0\\2\end{bmatrix}$ [6M]

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

c) Fit a straight line passing through points (0, 1), (1, 2), (2, 3), (3, 4, 5), (4, 6), (5, 7, 5) [8M]