Applied Mathematics-I-Mumbai-Winter-2016

Time: 3 hours
Maximum marks: 80

Q.1 a) If $latex \tan hx=\frac12,$ find sinh2x, cosh2x. [3M]

b) If $latex z=xyf\left(\frac yx\right),$ prove that $latex x\frac{\partial z}{\partial x}+y\frac{\partial y}{\partial y}=2z$ [3M]

c) If $latex x=u\left(1-v\right),\;y=uv-uvw,z=uvw\;find\;\frac{\partial\left(x,y,z\right)}{\partial\left(u,v,w\right)}$ [3M]

d) Using Maclaurins expansion, Prove that $latex e^xsecx=1+x+\frac{2x^2}{2!}+\frac{4x^3}{3!}+…..$ [3M]

e) Show that every square matrix A can be uniquely expressed as P+iQ, where P and Q are Hermitian Matrices. [4M]

f) Find n^th  derivative of $latex e^x\cos x\;\cos2x$ [4M]

Q.2 a) If $latex x=\cos\theta+\sin\theta,\;y=\cos\phi+1\sin\phi,$ show that $latex \frac{x-y}{x+y}=i\tan\frac{\theta-\phi}2$ [6M]

b) For the following matrix A, find non singular matrices P and Q such that PAQ is in normal form and hence find the rank of A, $latex A=\begin{bmatrix}1&1&2\\1&2&3\\0&-1&-1\end{bmatrix}$ [6M]

c) If $latex u=\cos ec^{-1}\sqrt{\left(\frac{x^{\displaystyle\frac12}+y^{\displaystyle\frac12}}{x^{\displaystyle\frac13}+y^{\displaystyle\frac13}}\right)},$ show that

$latex x^2\frac{\partial^2u}{\partial x^2}+2xy\frac{\partial^2u}{\partial xy}+y^2\frac{\partial^2u}{\partial x^2}=\frac{\tan u}{12}\left(\frac{13}{12}+\frac{\tan^2u}{12}\right)$ [8M]

Q.3 a) For what values of λ, the system of equations $latex 3x-y+4z=3,\;x+2y-3z=-2,\;6x+5y+\lambda z=-3$ has a unique solution. Determine the solution in each case.

b) Find the maxima and minima of the function $latex f\left(x,\;y\right)=x^3+y^3-3x-12y+20$ [6M]

c) show that $latex \tan^{-1}i\left(\frac{x-a}{x+a}\right)=\frac i2\log\left(\frac xa\right)$ [8M]

Q.4 a) Find $latex \frac{\partial z}{\partial x’}\frac{\partial x}{\partial y’}$ using derivatives for $latex xe^y+ye^x+\log x-2-3\log2=0\;at\;P\left(1,\;\log2,\;\log3\right)$ [6M]

b) Find the principal value of $latex \left(1+i\right)^{1-t}$ [6M]

c) Solve the following system of equation by crouts method

$latex x+y+z=3,\;2x-y+3z=16,\;3x+y-z=-3$ [8M]

Q.5 a) Show that $latex \frac{\sin6\theta}{\sin2\theta}=16\cos^4\theta-16\cos^2\theta+3$ [6M]

b) Find a and b such that $latex \lim_{x\rightarrow0}\frac{x\left(1-a\cos x\right)+b\sin x}{x^4}=\frac13$ [6M]

c) If $latex y=\left(1-x\right)^{-a}e^{-ax},$ show that

i. $latex \left(1-x\right)y_1=axy$

ii. $latex \left(1-x\right)y_{n+1}-\left(n+ax\right)y_n-nay_{n-1}=0$

Q.6 a) Show that the rows of the following matrix are linearly dependent and find the relationship between them $latex \begin{bmatrix}1&0&2&1\\3&1&2&1\\4&6&2&-4\\-6&0&-3&-4\end{bmatrix}$ [6M]

b) If $latex \phi\left(\frac z{x^3},\;\frac yx\right)=0$ prove that $latex px+qy=3z$ [6M]

c) Fit a second degree parabola to be the following data