Applied Mathematics 1 - Dec 2013

First Year Engineering (Semester 1)

TOTAL MARKS: 80
TOTAL TIME: 3 HOURS (1) Question 1 is compulsory.
(2) Attempt any three from the remaining questions.
(3) Assume data if required.
(4) Figures to the right indicate full marks. 1 (a) $$ If \ \alpha+i \beta=\tan h \ \left(x+i\frac{\pi}{4}\right),\ prove\ that \ \alpha^2+ \beta^2=1.$$(3 marks) 1 (b) $$ If \ u=x^2y+e^{xy^2}\ show\ that\ \frac{{\partial}^2u}{\partial x\partial y}=\frac{{\partial}^2u}{\partial y\partial x}$$(3 marks) 1 (c) $$If\ u=1-x,\ v=x\left(1-y\right),\ w=xy\left(1-z\right)\\show\ that\frac{\partial{}\left(u,\ v,\ w\right)}{\partial{}\left(x,\ y,\ z\right)}=x^2y.$$(3 marks) 1 (d) $$ prove\ that \log{\left(1-x+x^2\right)}=-x+\frac{x^2}{2}+\frac{2x^3}{3}\ ---- \ $$(3 marks) 1 (e) Express the relation in α , β ,γ , δ for which $$A=\begin{array}{cc}\alpha{}+i\gamma{} & -\beta{}+i\delta{} \\\beta{}+i\delta{} & \alpha{}-i\gamma{}\end{array}$$ is unitary.(4 marks) 1 (f) Find n^th derivative of 2^xcos²x sin x.(4 marks) 2 (a) $$ Z^3={\left(z+1\right)}^3,\ then\ show\ that\ z=\frac{-1}{2}+\frac{i}{2}\cot{\frac{\theta{}}{2}\ where\ \theta{}=\ 20\frac{\pi{}}{3}.} $$(6 marks) 2 (b) Find the non-singular matrices P and Q such that PAQ is in Normal Form. Also find rank of A. $$A=\ \left[\begin{array}{ccc}4 & 3 & 1 & 6 \\2 & 4 & 2 & 2 \\12 & 14 & 5 & 16\end{array}\right]$$(6 marks) 2 (c) State and Prove Euler's theorem for homogeneous function in two variables anf hence find the value of $$x^2\frac{{\partial{}}^2u}{\partial{}x^2}+2xy\frac{{\partial{}}^2u}{\partial{}x\partial{}y}+y^2\frac{{\partial{}}^2u}{\partial{}y^2}+x\frac{\partial{}u}{\partial{}x}+y\frac{\partial{}u}{\partial{}y}\\ for\ u=e^{x+y}+\log{\left(x^3+y^3-x^2y-xy^2\right)}$$(8 marks) 3 (a) For what values of λ the system of equations have non-trivial solution? Obtain the solution for real values of λ where $$ 3x+y-\lambda{}x=0,\ 4x-2y-3z=0,\ 2\lambda{}x+4y-\lambda{}z=0\} $$(6 marks) 3 (b) Find the stationary values of sin x sin (x+y).(6 marks) 3 (c) If cos(x+iy) cos(u+iv)=1, where x,y,u, v are real then show that tanh² cosh² v=sin²u.(8 marks) 4 (a) $$ if\ ux+vy=a,\frac{u}{x}+\frac{v}{y}=1,\ show\ that \ \frac{u}{x} \left(\frac{\partial x}{\partial u}\right)_v+\frac{v}{y} \left(\frac{\partial y}{\partial v}\right)_u=0 $$(6 marks) 4 (b) $$ If\ (1+i \left(\tan\alpha \right)^{(1+i\ tan\beta)} $$ is real then one of the principle values is $$ \left( sec\alpha \right)^{sec^2 \beta} $$(6 marks) 4 (c) Solve by Crout's Method the system of equaition 2x+3y+z=-1, 5x+y+z=9, 3x+2y+4z=11 (8 marks) 5 (a) If sin⁴θ cos³θ = a cosθ- bcos³θ + ccos 5θ + dcos 7θ then find a,b,c,d(6 marks) 5 (b) Use Taylor theorem and arrange the equaition in power of x.7+(x+2)+3(x+2)³+(x+2)⁴-(x+2)⁵(6 marks) 5 (c) $$ If\ y=\cos(m\sin^{-1}X)\ \\prove\ that\ \left(1-x^2 \right)y_{n+2}-\left(2n+1\right)xy_{n+1}+ \left(m^2-n^2\right)_n^y=0$$(8 marks) 6 (a) Solve correctly upto three iterations the following equaition by Gauss-Seidel method. 10x-5y-2z=3, 4x-10y+3z=-3, x+6y+10z=-3.(6 marks) 6 (b) $$If\ u=\sin{\left(x^2+y^2\right)}and\ a^2x^2+b^2y^2=c^2\ find\frac{du}{dx}.$$(6 marks) 6 (c) Fit a curve y=ax+bx² for the data;