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MU First Year Engineering (Semester 1)
Applied Mathematics 1
December 2013
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) $If \ \alpha+i \beta=\tan h \ \left(x+i\frac{\pi}{4}\right),\ prove\ that \ \alpha^2+ \beta^2=1.$
3 M
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 M
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 M
1 (d) $prove\ that \log{\left(1-x+x^2\right)}=-x+\frac{x^2}{2}+\frac{2x^3}{3}\ ---- \$
3 M
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 M
1 (f) Find nth derivative of 2xcos2x sin x.
4 M

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 M
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 M
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 M

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 M
3 (b) Find the stationary values of sin x sin (x+y).
6 M
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 M

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 M
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 M
4 (c) Solve by Crout's Method the system of equaition 2x+3y+z=-1, 5x+y+z=9, 3x+2y+4z=11
8 M

5 (a) If sin4θ cos3θ = a cosθ- bcos3θ + ccos 5θ + dcos 7θ then find a,b,c,d
6 M
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 M
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 M

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 M
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 M
6 (c) Fit a curve y=ax+bx2 for the data;
 x : 1 2 3 4 5 6 y : 2.51 5.82 9.93 14.84 20.55 27.06
8 M

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