Applied Mathematics 1 - May 2014

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) $$ prove\ that\ sech^{-1}\ \left(\sin\theta\right)=\log \left(\cot \frac{\theta}{2}\right) $$(3 marks) 1 (b) If x=cosθ-rsinθ, y=sinθ+rcosθ prove that dr/dx = x/r(3 marks) 1 (c) If x=e^v secu, y=e^v tanu find $$\ j\left(\frac{u,v}{x,y}\right)$$(3 marks) 1 (d) If y=sinpx+cospx prove that $$y_n={P^n\left[1+{\left(-1\right)}^n\sin{\ }\right(2px)]}^{\frac{1}{2}}$$(3 marks) 1 (e) Find the series expansion of log(1+x) in powers of x, Hence prove that $$logx=\left(x-1\right)-\frac{1}{2}{\left(x-1\right)}^2+\frac{1}{3}{\left(x-1\right)}^3\ .....$$(4 marks) 1 (f) If 'A' is skew-symmetric matrix of odd order then prove that it is singular.(4 marks) 2 (a) Show that the roots of the equation $$ \left(x+1\right)^6+\left(x-1\right)^6=0 $$ are given by $$ -icot \left( \frac{2n+1}{12}\right)\pi,n=0,1,2,3,4,5.\ $$(6 marks) 2 (b) Find two non-singular matrices P & Q such that PAQ is in normal form where $$A=\left[\begin{array}{ccc}1 & 2 & 3 & -4 \\2 & 1 & 4 & -5 \\-1 & -5 & -5 & 7\end{array}\right]$$(6 marks) 2 (c) $$ If\ x+y=2e^{\theta}\cos{\varnothing{}},\ x-y=2ie^{\theta{}}\sin{\varnothing{}}\ $$ & u is a function of x & y the prove that
$$ \frac{{\partial{}}^2u}{\partial{}{\theta{}}^2}+\frac{{\partial{}}^2u}{\partial{}{\varphi{}}^2}=4xy\frac{{\partial{}}^2u}{\partial{}x\partial{}y} $$(8 marks) 3 (a) Find the value of λ for which the equations $$x_1+2x_2+x_3=3,\ x_1+x_2+x_3=\lambda{},\ 3x_1+x_2+3x_3={\lambda{}}^2$$ has a solution & solve them completely for each value of λ(6 marks) 3 (b) Divide 24 into three parts such that the product of the first, square of the second & cube of the third is maximum.(6 marks) 3 (c) (i) $$ If\ cosec \left( \frac{\pi}{4}+ix\right)=u+iv\ \ prove\ that \ \left(u^2+v^2\right) ^2=2 \left(u^2-v^2 \right) $$(4 marks) 3 (c) (ii) $$Prove\ that\tan{\left(i\log{\left(\frac{a-ib}{a+ib}\right)}\right)}=\frac{2ab}{a^2-b^2}$$(4 marks) 4 (a) $$Show\ that\ \frac{\partial{}\left(u,v\right)}{\partial{}\left(x,y\right)}=6\ r^3\sin{2\ }\theta{}\\ given\ that\ u=x^2-y^2,\ v=2x^2-y^2\ \&\ x=r\cos{\theta{}},\ y=r\sin{\theta{}}.$$(6 marks) 4 (b) $$ If \ \alpha{}=1+i,\ \beta=1-i\ \ and \cot\theta =x+1\ \\prove\ that \ \left(x+\alpha \right)^n+ \left(x+\beta \right)^n=\left(\alpha +\beta \right)cosn \theta \ cosec^n\theta{}. $$(6 marks) 4 (c) Using Gauss-seidel method, solve the following system of equations upto 3^rd iteration.
5x-y=9
-x+5y-z=4
-y+5z=-6 (8 marks) 5 (a) Using De-Moivr's theorem, prove that $$ \frac{\sin 6\theta}{\sin\theta}=16\cos^4\theta -16\cos^2\theta +3\ $$(6 marks) 5 (b) Expland $$ \frac{x}{e^x-1}\ $$ in powers of x. hence prove that $$ \frac{x}{2}\left[ \frac{e^x+1}{e^x-1}\right]=1+\frac{1}{12}x^2-\frac{1}{720}x^4+\ ..... $$(6 marks) 5 (c) $$ If\ y=\frac{\sin^{-1}x}{\sqrt{1-x^2}}\ \\ prove\ that\\ \left(1-x^2 \right)y_{n+2}- \left(2n+3\right)xy_{n+1}- \left(n-1\right)^2y_n=0\ \ hence\ find\ y_n \left(0\right) $$(8 marks) 6 (a) Examine the linear dependence or independence of vector (1,2,-1,0), (1,3,1,3), (4,2,1,-1) & (6,1,0,-5) (6 marks) 6 (b) $$If\ u=f\left(\frac{x-y}{xy},\frac{z-x}{zx}\right)prove\ that\ \ x^2\frac{\partial{}u}{\partial{}x}+y^2\frac{\partial{}u}{\partial{}y}+z^2\frac{\partial{}u}{\partial{}z}=0$$(6 marks) 6 (c) (i) Fit a straight line to the following data with x-as independent variable.