Applied Mathematics 1 - Dec 2015

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 log tan x=y then prove that sinh(n+1)y+sinh(n-1)y=2 sinh ny⋅cosec2x(3 marks) 1 (b) If z=log (tan x+tan y) then prove that $$ \sin 2x \dfrac {\partial z}{\partial x} + \sin 2y \dfrac {\partial z}{\partial y} = 2 $$(3 marks) 1 (c) If x=r sin θ cos φ, y=r sin θ φ, z=r cos θ, then find $$ \dfrac {\partial (r, \theta , \varphi)}{\partial (x,y,z) } $$(3 marks) 1 (d) Prove that $$ \log \sec x=\dfrac {x^2}{2}+ \dfrac {x^4}{12}+ \dfrac {x^6}{45}+ \cdots \ \cdots $$(3 marks) 1 (e) Find the values of a,b,c and A^-1 when $$ A=\dfrac {1}{9} \begin{bmatrix} -8 &4 &a \\ 1 &4 &b \\4 &7 &c \end{bmatrix} $$ is or orthogonal.(4 marks) 1 (f) If y=sinθ+cos θ then prove that $$y_n = r^n \sqrt{1+(-1)^n \sin 2 \theta} $$ where θ rx(4 marks) 2 (a) If z=-1+i√3 then prove that $$ \left ( \dfrac {z}{2} \right )^n + \left (\dfrac {2}{z} \right )^n = \left\{\begin{matrix} 2 ,& \text{if n is multiple of 3} \ \ \ \ \ \\-1 , & \text {if n is not multiple of 3} \end{matrix}\right.$$(6 marks) 2 (b) $$ \text {if} A=\begin{bmatrix} 1 &2 &-2 \\-1 &3 &0 \\0 &-2 &1 \end{bmatrix} $$ then find two non-singular matrices P&Q such that PAQ is in normal form also find ρ(A) and A^-1.(6 marks) 2 (c) State and prove Euler's theorem for functions of two independent variable hence prove that $$ \left ( x \dfrac {\partial u}{\partial x}+ y \dfrac {\partial u} {\partial y} \right ) \left( x \dfrac {\partial v} {\partial x} + y\dfrac {\partial v} {\partial y} \right ) = 0 $$ if x=e^u \tan v, y=e^u, sec v.(8 marks) 3 (a) Determine the values of a and b such that system $$ \left\{\begin{matrix} 3x - 2y+z=b \\ 5x-8y+9z=3 \\ 2x+y+az=-1 \end{matrix}\right. $$ has i) no solution, ii) a unique solution, iii) infinite number of solutions(6 marks) 3 (b) Discuss the maximum and minimum of f(x,y)=x³+3xy²-15(x²+y²)+72x(6 marks) 3 (c) show that $$ \tan^{-1} \left ( \dfrac {x+iy}{x-iy} \right ) = \dfrac {\pi}{4} + \dfrac {i}{2}\log \left ( \dfrac {x+y}{x-y} \right ) $$(8 marks) 4 (a) If u=xyz, v=x²+y²+x², w=x+y+z then prove that $$ \dfrac {\partial x}{\partial u} = \dfrac {1}{(x-y)(x-z)} $$(6 marks) 4 (b) $$ \text {if} \sqrt {i}^{\sqrt{i}^{\sqrt{i}\cdots \cdots \infty}} = \alpha + i\beta $$ then prove that $$ i) \ \alpha^2 + \beta^2 = e\dfrac {\pi \beta}{2} \\ ii) \ \tan^{-1}\left ( \dfrac {\beta}{\alpha} \right ) = \dfrac {\pi \alpha} {4} $$(6 marks) 4 (c) Apply Crout's method to solve $$ \left\{\begin{matrix} x-y + 2z=2 \ \ \\ 3x+2y-3z=2 \\ 4x -4y+2z=2 \end{matrix}\right. $$(8 marks) 5 (a) If cos⁶θ+sin⁶θ = α cos 4θ+β then prove that α+β=1.(6 marks) 5 (b) Find the values of a,b & c such that $$ \lim_{x\to 0} \dfrac {a e^x - be^{-x}+cx}{x-\sin x}=4 $$(6 marks) 5 (c) $$ \text {if } x =\cos \Big [ \log \big (y^{1/m} \big) \Big ] $$ then prove that
(1-x²)y_n+2-(2n+1)xy_n+1-(m²+n²)y_n = 0(8 marks) 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 $$ X_1 = \begin{bmatrix}1\\-1 \\1\end{bmatrix}, X_2 = \begin{bmatrix}2\\1 \\1\end{bmatrix}, X_3 = \begin{bmatrix}3\\0 \\2\end{bmatrix} $$(6 marks) 6 (b) If z=f(u,v), u=x²-y², v=2xy then prove that $$ \dfrac {\partial ^2 z}{\partial x^2} + \dfrac {\partial ^2 z} {\partial y^2} = 4 \sqrt{u^2 + v^2} \left ( \dfrac {\partial ^2 z} {\partial u^2} + \dfrac {\partial ^2 z}{\partial v^2} \right ) $$(6 marks) 6 (c) Fit a straight line passing through points (0,1), (1,2), (2,3), (3,4,5), (4,6), (5,7,5).(8 marks)