Applied Mathematics 1 - Dec 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) If $$ \ \tanh x=\dfrac {2}{3} $$. Find the value of x and then cosh 2x.(3 marks) 1(b) $$ if \ u = \tan^{-1} \left ( \dfrac{y}{x} \right ), $$ Find these value of $$ \dfrac {\partial ^2 u}{\partial x^2} + \dfrac {\partial ^2 u}{\partial y^2} $$(3 marks) 1(c) If x=r cos θ, y=r sin θ Find $$ \dfrac {\partial (x,y)}{\partial (r, \theta)} $$(3 marks) 1(d) Prove that $$\log \sec x = \dfrac {1}{2}x^2 + \dfrac {1}{1z}x^2 + \dfrac {1}{45}x^6 \cdots \ \cdots $$(3 marks) 1(e) Show that every square matrix can be uniquely expressed as the sum of Hermitian matrix and a skew Hermitian matrix.(4 marks) 1(f) Find the n^th derivative of y =sin x sin 2x sin 3x.(4 marks) 2(a) Solve the equation x⁶+1=0(6 marks) 2(b) Reduce the matrix to normal form and find its rank where, $$ A= \begin{bmatrix} 1 &-1 &3 &6 \\1 &3 &-3 &-4 \\5 &3 &3 &11 \end{bmatrix} $$(6 marks) 2(c) State and prove Eulers theorem for a homogeneous function in two variables: Hence verify the Eulers theorem for $$ u = \dfrac {\sqrt{xy}}{\sqrt{x+} \sqrt{y}} $$(8 marks) 3(a) Test the consistancy of the following equations and solve them if they are consistent.
2x-y+z=8, 3x-y+z=6
4x-y+2z=7, -x+y-z=4(6 marks) 3(b) Find the stationary values
x³+3xy²-3x²-3y²+4(6 marks) 3(c) Separatic into real and imaginary parts of sin^-1 (e^io).(8 marks) 4(a) $$ if \ x=uv, \ y=\dfrac {u}{v} $$ prove that J.J=1(6 marks) 4(b) Show that for real values of a and b, $$ e^{2 \ ai \cot ^{-1}b} \left [ \dfrac {bi-1}{bi+1} \right ]^{-2} = 1 $$(6 marks) 4(c) Solve the following equations by Gauss-seidal method
27x + 6y-z=85
6x+15y+2z=72
x+y+54z=110(8 marks) 5(a) Expond cos⁷θ in a series of cosine of multiple of θ(6 marks) 5(b) $$ If \lim_{x \to 0} \dfrac {a\sin hx + b \sin x}{x^3} = \dfrac {5}{3}, $$ find a and b(6 marks) 5(c) $$ If \ y = \dfrac {\sin^{-1}x}{\sqrt{1-x^2}} $$ then prove that (1-x²)y_n+1 - (2n+1)xy_n-n²y_n-1=0(8 marks) 6(a) Examine whether the vectors
x₁= [3,1,1] x₂=[2,0,-1]
x₃=[4,2,1] are linearly independent.(6 marks) 6(b) If u=f(x-y, y-z, z-x) then show that $$ \dfrac {\partial u}{\partial x}+ \dfrac {\partial u}{\partial y} + \dfrac {\partial u}{\partial z} = 0 $$(6 marks) 6(c) Fit a straight line for the following data