## 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 x^{6}+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

x^{3}+3xy^{2}-3x^{2}-3y^{2}+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 cos^{7}θ 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-x^{2})y_{n+1} - (2n+1)xy_{n}-n^{2}y_{n-1}=0(8 marks)
**6(a)** Examine whether the vectors

x_{1}= [3,1,1] x_{2}=[2,0,-1]

x_{3}=[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

x | 1 | 2 | 3 | 4 | 5 | 6 |

y | 49 | 54 | 60 | 73 | 80 | 86 |