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MU First Year Engineering (Semester 1)
Applied Mathematics 1
December 2014
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 $\ \tanh x=\dfrac {2}{3}$. Find the value of x and then cosh 2x.
3 M
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 M
1(c) If x=r cos θ, y=r sin θ Find $\dfrac {\partial (x,y)}{\partial (r, \theta)}$
3 M
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 M
1(e) Show that every square matrix can be uniquely expressed as the sum of Hermitian matrix and a skew Hermitian matrix.
4 M
1(f) Find the nth derivative of y =sin x sin 2x sin 3x.
4 M

2(a) Solve the equation x6+1=0
6 M
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 M
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 M

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 M
3(b) Find the stationary values
x3+3xy2-3x2-3y2+4
6 M
3(c) Separatic into real and imaginary parts of sin-1 (eio).
8 M

4(a) $if \ x=uv, \ y=\dfrac {u}{v}$ prove that J.J=1
6 M
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 M
4(c) Solve the following equations by Gauss-seidal method
27x + 6y-z=85
6x+15y+2z=72
x+y+54z=110
8 M

5(a) Expond cos7θ in a series of cosine of multiple of θ
6 M
5(b) $If \lim_{x \to 0} \dfrac {a\sin hx + b \sin x}{x^3} = \dfrac {5}{3},$ find a and b
6 M
5(c) $If \ y = \dfrac {\sin^{-1}x}{\sqrt{1-x^2}}$ then prove that (1-x2)yn+1 - (2n+1)xyn-n2yn-1=0
8 M

6(a) Examine whether the vectors
x1= [3,1,1] x2=[2,0,-1]
x3=[4,2,1] are linearly independent.
6 M
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 M
6(c) Fit a straight line for the following data
 x 1 2 3 4 5 6 y 49 54 60 73 80 86
8 M

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