MU Applied Mathematics 1 - December 2014 Exam Question Paper

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 n^th 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 (e^io).

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)y_n+1 - (2n+1)xy_n-n²y_n-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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