Question Paper: Applied Mathematics 1 : Question Paper Dec 2014 - First Year Engineering (Semester 1) | Mumbai University (MU)
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## 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 nth 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 (eio).(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)yn+1 - (2n+1)xyn-n2yn-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

 x 1 2 3 4 5 6 y 49 54 60 73 80 86
(8 marks)