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Applied Mathematics 1 : Question Paper May 2013 - First Year Engineering (Semester 1) | Mumbai University (MU)

## Applied Mathematics 1 - May 2013

### 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 cosh x = sec θ, prove that x = log (sec θ + tan θ)(3 marks) 1(b) If u = log (x2+y2), prove that
$$\dfrac{\partial^2 u}{\partial x \ \partial y} = \dfrac{\partial^2 u}{\partial y \ \partial x}$$
(3 marks)
1(c) If x = r cosθ, y = r sinθ;
$$\dfrac{\partial(x,y)}{\partial(r, \theta)}$$
(3 marks)
1(d) Expand log(1 + x + x2 + x3) in powers of x upto x8.(3 marks) 1(e) Show that every square matrix can be uniquely expressed as sum of a symmetric and Skew-symmetric matrix.(4 marks) 1(f) If y = cos x.cos 2x.cos 3x then find its nth order derivative
(4 marks)
2(a) Solve the equation x6-i=0.(6 marks) 2(b) Reduce matrix A to normal form and find its rank where
$$A={ \left[ \begin{array}{ccc} 1 & 2 & 3 &2 \\ 2 & 3 & 5 & 1 \\ 1 & 3 & 4 &5 \end{array} \right]}$$
(6 marks)
2(c) State and prove Euler's theorem for a homogeneous function in two variable. And hence find
$$x\dfrac{\partial u}{\partial x} + y\dfrac{\partial u}{\partial y} \ \ \ where \ \ u = \dfrac{\sqrt{x}+\sqrt{y}}{x+y}$$
(8 marks)
3(a) Determine the values of λ so that the equations
x+y+z=1,
x+2y+4z= λ ,
x+4y+10z=λ2
have a solution and solve them completely in each case.
(6 marks)
3(b) Find the stationary values of
x3 + y3 - 3axy, a > 0
(6 marks)
3(c) Separate into real and imaginary parts
tan-1(e)
(8 marks)
4(a) If x = u cos v and y = u sin v
$$\dfrac{\partial(x,y)}{\partial(u,v)}.\dfrac{\partial(u,v)}{\partial(x,y)} =1$$
(6 marks)
4(b) If tan[log(x + iy)] = a+ib,
$$prove \ that \ tan[log(x^2+y^2)]=\dfrac{2a}{1-a^2-b^2}$$
where a2 + b2 ≠ 1.
(6 marks)
4(c) Using Gauss- Seidel iteration method solve,
10x1 + x2 + x3 = 12,
2x1 + 10x2 + x3=13,
2x1 + 2x2 + 10x3 = 14
Upto three iterations.
(8 marks)
5(a) In a series of sines of multiple of θ, expand sin7 θ(6 marks) 5(b) Evaluate the following:
$$\lim_{x\rightarrow 1}\dfrac{x^x-x}{x-1-logx}$$
(6 marks)
5(c) Prove the following if y1/m + y-1/m = 2x;
(x2 -1) y(n+2) + (2n+1)xy(n+1) + (n2-m2)yn = 0
(8 marks)
6(a) Examine the following vectors for linear dependence/independence
X1 = (a,b,c), X2 = (b,c,a), X3 = (c,a,b)
where a+b+c ≠ to zero.
(6 marks)
6(b) If z = f(x,y) , x=(eu + e-v), y=(e-u - ev)
$$\dfrac{\partial z}{\partial u}-\dfrac{\partial z}{\partial v} = x\dfrac{\partial z}{\partial x}-y\dfrac{\partial z}{\partial y}$$
(6 marks)
6(c) Fit a straight line to following data and also estimate the production in 1957.

 Year 1951 1961 1971 1981 1991 Production in Thousand Tones 10 12 8 10 13
(8 marks)

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