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MU First Year Engineering (Semester 1)
Applied Mathematics 1
May 2013
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 cosh x = sec θ, prove that x = log (sec θ + tan θ)
3 M
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 M
1(c) If x = r cosθ, y = r sinθ;
$\dfrac{\partial(x,y)}{\partial(r, \theta)}$
3 M
1(d) Expand log(1 + x + x2 + x3) in powers of x upto x8.
3 M
1(e) Show that every square matrix can be uniquely expressed as sum of a symmetric and Skew-symmetric matrix.
4 M
1(f) If y = cos x.cos 2x.cos 3x then find its nth order derivative
4 M

2(a) Solve the equation x6-i=0.
6 M
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 M
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 M

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 M
3(b) Find the stationary values of
x3 + y3 - 3axy, a > 0
6 M
3(c) Separate into real and imaginary parts
tan-1(e)
8 M

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 M
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 M
4(c) Using Gauss- Seidel iteration method solve,
10x1 + x2 + x3 = 12,
2x1 + 10x2 + x3=13,
2x1 + 2x2 + 10x3 = 14
Upto three iterations.
8 M

5(a) In a series of sines of multiple of θ, expand sin7 θ
6 M
5(b) Evaluate the following:
$\lim_{x\rightarrow 1}\dfrac{x^x-x}{x-1-logx}$
6 M
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 M

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 M
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 M
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 M

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