Question Paper: Applied Mathematics 1 : Question Paper Dec 2012 - First Year Engineering (Semester 1) | Mumbai University (MU)
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## Applied Mathematics 1 - Dec 2012

### 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) Prove the following:
$$\dfrac{1}{1- \dfrac{1}{1-\dfrac{1}{1-\cos h^2x}}} = \cos h^2 x$$
(3 marks)
1(b) If u = log (tanx + tany)
$$2x\dfrac{\partial u}{\partial x}+sin2y\dfrac{\partial u}{\partial y} = 2$$
(3 marks)
1(c) If the following expression is true,
$$u=\dfrac{x+y}{1-xy} \ \ , \ v = tan^{-1}x + tna^{-1}y \\ Find \ \ \dfrac{\partial (u,v)}{\partial (x,y)}$$
(3 marks)
1(d) Expand log (1+sinx) = (x - x2/2 + x3/6 +...)(3 marks) 1(e) Show that every square matrix can be uniquely expressed as P+iQ where P and Q are Hermitian Matrices.(4 marks) 1(f) Find nth order derivative of
$$y= \dfrac{x^2+4}{(2x+3)(x-1)^2}$$
(4 marks)
2(a) Show that roots of the equation (x+1)6 + (x-1)6 = 0 are given by
$$-i\cot\Big[ \dfrac{(2k+1)\pi}{12} \Big] \ \ \ , \ k=0,1,2,3,4,5$$
(6 marks)
2(b) Reduce the following matrix into normal form and find its rank
$$\left[ {\begin{array}{cc} 2 & -1 & 1 & 1\\ 1 & 0 & 1 & 2\\ 3 & 3 & 3 & 1\\ 0 & -4 & -1 & 2\\ \end{array} } \right]$$
(6 marks)
2(c) State and prove Euler's theorem for a homogeneous function in two variables. And hence
$$Find \ \ x \dfrac{\partial u}{\partial x}+ y\dfrac{\partial u}{\partial y} \ \ where \ u=\dfrac{\sqrt{xy}}{\sqrt{x}+\sqrt{y}}$$
(8 marks)
3(a) Test for consistency and solve if consistent -
x1-2x2+x3-x4=2;
x1+2x2+2x4=1;
4x2-x3+3x4=-1.
(6 marks)
3(b) Find all stationary value of x2 + 3xy - 15x2 - 15y2 + 72x.(6 marks) 3(c) If tan[(π/4)+iv] = re show that
(i) r=1
(ii) tanθ = sinh 2v
(iii) tanhv = tan(θ/2)
(8 marks)
4(a) If x = u+e(-v)sin u, and y = v+e(-u)cos u,
$$Find \ \ \dfrac{\partial u}{\partial y}, \dfrac{\partial v}{\partial x} \ \ using \ \ Jacobian$$
(6 marks)
4(b) Considering only the principal value,
if (1 + i tanθ)(1+i tanθ) is real, prove that its value is (sec?)(sec2θ).
(6 marks)
4(c) Solve the system of linear equation by Crout's method
x - y + 2z = 2;
3x + 2y - 3z = 2;
4x - 4y + 2z = 2
(8 marks)
5(a) Expand cos7θ in a series of cosines of multiple of θ .(6 marks) 5(b) Evaluate the following:
$$\displaystyle\lim_{x \to 0}\Big[ \dfrac{1}{x^2} - cot^2x \Big]$$
(6 marks)
5(c) If y = (sin-1x)2, obtain yn(0).(8 marks) 6(a) Show that the vectors are linearly dependent and find the relation between them
X1=[1,2,-1,0],
X2=[1,3,1,2],
X3=[4,2,1,0],
X4=[6,1,0,1].
(6 marks)
6(b) If the expression is
$$\dfrac{x^2}{(1+u)}+\dfrac{y^2}{(2+u)} +\dfrac{z^2}{(3=1+u)}$$
prove that
$$\Big[\Big(\dfrac{\partial u}{\partial x}\Big)^2+ \Big(\dfrac{\partial u}{\partial y}\Big)^2 +\Big(\dfrac{\partial u}{\partial z}\Big)^2 \Big] = 2\Big[x\dfrac{\partial u}{\partial x}+ y\dfrac{\partial u}{\partial y} +z\dfrac{\partial u}{\partial z} \Big]$$
(6 marks)
6(c) Fit a second degree parabolic curve to following data:-

 X 1 2 3 4 5 6 7 8 9 Y 2 6 7 8 10 11 11 10 9

(8 marks)