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 n^th 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^iθ 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 y_n(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:-

(8 marks)