## 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 - x^{2}/2 + x^{3}/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 -

x_{1}-2x_{2}+x_{3}-x_{4}=2;

x_{1}+2x_{2}+2x_{4}=1;

4x_{2}-x_{3}+3x_{4}=-1.(6 marks)
**3(b)** Find all stationary value of x^{2} + 3xy - 15x^{2} - 15y^{2} + 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 cos^{7}θ 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^{-1}x)^{2}, obtain y_{n}(0).(8 marks)
**6(a)** Show that the vectors are linearly dependent and find the relation between them

X_{1}=[1,2,-1,0],

X_{2}=[1,3,1,2],

X_{3}=[4,2,1,0],

X_{4}=[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)