Applied Mathematics 1 - May 2015

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) $$ If \ \tan \dfrac {x}{2} = \tan h \ \dfrac {u}{2} \ then \ S.T. \\ u=\log \tan \left ( \dfrac {\pi}{4} + \dfrac {x}{2} \right ) $$(3 marks) 1 (b) $$ If \ u = x^y \ find \ \dfrac {\partial^3 u}{\partial x \partial y \partial x} $$(3 marks) 1 (c) If ux=yz,vy=zx, wz=xy find $$ j \left [ \dfrac {u,v,w}{x,y,z} \right ] $$(3 marks) 1 (d) $$ If y = (x-1)^n \ then \ P.T. \ y+ \dfrac {y_1}{1!} + \dfrac{y_2}{2!}+ \dfrac {y_3}{3!}+ \cdots \ \cdots \dfrac {y_n}{n!}= x^n $$(3 marks) 1 (e) $$ P.T.\ sinhx = X + \dfrac {x^3}{3!} + \dfrac {x^5}{5!} + \dfrac {x^7}{7!}+ $$(4 marks) 1 (f) Express the matrix A as sum of Hermition and skew Hermition matrix where $$ \begin{bmatrix} 3i &-1+i &3-2i \\1+i &-i &1+2i \\-3-2i &-1+2i &0 \end{bmatrix} $$(4 marks) 2 (a) Solve x⁷+x⁴+i(x³+1)=0(6 marks) 2 (b) Reduce the matrix A to normal form and hence find its rank where $$ A=\begin{bmatrix}0 &1 &-3 &-1 \\1 &0 &4 &3 \\3 &1 &0 &2 \\1 &1 &-2 &0 \end{bmatrix} $$(6 marks) 2 (c) State and prove Euler's theorem for three variable and hence find $$ x \dfrac {\partial u}{\partial x} + y \dfrac {\partial u}{\partial y}+ z \dfrac {\partial u}{\partial z} \ where \\ u= \dfrac {x^3 y^3 z^3}{x^3+ y^3 +z^3} $$(8 marks) 3 (a) Solve the following system of equations
2x-2y-5z=0
4x-y+z=0
3z-2y+3z=0
x-3y+7z=0(6 marks) 3 (b) Find the maximum and minimum values of
x³+3xy_{2-3x²-3y²+4}(6 marks) 3 (c) Separate into real and imaginary parts of tanh^-1 (x+iy).(8 marks) 4 (a) If u=2xy, v=x²-y² and x=rcos?, y=rsin? then find $$ \dfrac {\partial (u_1v)}{\partial (\partial_1 \theta)} $$(6 marks) 4 (b) If i^{ii... ?} =A+i B, prove that $$ \tan \left ( \dfrac {\pi A}{2} \right )= \dfrac {B}{A} \ and \ A^2 + B^2 = e^{-\pi B} $$(6 marks) 4 (c) Solve by crouts methods the system of equations
3x+2y+7z=4
2x+3y+z=5
3x+4y+z=7.(8 marks) 5 (a) By using De Moivre's theorem Express $$ \dfrac {\sin 7\theta }{\sin \theta} $$ in powers of sinθ only.(6 marks) 5 (b) By using Taylor's series expand tan^-1 x in positive powers of (x-1) upto first four non-zero terms.(6 marks) 5 (c) if y=sin [log (x²+2x+1)] prove that (x+1)² y_n+2 + (2n+1 (x+1) )y_n+1+ (n²+4)y_n=0(8 marks) 6 (a) Determine linear dependance or independance of vectors
x₁=[1,3,4,2] x₂==[3,-5,2,6]
x_{=[2,-1,3,4] and if dependent find the relation between them.}(6 marks) 6 (b) If u =x²-y², v=2xy and z=f(u,v) prove that $$ \left ( \dfrac {\partial z}{\partial x} \right )^2 + \left ( \dfrac {\partial z}{\partial y} \right )^2 = 4\sqrt{u^2+v^2} \left [ \left ( \dfrac{\partial z}{\partial u} \right )^2 + \left ( \dfrac {\partial z}{\partial v} \right )^2 \right ] $$(6 marks) 6 (c) Evaluate $$ i) \ \lim_{x\to 0} \dfrac {\sin x. \sin^{-1} x-x^2}{x^6} $$ ii) Fit straight line to the following data
(x,y)= (-1, -5), (1,1), (2,4), (3,7), (4, 10)
Estimate y when x=7.(4 marks)