Engineering Mathematics-1 - Jun 2015

First Year Engg (Semester 1)

TOTAL MARKS:
TOTAL TIME: HOURS

Answer any one question from Q1 and Q2

1 (a) Find the Eigen values and Eigen vector corresponding to minimum Eigen value for the matrix: $$ A= \begin{bmatrix}1 &0 &-1 \\1 &2 &1 \\2 &2 &3 \end{bmatrix} $$(4 marks) 1 (b) Determine the value for λ for which the equations: $$ 3x_1 + 2x_2 + 4x_3 =3 \\ x_1 + x_2 + x_3= \lambda \\ 5x_1 + 4x_2 + 6x_3 = 15 $$ are consistent. Find also the corresponding solution.(4 marks) 1 (c) If z₁, z₂, z₃ are the vertices of an isosceles triangle right angled at z₂, prove that: $$ z^2_1 + z^2_3 + 2 z^2_2 = 2z_2 (z_1 + z_3) $$(4 marks) 2 (a) If $$ 2 \cos \theta = x + \dfrac {1} {x} , $$ Prove that: $$ 2 \cos r \theta = x^r + \dfrac {1}{x^r} $$(4 marks) 2 (b) if Y-log tan x, prove that: $$ i) \ \sin h ny = \dfrac {1}{2} (\tan^n x - \cot ^n x), $$ (ii) 2coshny cosec2x=cosh (n+1) y+cosh(n-1)y.(4 marks) 2 (c) Examine for linear dependence or independence for the given vectors and if dependence, find the relation between Them:
X₁ = (1, ?1, 2, 2),
X₂ = (2, -3, 4, -1),
X₃ = (-1, 2, -2, 3).(4 marks)

Answer any one question from Q3 and Q4

3 (a) Test convergence of the series (any one): $$ i) \ \ \sum^{\infty}_{n=1} \left ( \dfrac {2n+1}{3n+4} \right )5^n \\ ii) \ \dfrac {1}{1+\sqrt{2}} + \dfrac {2}{1+2\sqrt{3}} + \dfrac {3}{1+3 \sqrt{4}}+ \cdots \ \cdots $$(4 marks) 3 (b) Prove that $$ \sin x \cos h \ x= x+ \dfrac {1}{3}x^3 - \dfrac {1}{30}x^5 - \cdots \ \cdots $$(4 marks) 3 (c) Find n^th derivative of:
e² sin h 3x cos 4x.(4 marks) 4 (a) Solve any one: $$ i) \ \ \lim_{x \to a}(x-a)^{(x-a)} \\ ii) \ \lim_{x\to \pi /2} (\sec x - \tan x) $$(4 marks) 4 (b) Using Taylor's theorem expand: 2x³+3x²-8x+7 in powers of x-2.(4 marks) 4 (c) If y=e^tan-1x, then show that: $$ \left ( 1+x^2 \right )y_{n+1} + (2nx-1)y_n + n(n-1)y_{n-1}=0 $$(4 marks)

Solve any two of the following:

5 (a) Find the value of n for which:
z=A e^-gx sin (nt - gx),
satisfies the partial differential equation: $$ \dfrac {\partial z}{\partial t} = \dfrac {\partial^2 z}{\partial x^2} $$(6 marks)

Answer any one question from Q5 and Q6

5 (b) $$ If \ u=\dfrac {x^4 + y^4}{x^2y^2}+ x^6 \tan ^{-1} \left [ \dfrac {x^2 + y^2} {x^2 + 2xy} \right ], $$ find the value of: $$ x^2 \dfrac {\partial ^2 u}{\partial x^2}+ 2xy \dfrac {\partial ^2 u}{\partial x \partial y} + y^2 \dfrac {\partial ^2 u}{\partial y^2}+ \dfrac {\partial u}{\partial x} + y \dfrac {\partial u}{\partial y} $$ at x=1, y=2.(7 marks) 5 (c) If z=f(u,v) and $$ u=\log (x^2 + y^2), \ v=\dfrac {y}{x}, $$ show that: $$ x \dfrac {\partial y}{\partial y} - y \dfrac {\partial z}{\partial x} = (1+v^2) \dfrac {\partial z}{\partial v} $$(6 marks)

Solve any two of the following:

6 (a) $$ If \ x=\dfrac {\cos \theta}{r}, \ y=\dfrac {\sin \theta}{r}, $$ find the value of: $$ \left ( \dfrac {\partial x}{\partial r} \right )_\theta \left ( \dfrac {\partial r}{\partial x} \right )_y + \left( \dfrac {\partial y} {\partial r} \right )_\theta \left ( \dfrac {\partial r}{\partial y} \right )_x $$(6 marks) 6 (b) $$ If \ u=\sin^{-1}\sqrt{\dfrac {x^2+ y^2}{x+y}} $$ Show that: $$ x^2 \dfrac {\partial^2 u}{\partial x^2} + 2xy \dfrac {\partial^2 u}{\partial x \partial y} + y^2 \dfrac {\partial^2 u}{\partial y^2} = \dfrac {1}{4} \tan u [\tan^2 u-1] $$(7 marks) 6 (c) If z=f(u,v) and u=x cos t ? y sin t, v=x sin t + y cos t, where t is a constant, prove that: $$ x \dfrac {\partial z}{\partial x} + y \dfrac {\partial z}{\partial y}= u \dfrac {\partial z}{\partial u}+ v \dfrac {\partial z}{\partial v}. $$(6 marks)

Answer any one question from Q7 and Q8

7 (a) $$ If \ u^3 + v^3 = x + y, \ u^2+v^2 = x^3 + y^3, \ find \ \dfrac {\partial (u,v)}{\partial (x, y)}$$(4 marks) 7 (b) Determine whether the following functions are functionally dependent. If functionally dependent, find the relation between them:
u=sin x + sin y, v=sin (x+y).(4 marks) 7 (c) Examine maxima and minima of the following function and find their extreme values:
(x²+y²+6x+12).(5 marks) 8 (a) $$ If \ x=u+v, \ y=v^2+w^2, \ z=w^3+u^3, \ show \ that: \\ \dfrac {\partial u}{\partial x}= \dfrac {vw}{vw+u^2} $$(4 marks) 8 (b) If e^z=sec x cos y and error of magnitude h and -h are made in estimating x and y, where x and y are found to be π/3 and π6 respectively, find the corresponding error in z.(5 marks) 8 (c) Find the minimum distance from origin to the plane:
3x+2y+z=12.(4 marks)