MORE IN Engineering Mathematics-1
SPPU First Year Engineering (Semester 1)
Engineering Mathematics-1
June 2015
Total marks: --
Total time: --
INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

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 M
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 M
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 M

2 (a) If $2 \cos \theta = x + \dfrac {1} {x} ,$ Prove that: $2 \cos r \theta = x^r + \dfrac {1}{x^r}$
4 M
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 M
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 M

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 M
3 (b) Prove that $\sin x \cos h \ x= x+ \dfrac {1}{3}x^3 - \dfrac {1}{30}x^5 - \cdots \ \cdots$
4 M
3 (c) Find nth derivative of:
e2 sin h 3x cos 4x.
4 M

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 M
4 (b) Using Taylor's theorem expand: 2x3+3x2-8x+7 in powers of x-2.
4 M
4 (c) If y=etan-1x, then show that: $\left ( 1+x^2 \right )y_{n+1} + (2nx-1)y_n + n(n-1)y_{n-1}=0$
4 M

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 M
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 M
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 M

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 M
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 M
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 M

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 M
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 M
7 (c) Examine maxima and minima of the following function and find their extreme values:
(x2+y2+6x+12).
5 M

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 M
8 (b) If ez=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 M
8 (c) Find the minimum distance from origin to the plane:
3x+2y+z=12.
4 M

More question papers from Engineering Mathematics-1