Engineering Maths 1 - Dec 2014

First Year Engineering (P Cycle) (Semester 1)

TOTAL MARKS: 100
TOTAL TIME: 3 HOURS (1) Question 1 is compulsory.
(2) Attempt any four from the remaining questions.
(3) Assume data wherever required.
(4) Figures to the right indicate full marks. 1 (a) If Y=cos(m log x), prove that $$ x^2 y_{n+2}+ (2n+1)xy_{n+1} + (m^2 + n^2)y_s=0 $$(7 marks) 1 (b) Find the angle intersection between the curves $$ r=a \log \theta \ and \ r = \dfrac {n}{\log \theta} $$(6 marks) 1 (c) Derive an expression to find radius of curvature in Cartesian form.(7 marks) 10 (a) Show that the transformation $$ y_1 = x_1 + 2x_2 +5x_3 , \ y_2 = 2x_3 + 4x_2 + 11 x_3 , \ y_3 = -x_2 + 2x_3 $$ is regular and find the inverse transformation.(6 marks) 10 (b) Solve by LU decomposition method 2x+y+4z=12, 8x-3y+2z=20, 4x+11y-z=33(7 marks) 10 (c) Reduce the quadratic form 2x²+2y²-2xy-2yz-2zx into canonical form. Hence indicate its nature, rank, index and signature.(7 marks) 2 (a) $$ If \ \sin^{-1} y=2 \log (x+1) \ prove \ that \ (x^2 +1) y_{n+2} + (2n+1) (x+1)y_{n+1}+ (n^2 + 4)y_n=0 $$(7 marks) 2 (b) Find the pedal equation rⁿ=sec hn?(6 marks) 2 (c) Show that the radius of curvature of the curve $$ x^3 + y^3 = 3xy \ at \ \left ( \dfrac {3}{2}, \dfrac {3}{2} \right ) \ is \ \dfrac {-3}{8\sqrt{2}} $$(7 marks) 3 (a) Find the first four non zero terms in the expansion of $$ f(x) = \dfrac {x}{e^{x-1}} $$(7 marks) 3 (b) $$ If \ \cos u = \dfrac {x+y}{\sqrt{x}+ \sqrt{y}} \ show \ that \ x\dfrac {\partial u}{\partial x} + y \dfrac {\partial u}{\partial y} = - \dfrac {\cot u}{2} $$(6 marks) 3 (c) $$ Find \ \dfrac {\partial (u,v,w)}{\partial (x,yz)} \ where \ u=x^2+y^2+z^2, \ v=xy+yz + zx \ and \ w=x+y+z $$ Hence interpret the result.(7 marks) 4 (a) If w=f(x,y), x=r cos &theta, y=r sin ? show that $$ \left ( \dfrac {\partial t}{\partial x} \right )^2 + \left ( \dfrac {\partial t}{\partial y} \right )^2 - \left ( \dfrac {\partial w}{\partial r} \right )^2 = \dfrac {1}{r^2} \left ( \dfrac {\partial w}{\partial \theta} \right )^2 $$(7 marks) 4 (b) Evaluate $$ \lim_{x\to 0} \left ( \dfrac {\sin x} {x}\right )^{\frac {1}{x}} $$(6 marks) 4 (c) Examine the function f(x,y)=1+sin(x²+y²) for extremum.(7 marks) 5 (a) A particle moves along the curve x=2t², y=t²-4t, z=3t-5. Find the components of velocity and acceleration at t=1 in the direction $$ \widehat{i}-2\widehat{j}+2\widehat{k} $$(7 marks) 5 (b) Using differentiation under integral sign, evaluate $$ \int^\infty_0 \dfrac{e^{-\alpha x}\sin x}{x}dx $$(7 marks) 5 (c) Use general rules to trace the curve y²(a-x)=x³, a>0(6 marks) 6 (a) $$ If \ \overrightarrow{r} = x\widehat{i} + y\widehat{j}+ z\widehat{k} \ and \ |\overrightarrow{r}|= r. \ Find \ grad \ div \left ( \dfrac {\overrightarrow{r}}{r} \right ) $$(7 marks) 7 (a) Obtain the reduction formula for $$ \int^{\frac {\pi}{2}}_0 \cos^n xdx $$(7 marks) 7 (b) Solve (xy³+y)dx+2(x²y²+x+y⁴)dy=0(6 marks) 7 (c) Show that the orthogonal trajectories of the family of cardioids $$ r= a \cos^2 \left ( \dfrac {\theta}{2} \right ) $$ is another family of cardioids $$ r= b \sin^2 \left ( \dfrac {\theta}{2} \right ) $$(7 marks) 8 (a) $$ Evaluate \ \int^\pi_0 x \sin^2 x \cos^4 xdx $$(7 marks) 8 (b) $$ Solve \ \dfrac {dy}{dx}- y \tan x=y^2 \sec x $$(6 marks) 8 (c) If the temperature of the air is 30°C and the substance cools from 100°C to 70°C in 15 minutes, find when the temperature will be 40°C.(7 marks) 9 (a) Solve 3x-y+2z=12, x+2y+3z=11, 2x-2y-z=2 by Gauss elimination method.(6 marks) 9 (b) Diagonalize the matrix $$ A= \begin{bmatrix}-1 &1 &2 \\0 &-2 &-1 \\0 &0 &-3 \end{bmatrix} $$(7 marks) 9 (c) Determine the largest eigen value and the corresponding eigen vector of $$ A= \begin{bmatrix}1 &3 &-1 \\3 &2 &4 \\-1 &4 &10 \end{bmatrix} $$ Staring with [0, 0, 1]^? as the initial eigenvector. Perform 5 iterations.(7 marks)