Engineering Maths 1 - Jun 2015

First Year Engineering (C 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.

Answer any one question from Q1 and Q2

1 (a) If y^1/m + y^-1/m=2x prove that $$ (x^2 - 1) y_{n+2} + (2n+1)xy_{n+1} + (n^2 - m^3) y_n = 0 $$(7 marks) 1 (b) Find the pedal equation for the curve
r^infin;=a^∞ sin mθ+b^∞ cos mθ.(6 marks) 1 (c) Derive an expression to find radius of curvature in Cartesian form.(7 marks) 10 (a) Solve by L U decomposition method
x+5y+z=14, 2x+y+3z=14, 3x+y+4z=17(7 marks) 10 (b) Show that the transformation y₁=2x₁- 2x₂-x₃, y₂=-4x₁ + 5x₂ + 3x₃, y₃= x₁-x₂-x₃ is regular and find the inverse transformation.(6 marks) 10 (c) Reduce the quadratic form $$ 2x^2_1 + 2x^2 _2 + 2x^2_3 + 2x_1x_3 $$ into canonical form by orthogonal transformation.(7 marks) 2 (a) Find the n^th derivative of sin²x cos³x.(7 marks) 2 (b) Show that the curves r=a(1+cos θ) and r=b (1-cos &theta) intersect at right angles.(6 marks) 2 (c) Find the radius of curvature when x=a log (sec t + tan t) y=a sect.(7 marks)

Answer any one question from Q3 and Q4

3 (a) Using McLaurin's series expand tan x upto the term containing x⁵.(7 marks) 3 (b) Show that $$ x \dfrac {\partial u}{\partial x} + y \dfrac {\partial u}{\partial y} = 2u \log u \ where \log u = \dfrac {x^3 + y^3}{3x+4y} $$(6 marks) 3 (c) Find the extreme values of x⁴+y⁴-2(x-y)².(7 marks) 4 (a) $$ Evaluate \ \lim_{x \to \inty} \left \{ \dfrac {e^x \sin x-x-x^2}{x^2 + x \log (1-x)} \right \} $$(7 marks) 4 (b) If u=x log xy where x³+y³3xy=1 Find fu/dx.(6 marks) 4 (c) $$ If \ u=\dfrac {yz}{x}, \ v= \dfrac {xz}{y}, \ w=\dfrac {xy}{z}, \ find \ \dfrac {\partial (u,v,w)}{\partial (x,y,z)} $$(7 marks)

Answer any one question from Q5 and Q6

5 (a) Find div $$ \overrightarrow {F} $$ and Curl $$ \overrightarrow {F} $$ where $$overrightarrow {F} = grad (x^3 + y^3 + z^3 - 3xyz) $$(7 marks) 5 (b) Using differentiation under integral sign,
Evaluate $$ \int^1_0 \dfrac {x^\alpha - } {\log x} dx (\alpha \ge 0) $$ Hence find $$ \int^1_0 \dfrac {x^3 -1} {\log x} dx $$(6 marks) 5 (c) Trace the curve y²(a-x)=x³, a>0 use general rules.(7 marks) 6 (a) $$ if \ \overrightarrow {r} = xi + yj + zk \ and \ r=|\overrightarrow {r} | $$ then prove that $$ \nabla r^n = nr^{n-2} \overrightarrow{r} $$(7 marks) 6 (b) Find the constants a, b, c such that $$ \overrightarrow {F} = (x+y+az)i + (bx+2y-z)j + (x+cy + 2z)k $$ is irrotational. Also find ϕ such that $$ \overrightarrow {F} = \nabla \phi $$(6 marks) 6 (c) Using differentiation under integral sign, evaluate $$ int^\infty _0 e^{-\alpha x } \dfrac {\sin x } {x } dx $$(7 marks)

Answer any one question from Q7 and Q8

7 (a) Obtain reduction formula for $$ int^{1/2}_0 \cos^n x \ dx $$(7 marks) 7 (b) Solve: (1+2xy \cos x^2 - 2xy) dx + (\sin x^2 - x^2) dy = 0.(6 marks) 7 (c) A body originally at 80°C cools down to 60°C in 20 minutes, the temperature of the air being 40°C. What will be temperature of the body after 40 minutes from the original?(7 marks) 8 (a) Evaluate $$ \int^{2a}_0 x^2 \sqrt { 2ax - x^2 } dx $$(7 marks) 8 (b) Solve: $$ xy \left (1+x \ y^2 \right ) \dfrac {dy}{dx}= 1 $$(6 marks) 8 (c) Fid the orthogonal trajectories of the family of confocal conics $$ \dfrac {x^2}{b^2}+ \dfrac {y^2}{b^2 + \lambda}= 1 $$ where λ is parameter.(7 marks)

Answer any one question from Q9 and Q10

9 (a) Solve by Gauss elimination method.
$$ 5x_1 + x_2 + x_3 + x_4 =4, \ x_1+7x_2 + x_3 + x_4 =12, \ x_1 + x_2 + 6x_3 + x_4 = -5, \ x_1 + x_2 + x_3 + 4x_4 = 6 $$(7 marks) 9 (b) Diagonalize the matrix $$ A= \begin{bmatrix}-19 &7 \\-42 &16 \end{bmatrix} $$(6 marks) 9 (c) Find the dominant Eigen value and the corresponding Eigen vector of the matrix $$ A= \begin{bmatrix}6 &-2 &2 \\-2 &3 &-1 \\2 &-1 &3 \end{bmatrix} $$ by power method taking the initial Eigen vector (1,1,1)¹.(7 marks)