## 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 y_{1}=2x_{1}- 2x_{2}-x_{3}, y_{2}=-4x_{1} + 5x_{2} + 3x_{3}, y_{3}= x_{1}-x_{2}-x_{3} 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 sin^{2}x cos^{3}x.(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 x^{5}.(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 x^{4}+y^{4}-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 x^{3}+y^{3}3xy=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 y^{2}(a-x)=x^{3}, 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)