VTU Engineering Maths 1 - June 2015 Exam Question Paper

VTU First Year Engineering (P Cycle) (Semester 1)
Engineering Maths 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) 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 M

1 (b) Find the pedal equation for the curve
r^infin;=a^∞ sin mθ+b^∞ cos mθ.

6 M

1 (c) Derive an expression to find radius of curvature in Cartesian form.

7 M

2 (a) Find the n^th derivative of sin²x cos³x.

7 M

2 (b) Show that the curves r=a(1+cos θ) and r=b (1-cos &theta) intersect at right angles.

6 M

2 (c) Find the radius of curvature when x=a log (sec t + tan t) y=a sect.

7 M

Answer any one question from Q3 and Q4

3 (a) Using McLaurin's series expand tan x upto the term containing x⁵.

7 M

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 M

3 (c) Find the extreme values of x⁴+y⁴-2(x-y)².

7 M

4 (a)

$Evaluate \ \lim_{x \to \inty} \left \{ \dfrac {e^x \sin x-x-x^2}{x^2 + x \log (1-x)} \right \}$

7 M

4 (b) If u=x log xy where x³+y³3xy=1 Find fu/dx.

6 M

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 M

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 M

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 M

5 (c) Trace the curve y²(a-x)=x³, a>0 use general rules.

7 M

6 (a)

$if \ \overrightarrow {r} = xi + yj + zk \ and \ r=|\overrightarrow {r} |$ then prove that

$\nabla r^n = nr^{n-2} \overrightarrow{r}$

7 M

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 M

6 (c) Using differentiation under integral sign, evaluate

$int^\infty _0 e^{-\alpha x } \dfrac {\sin x } {x } dx$

7 M

Answer any one question from Q7 and Q8

7 (a) Obtain reduction formula for

$int^{1/2}_0 \cos^n x \ dx$

7 M

7 (b) Solve: (1+2xy \cos x^2 - 2xy) dx + (\sin x^2 - x^2) dy = 0.

6 M

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 M

8 (a) Evaluate

$\int^{2a}_0 x^2 \sqrt { 2ax - x^2 } dx$

7 M

8 (b) Solve:

$xy \left (1+x \ y^2 \right ) \dfrac {dy}{dx}= 1$

6 M

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 M

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 M

9 (b) Diagonalize the matrix

$A= \begin{bmatrix}-19 &7 \\-42 &16 \end{bmatrix}$

6 M

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 M

10 (a) Solve by L U decomposition method
x+5y+z=14, 2x+y+3z=14, 3x+y+4z=17

7 M

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 M

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 M

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