First Year Engineering (Semester 2)
Total marks: 80
Total time: 3 Hours
INSTRUCTIONS
(1) Question 1 is compulsory.
(2) Attempt any three from the remaining questions.
(3) Draw neat diagrams wherever necessary.
1.a.
Evaluate $\int^{\infty}_0 \frac{e^{-x^3}}{\sqrt x} \ dx$
(3 marks)
12709
1.b.
Find the length of the curve $x = \frac{y^3}{3} + \frac{1}{4y}$ from y = 1 to y = 2
(3 marks)
12710
1.c.
Solve $(D^2 + D)y = e^{4x}$
(3 marks)
12711
1.d.
Evaluate $\int^1_0 \int^x_{x^2} xy (x + y) dy \ dx$
(3 marks)
12713
1.e.
Solve (4x + 3y – 4) dx + (3x – 7y – 3) dy = 0
(4 marks)
12708
1.f.
Solve $\frac{dy}{dx} = 1 + xy $ with initial condition.
$x_0 = 0, y_0 = 0.2$ by Taylor's series method. Find the approximate value of y for x = 0.4 (step size 0.4)
(4 marks)
12707
2.a.
Solve $\frac{d^2y}{dx^2} – 16y = x^2 e^{3x} + e^{2x} – cos3x + 2^x$
(6 marks)
12706
2.b.
Show that $\int^\pi _0 \frac{log(1 + acosx)}{cosx} dx = \pi \ sin^{-1} a \ 0 \leq a \leq 1$
(6 marks)
12705
2.c.
Change the order of integration and evaluate $\int^2_0 \int^{2+\sqrt{4.y^2}}_{2-\sqrt{4 – y^2}} dxdy$
(8 marks)
12704
3.a.
Evaluate $\int \int \int (x + y + z) dx \ dy \ dz$ over the tetrahedron bounded by the planes x = 0, y = 0, z = 0 and x + y + z = 1
(6 marks)
12703
3.b.
Find the mass of the lamina bounded by the curves $ y = x^2 – 3x$ and y = 2x if the density of the lamina at any point is given by $\frac{24}{25} xy$
(6 marks)
12702
3.c.
Solve $x^2 \frac{d^2y}{dx^2} + 3x \frac{dy}{dx} + 3y = \frac{logx.cos(logx)}{x}$
(8 marks)
12700
4.a.
Find by double integration the area bounded by the parabola $y^2 = 4x$ and the line y = 2x – 4
(6 marks)
12699
4.b.
Solve $\frac{dy}{dx} + xsin2y = x^3 cos^2 y$
(6 marks)
12698
4.c.
Solve $\frac{dy}{dx} = x^3 + y$ with initial conditions y(0) = 2 at x = 0.2 in steps of h = 0.1 by Runge Kutta method of fourth order.
(8 marks)
12697
5.a.
Evaluate $\int^1_0 x^5 sin^{-1} x \ dx$ and find the value of $\beta (\frac{9}{2} , \frac{1}{2})$
(6 marks)
12696
5.b.
In a circuit containing inductance L, resistance R, and voltage E, the current I is given by $L \frac{di}{dt} + Ri = E$
find the current i at time t if at t = 0, i = 0 and L,R,E are constants.
(6 marks)
12695
5.c.
Evaluate $\int^6_0 \frac{dx}{1+3x}$ by using
i) Trapezoidal
ii) Simpson’s $(1/3)^{rd}$ and
iii) Simpsons $(3/8)^{th}$ rule
(8 marks)
12694
6.a.
Find the volume bounded by the paraboloid.
$x^2 + y^2 = az$ and the cylinder $x^2 + y^2 = a^2$
(6 marks)
7652
6.b.
Change to polar co-ordinates and evaluate.
$\int^1_0 \int^x_0 (x + y ) dy \ dx$
(6 marks)
12693
6.c.
Solve by method of variation of parameters.
$\frac{d^2y}{dx^2} + 3 \frac{dy}{dx} + 2y = e^{e^x}$
(8 marks)
12690