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Applied Mathematics 2 Question Paper - Dec 18 - First Year Engineering (Semester 2) - Mumbai University (MU)
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Applied Mathematics 2 - Dec 18

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

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