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Applied Mathematics 2 Question Paper - Dec 17 - First Year Engineering (Semester 2) - Mumbai University (MU)
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Applied Mathematics 2 - Dec 17

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 $\int^\infty_0 \frac {e^{-x^2}}{\sqrt x} \ dx$
(3 marks) 12650

1.b. Solve $(D^3 + 1)^2 y = 0$
(3 marks) 12651

1.c. Solve the ODE $( y + \frac{1}{3} y^3 + \frac{1}{2} x^2 ) dx + (x + x y^2) dy = 0$
(3 marks) 12653

1.d. Use Taylor’s series method to find a solution of $\frac{dy}{dx} = 1 + y^2, y(0) = 0$

at x = 0.1 taking h = 0.1 correct to three decimal value.

(3 marks) 12654

1.e. Given $\int^x_0 \frac{dx}{x^2 + a^2} = \frac{1}{a} tan^-1 (\frac{x}{a})$, using DUIS find the value of $\int^x_0 \frac{dx}{(x^2 + a^2)^2}$
(4 marks) 12655

1.f. Find the perimeter of the curve $r = a (1- cos\theta$)

(4 marks) 12656

2.a. Solve $(D^3 + D^2 + D + 1) y = sin^2 x$
(6 marks) 12657

2.b. Change the order of integration $\int^a_0 \int^{x+3a}_{\sqrt{a^2-x^2}} f (x,y) \ dx \ dy$
(6 marks) 12658

2.c. Evaluate $\int \int_R \frac{2x \ y^5}{\sqrt{1+ x^2\ y^2 – y^4}} \ dx \ dy$, where R is a triangle whose vertices are (0, 0), (1,1), (0,1).
(8 marks) 12659

3.a. Find the volume enclosed by the cylinder $y^2 = x \ and\ y = x^2$ Cut off by the planes z = 0, x + y + z =2
(6 marks) 12660

3.b. Using modified Euler’s method, find an appropriate value of y at x = 0.2 in two step taking h = 0.1 and using iteration, given that $\frac{dy}{dx} = x + 3y, y = 1 \ when \ x = 0.$
(6 marks) 12661

3.c. Solve $(1+x)^2 \frac{d^2y}{dx^2} + (1+x) \frac{dy}{dx} + y = 4 \ cos log (1 + x) $
(8 marks) 12662

4.a. Show that $\int^\alpha_0 \sqrt\frac{x^3}{a^3 – x^3} dx$

$= \frac{a\sqrt\pi \sqrt{5/6}}{\sqrt{1/3}}$

(6 marks) 12663

4.b. Solve $(D^2 + 2)y = e^x \ cos x + x^2 e^{3x}$
(6 marks) 12664

4.c. Use polar co-ordinates to evaluate $\int \int \frac{(x^2+y^2)^2}{x^2 \ y^2} dx \ dy$ over the area

Common to the circle $x^2 + y^2 = ax$ and $x^2 + y^2 = by, a \gt b \gt 0$

(8 marks) 12665

5.a. Solve $y \ dx + x (1 – 3 x^2 y^2) dy = 0$
(6 marks) 12666

5.b. Find the mass of a lamina in the form of an eclipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} - 1$, if the density at any point varies as the product of the distance from the azes of the ellipse.
(6 marks) 12667

5.c. Compute the value of $\int^{\pi/2}_0 \sqrt{sinx + cosx} \ dx$ using

(i) Trapezoidal rule

(ii) Simpson’s $(1/3)^{rd}$ rule

(iii) Simpson’s $(3/8)^{th}$ rule by dividing into six subintervals

(8 marks) 12668

6.a. Evaluate $\int\int\int_v x^2 dx \ dy \ dz$ over the volume bounded by the planes $x = 0, y = 0, z = 0 \ and \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$
(6 marks) 12669

6.b. Change the order of integration and evaluate $\int^2_0 \int^2_\sqrt{2y} \frac{x^2}{\sqrt{x^4 – 4 y^2}} dx \ dy$
(6 marks) 12670

6.c. Solve by the method of variation of parameters $\frac{d^2y}{dx^2} – 6 \frac{dy}{dx} + 9y = \frac{e^{3x}}{x^2}$
(8 marks) 12671

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