Applied Mathematics 2 Question Paper - May 17 - First Year Engineering (Semester 2)

0views

written 5.0 years ago by

Applied Mathematics 2 - May 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. Show that $\int^\infty_0 3^{-4x^2}dx = \frac{\sqrt{\pi }}{4\sqrt{log 3}}$

(3 marks) 12629

1.b. Solve $( 2y^2 – 4x + 5) dx = (y – 2y^2 – 4xy) dy$

(3 marks) 12630

1.c. Solve the ODE $(D-1)^2 (D^2 + 1)^2 y = 0$

(3 marks) 12631

1.d. Evaluate $\int^1_0 \int_0^{x^2} e^ \frac{y}{x} dy dx $

(3 marks) 12632

1.e. Evaluate $\int^1_0 \frac{x^a – 1}{log x} dx$

(4 marks) 12633

1.f. Find the length of the cycloid from one cusp to the next, where $x = a (\theta + sin \theta ), y = a (1 – cos \theta)$

(4 marks) 12634

2.a. Solve $(D^2 – 3D + 2) y = 2e^x sin (\frac{x}{2})$

(6 marks) 12635

2.b. Using DUIS prove that $\int^{\infty}_0 e^{-(x^2+\frac{a^2}{x^2})} dx=\frac{\sqrt\pi}{2} e^{-2a}, a \gt 0$

(6 marks) 12636

2.c. Change the order of integration and evaluate $ \int^1_0 \int^\sqrt{2-x^2}_x \frac{x}{\sqrt{x^2 + y^2}} dx \ dy$

(8 marks) 12637

3.a. Evaluate$\int^1_0 \int^{1-x}_0 \int^{1-x-y}_0 \frac{1}{(x + y + z +1)^3} dz \ dy \ dx$

(6 marks) 12638

3.b. Find the mass of the leminiscate $r^2 = a^2 cos 2\theta$ if the density at any point is proportional to the square of the distance from the pole.

(6 marks) 12639

3.c. Solve $x^2 \frac{d^3y}{dx^3} + 3x \frac{d^2y}{dx^2} + \frac{dy}{dx} + \frac{y}{x} = 4 \ log x$

(8 marks) 12640

4.a. Prove for an asteroid $x ^{2/3} + y ^{2/3} = a^{2/3}$, the line $\theta = \frac{\pi }{6}$ divide the arc in the first quadrant in a ratio 1:3

(6 marks) 12641

4.b. Solve $(D^2 – 7D – 6) y = (1 + x^2) e^2x$

(6 marks) 12642

4.c. Apply Runge Kutta method of fourth order to find an appropriate value of y when x = 0.4 given that $\frac{dy}{dx} = \frac{y-x}{y+x}, \ y = 1$ when x = 0 taking h = 0.2

(8 marks) 12643

5.a. Use Taylor’s series method to find a solution of $\frac{dy}{dx} = 1 + x y, \ y (0) = 0 for \ x = 0.2$ taking h = 0.1 correct to four decimal places

(6 marks) 12644

5.b. Solve by the method of variation of parameters $\frac{d^2y}{dx^2} + y = \frac{1}{1+sin x}$

(6 marks) 12645

5.c. Compute the value of $\int^{1.4}_{0.2} (sin x – ln x + e^x) dx$ using

(i) Trapezoidal rule

(ii) Simpson’s $(1/3)^{rd}$ rule

(iii) Simpson’s $(3/8)^{th}$ rule by dividing into six sub-intervals.

(8 marks) 12646

6.a. Using Beta function evaluate $\int^{\frac{\pi}{6}}_0 cos^6 3\theta \ sin^2 6 \theta \ d\theta$

(6 marks) 12647

6.b. Evaluate $\int_0^{\frac{a}{\sqrt2}} \int_0^{\sqrt{a^2-y^2}} log (x^2 + y^2) dx \ dy$ changing to polar co-ordinates

(6 marks) 12648

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

(8 marks) 12649

ADD COMMENT EDIT