Applied Mathematics 2 Question Paper - May 18 - First Year Engineering (Semester 2)

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Applied Mathematics 2 - May 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 $\int^{\infty}_0 5^{-4x^2} \ dx$

(3 marks) 12629

1.b. Solve $\frac{dy}{dx} = xy$ with the help of Euler’s method, given that y(0) = 1 and find y when x = 0.3 (h = 0.1)

(3 marks) 12672

1.c. Evaluate $\frac{d^4y}{dx^4} + 2 \frac{d^2y}{dx^2} + y = 0$

(3 marks) 12673

1.d. Evaluate $\int^1_0 \sqrt{\sqrt{x} – x } dx$

(3 marks) 12674

1.e. Solve $(1 + log xy) dx + (1 + \frac{x}{y}) dy = 0$

(4 marks) 12675

1.f. Evaluate $\int^1_0 \int^\sqrt{1+x^2}_0 \frac{dxdy}{1 + x^2 + y^2}$

(4 marks) 12676

2.a. Solve $xy (1+ x y^2) \frac{dy}{dx} = 1$

(6 marks) 12677

2.b. Find the area inside the circle $r = a sin \theta$ and outside the cardioide $r = a (1 + cos \theta)$

(6 marks) 12678

2.c. Apply Runge-kutta Method of fourth order to find an approximate value of y when x = 0.2 given that $\frac{dy}{dx} = x + y$ when y = 1 at x = 0 with step size h = 0.2

(8 marks) 12679

3.a. Show that the length of the curve $9ay^2 = x(x-3a)^2$ is $4\sqrt3a$

(6 marks) 12680

3.b. Change the order of the integration of $\int^1_0 \int^{1+\sqrt{1-y^2}}_{-\sqrt{2y – y^2}} f (x,y) dx \ dy$

(6 marks) 12681

3.c. Find the volume of the paraboloid $x^2 + y^2 = 4z$ cut off by the plane z = 4

(8 marks) 12682

4.a. Show that $\int^1_0 \frac{x-1}{log x} dx = log (a+1)$

(6 marks) 12683

4.b. If Y satisfies the equation $\frac{dy}{dx} = x^2 y -1$ with $x_0 = 0, y_0 = 1$, using Taylor’s series method find y at x = 0.1 (take h = 0.1)

(6 marks) 12684

4.c. Find the value of the integral $\int^1_0 \frac{x^2}{1 +x^2} dx$ using

(i) Trapezoidal rule

(ii) Simpsons $1/3^{rd}$ rule

(iii) Simpsons $3/8^{th}$ rule.

(8 marks) 12685

5.a. Solve $(y – xy^2) dx – (x + x^2y)dy = 0$

(6 marks) 12686

5.b. Evaluate $\int\int\int \sqrt{1 - \frac{x^2}{a^2} - \frac{y^2}{b^2} - \frac{z^2}{c^2}}$ dx dy dz over the ellipsoid $\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$

(6 marks) 12687

5.c. Evaluate $(2x + 1)^2 \frac{d^2y}{dx^2} – 2(2x +1) \frac{dy}{dx} – 12y = 6x$.

(8 marks) 12688

6.a. A resistance of 100 ohms and inductance of 0.5 henneries are connected in series with a battery of 20 volts. Find the current at any instant if the relation between L, R, E is $L \frac{di}{dt} + Ri = E$

(6 marks) 12689

6.b. Solve by variation parameter method $\frac{d^2y}{dx^2} + 3 \frac{dy}{dx} + 2y = e^{e^x}$

(6 marks) 12690

6.c. Evaluate $\int\int xy (x – y)$ dx dy over the region bounded by xy = 4, y = 0, x = 1 and x = 4.

(8 marks) 12691

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