Q.P. Code: 16281

Max Marks: 80 Time: 3 hrs.

Q1. A) show that $\int^infty_0 3_-4x^2 dx = \frac{\sqrt{\pi }}{4\sqrt{log 3}}$ (3marks)

b) Solve $( 2y^2 – 4x + 5) dx (y – 2y^2 – 4xy) dy$ 3(marks)

c) Solve the ODE $(D-1)^2 (D^2 + 1)^2 y = 0$ (3marks)

d) Evaluate $\int^1_0 \int^x^2_0 e^ \frac{x}{y} dy dx $ (3marks)

e) Evaluate $\int^1_0 \frac{x^a – 1}{log x} dx$ (4marks)

f) Find the length of the cycloid from one cusp to the next, where $x = a (\theta + sin \theta ), y = a (1 – coz \theta)$ (4 marks)

Q2. A) Solve $(D^2 – 3D + 2) y = 2e^\lambda sin (\frac{x}{2})$ (6 marks)

b) using DUIS prove that $\int^infty_0 e –(x^2 + \frac{2^2}{x^2} ) dx = \frac{\sqrt\pi}{2} e^-2n, a \gt 0$ (6 marks)

c) change the order of integration and evaluate $ \int^1_0 \int^\sqrt{2-x^2}_x \frac{x}{\sqrt{x^2 + y^2}} dxdy$ (8 marks)

Q3. A) evaluate$\int^1_0 \int^1-x_0 \int^1-x-y_0 \frac{1}{(x + y + z +1})^3 dzdydx$ (6marks)

b) Find the mass of the leminiscate $r^2 = a^2$ cos 20 if the density at any point is proportional to the square of the distance from the pole. (6marks)

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)

Q4. A) Prove for an asteroid $x 2/3 + y 2/3$, the line $\theta = \frac{\pi }{6}$ divide the arc in the first quadrant in a ratio 1:3 (6marks)

b) Solve $(D^2 – 7D – 6) y = (1 + x^2) c^2x$ (6marks)

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 x = 0.2 (8marks)

Q5. 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 (6marks)

b) Solve by the method of variation of parameters $\frac{d^2y}{dx^2} + y = \frac{1}{1+sin x}$ (6marks)

c) Compute the value of $\int^4_2 (sin x – ln x + e) dx$ using (i) Trapezoidal rule (8marks)

(ii) Simpson’s (1/3)rd rule (iii) Simpson’s (3/8)th rule by dividing into six subintervals.

Q6. A) Using Beta function evaluate $\int^6_0 cos^6 30 sin^2 6 \theta d\theta$ (8 marks)

b) Evaluate $\int log (x^2 + y^2) dx dy$ changing to polar co-ordinates (6marks)

c) Evaluate $\int\int\int_v x^2 y z dx dy dz$ over the volume bounded by the planes (8marks) x = 0, y = 0, z = 0 and $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$