Applied Mathematics 2 - Dec 2015

First Year Engineering (Semester 2)

TOTAL MARKS: 80
TOTAL TIME: 3 HOURS (1) Question 1 is compulsory.
(2) Attempt any three from the remaining questions.
(3) Assume data if required.
(4) Figures to the right indicate full marks. 1 (a) Evaluate $ \int^2_0 x^2 (2-x)^3 dx $(3 marks) 1 (b) Solve $ \dfrac {d^3y}{dx^3}- 6 \dfrac {d^2y}{dx^2}+11 \dfrac {dy}{dx} - 6y = 0 $(3 marks) 1 (c) Prove that E=1+Δ(3 marks) 1 (d) Solve $ \left [ y \left ( 1+ \dfrac {1}{x} \right )+ \text{cosy} \right ] dx + (x+ {\log x} - x\sin y)dy=0 $(3 marks) 1 (e) Change to polar coordinates and evaluate $ \int^a_0 \int^{\sqrt{a^2 - x^2}}_0 (x^2 + y^2) dy \ dx $(4 marks) 1 (f) Evaluate $ \int^1_0 \int^x_0 x\ y\ dy \ dx $(4 marks) 2 (a) Solve $ \dfrac {dy}{dx} + \dfrac {4x}{x^2 +1} y = \dfrac {1}{(x^2 +1)^3} $(6 marks) 2 (b) Change the order of integration and evaluate $$ \int^2_0 \int^2_{\sqrt{2x}} \dfrac {y^2 \ dx \ dy}{\sqrt{y^2 - 4x^2}} $$(6 marks) 2 (c) Prove that $ \int^{\pi / 2}_0 \dfrac {\log (1+ a \sin ^2 x) }{\sin^2 x}dx = \pi \big [ \sqrt{a+1}-1 \big ] a>-1 $(8 marks) 3 (a) Evaluate $ \int^1_0 \int^{1-y}_0 \int^{1-x-y}_0 \dfrac {1}{(x+y+z+1)^3} dz \ dy \ dx $(6 marks) 3 (b) Find by double integration the area enclosed by the curve
9xy=4 and the line 2x+y=2.(6 marks) 3 (c) Using method of Variation of Parameter solve $ \dfrac {d^2 y}{d x^2} + a^2 y = \sec ax $(8 marks) 4 (a) Find the perimeter of the cardioide r=a (1+ cos θ).(6 marks) 4 (b) Solve (D²+4)y=cos 2x(6 marks) 4 (c) Apply Runge-kutta Method of fourth order to find an approximate value of y for $ \dfrac {dy}{dx} = \dfrac {1}{x+y} $ with x₀ = 0, y₀=1 at x=1 taking h=0.5.(8 marks) 5 (a) Solve (y-x y²) dx - (x+x²y) dy = 0.(6 marks) 5 (b) Using Taylor Series Method obtain the solution of following differential equation $ \dfrac {dy}{dx}= 1+ y^2 $ with y₀=0 when x_{0=0 for x=0.2.}(6 marks) 5 (c) Find the approximate value of $ \int^6_0 e^x dx $ by i) Trapezoidal Rule, ii) Simpson's 1/3^rd Rule, iii) Simpson's 3/8^th Rule.(8 marks) 6 (a) A resistance of 100 Ohm and inductance of 0.5 Henry are connected in series with a battery of 20 Volt. Find the current at any instant if the relation between L. $ R, \ E \ is \ L\dfrac {di}{dt} + Ri = E $(6 marks) 6 (b) $ \int \int y \ dx \ dy $ over the area bounded by the x=0, y=x², x+y=2.(6 marks) 6 (c) Find the volume bounded by the paraboloid x²+y²=az and the cylinder x²+y²=a².(8 marks)