## 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 (D^{2}+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 x_{0} = 0, y_{0}=1 at x=1 taking h=0.5.(8 marks)
**5 (a)** Solve (y-x y^{2}) dx - (x+x^{2}y) dy = 0.(6 marks)
**5 (b)** Using Taylor Series Method obtain the solution of following differential equation $ \dfrac {dy}{dx}= 1+ y^2 $ with y_{0}=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=x^{2}, x+y=2.(6 marks)
**6 (c)** Find the volume bounded by the paraboloid x^{2}+y^{2}=az and the cylinder x^{2}+y^{2}=a^{2}.(8 marks)