Applied Mathematics 2 - Dec 2013

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 the following:
(3 marks) 1(b) Solve (D⁴+4)y=0(3 marks) 1(c) Prove that E∇ = Δ =∇E(3 marks) 1(d) Solve the following:
(3 marks) 1(e) Evaluate ∫∫r³drdθ over the regions between the circles r=2sinθ and r=4sinθ(4 marks) 1(f) Evaluate the following:
(4 marks) 2(a) Solve (x³y⁴+x²y³+xy²+y)dx + (x⁴y³-x³y²-x²y+x)dy=0(6 marks) 2(b) Change the order of integral and evaluate:
(6 marks) 2(c) Prove that:
(8 marks) 3(a) Evaluate the following:
(6 marks) 3(b) Find the area of one loop of the lemniscate r²=a²cos2θ(6 marks) 3(c) Solve (D³+2D²+D)y = x²e^3x+sin²x+2^x(8 marks) 4(a) Show that the length of the arc of the parabola y²=4ax cut off by the line 3y=8x is alog2+15/16(6 marks) 4(b) Using the method of variation of parameters, solve:
(6 marks) 4(c) Compute y(0.2) given (dy/dx)+y+xy²=0, y(0)=0, By taking h=0.1 using Runge-Kutta method of fourth order correct to 4 decimals.(8 marks) 5(a) Solve the following:
(6 marks) 5(b) Solve (dy/dx)-2y=3e^x,y(0)=0 using Taylor Series method. Find approximate value of y for x=0.1 and 0.2(6 marks) 5(c) Evaluate

using Trapezoidal rule, Simpson's 1/3rd rule and Simpson's 3/8th rule. Compare the result with exact values.(8 marks) 6(a) The current in a circuit containing an inductance L, resistance R and a voltage Esinωt is given by L (di/dt)+Ri=E sinωt. If i=0 at t=0, find i.(6 marks) 6(b) Evaluate ∫∫e^(2x-3y)dxdy over the triangle bounded by x+y=1, x=1 and y=1(6 marks) 6(c)(1) Find the volume of solid bounded by the surfaces y²=4ax, x²=4ay and the planes z=0, z=3. (4 marks) 6(c)(2) Change to polar co-ordinates and evaluate:
(4 marks)