Applied Mathematics 2 - May 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 (D2-1)(D-1)2y=0(3 marks) 1(c) Prove that E=1+Δ=ehD(3 marks) 1(d) Solve the following:
(3 marks) 1(e) Change into polar co-ordinates and evaluate:
(4 marks) 1(f) Evaluate the following:
(4 marks) 2(a) Solve (x3y3-xy)dy=dx(6 marks) 2(b) Change the order of Integration and evaluate:
(6 marks) 2(c)(1) Prove that:
(4 marks) 2(c)(2) Evaluate , where a>0(4 marks) 3(a) Evaluate the following:
(6 marks) 3(b) Find the area bounded by 9xy=4 and 2x+y=2(6 marks) 3(c)(1) Solve the following:
(4 marks) 3(c)(2) Solve the equation by variation of parameters:
(4 marks) 4(a)
Show that for the parabola
from θ=0 to θ=π/2, length of the arc is
(6 marks) 4(c) Apply Runge-Kutta method of fourth order to find an approximation value of y at x=0.2, if dy/dx=x+y2, given y=1 when x=0, in steps of h=0.1(8 marks) 5(a) Solve: (2xy4ey+2xy3+y)dx+(x2y4ey-x2y2-3x)dy=0(6 marks) 5(b) Solve dy/dx=2x+y with x0=0,y0=0 by Taylor’s method. Obtain y as a series in power of x. Find approximation value of y for x=0.2,0.4. Compare your result with exact values.(6 marks) 5(c) Evaluate the following equation
by Trapezoidal method, Simpson's 1/3rd and 3/8th methods. Compare result with exact values.(8 marks) 6(a) In a circuit containing inductance L, resistance R, voltage E, the current i is given by L(di/dt)+Ri=E. Find i at a time t if at t=0,i=0, and if L, R, E are constants.(6 marks) 6(b) Evaluate ∫∫xy dxdy bounded by y=x, x2+y2-2x=0, and y2=2x(6 marks) 6(c)(1) Find the volume of a tetrahedron bounded by the plane x=0,y=0,z=0 and x+y+z=a(4 marks) 6(c)(2) Find the volume bounded by the cone z2=x2+y2 and paraboloid z=x2+y2.(4 marks)