## 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 (D^{4}+4)y=0(3 marks)
**1(c)** Prove that E∇ = Δ =∇E(3 marks)
**1(d)** Solve the following:

(3 marks)
**1(e)** Evaluate ∫∫r^{3}drdθ over the regions between the circles r=2sinθ and r=4sinθ(4 marks)
**1(f)** Evaluate the following:

(4 marks)
**2(a)** Solve (x^{3}y^{4}+x^{2}y^{3}+xy^{2}+y)dx + (x^{4}y^{3}-x^{3}y^{2}-x^{2}y+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 r^{2}=a^{2}cos2θ(6 marks)
**3(c)** Solve (D^{3}+2D^{2}+D)y = x^{2}e^{3x}+sin^{2}x+2^{x}(8 marks)
**4(a)** Show that the length of the arc of the parabola y^{2}=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+xy^{2}=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 y^{2}=4ax, x^{2}=4ay and the planes z=0, z=3.
(4 marks)
**6(c)(2)** Change to polar co-ordinates and evaluate:

(4 marks)