Applied Mathematics 2 : Question Paper May 2012 - First Year Engineering (Semester 2)

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Applied Mathematics 2 - May 2012

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:
(5 marks) 1(b) Solve the following:
(5 marks) 1(c) Solve the following:
(5 marks) 1(d) Find by double integration the area enclosed by y² = x³, y = x.(5 marks) 2(a) Solve (4xy + 3y² - x) dx +x(x+2y)dy(6 marks) 2(b) Change the order of integration:
(6 marks) 2(c) Prove that:

Hence evaluate:
(8 marks) 3(a) Using Euler's method find approximate value of y at x=1 in five steps taking h=0.2 given dy/dx = x + y, and y(0) = 1.(6 marks) 3(b) Evaluate the following
(6 marks) 3(c) Solve the following:
(8 marks) 4(a) Show that the following holds true: :
(6 marks) 4(b) Evaluate the following, where R is the region bounded by y²=ax and y = x.
$$\displaystyle\int\limits_R\displaystyle\int \dfrac{y\ dx\ dy}{(a-x)\sqrt{ax-y^2}}$$(6 marks) 4(c) Solve by method of variation of parameters (D² - 2D + 2)y = e^xtan(x)(8 marks) 5(a) Solve the following: (D² + 2)y = e^xcos(x) + x²e^3x

(6 marks) 5(b) Using Taylor's Method solve the following: dy/dx = x² - y with y(0) = 1. Also find y at x - 0.1.(6 marks) 5(c) Find the Volume of the Tetrahedron bounded by the planes: x = 0, y = 0, z = 0 and x + y + z = a(8 marks) 6(a) In a single closed circuit, the current i at any time t, is given by: R i + L (di/dt) = E.
Find the current i at a time t if at t=0, i=0 and L, R, E are constants.(6 marks) 6(b) Find the mass of the octant of the ellipsoid x²/a² + y²/b² + z²/c² =1

, the density at any point being kxyz.(6 marks) 6(c) Using Runge-Kutta's Fourth order method find y at x = 0.2 if dy/dx = x + y² given that y = 1, when x = 0 in steps of h = 0.1.(8 marks) 7(a) State and prove Duplication formula for gamma functions.(6 marks) 7(b) Find the length of the cardiode r = a(1 + cosθ) which lies outside the circle r + acosθ = 0(6 marks) 7(c) Solve the following:

(8 marks)

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