Applied Mathematics 2 : Question Paper Dec 2011 - First Year Engineering (Semester 2)

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Applied Mathematics 2 - Dec 2011

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)

prove that:$\Gamma({\dfrac {3}{4}-x}).\Gamma({\dfrac {3}{4}+x})=(\dfrac{1}{4}-x^2)\pi\sec x.$ Provided -1<2x<1

(5 marks) 1(b) Solve (D⁴ - 4D³ + 8D² - 8D + 4)y = e^x + 1.(5 marks) 1(c) Find the length of the curve y = log(e^x + 1) - log(e^x - 1) from x=1 to x=2.(5 marks) 1(d) Find the area bounded by the curves : xy =2, 4y = x², y=4(5 marks) 2(a) Change the order of integration:

(6 marks) 2(b) Solve by the method of variation of parameters:

(6 marks) 2(c) Solve dy/dx = 2 + (xy)^xy with x₀ = 1.2 and y₀ =1.6403 by Euler's modifies formula for x = 1.6. Correct the four places of decimal. Take h = 0.2(8 marks) 3(a) Evaluate:

(6 marks) 3(b) Change to polar co-ordinates:

(6 marks) 3(c) Solve the differential equation dy/dx = x + y², y(0) = 1 by Runge-Kutta method of fourth order, for the interval (0,0.2) in steps of h =0.1(8 marks) 4(a) Solve (D² - 2D + 1)y = xe^xsin(x)(6 marks) 4(b) Evaluate:

(6 marks) 4(c) Solve:

(8 marks) 5(a) Solve dy/dx = e^x-y(e^x-e^y)(6 marks) 5(b) Solve:

(6 marks) 5(c) Find the volume of the tetrahedron bounded by the planes x = 0, y = 0, z=a,x + y + z = a(8 marks) 6(a) Find the mass of the lamina bounded by the curves ay² = x³ and the line by=x, if the density at a point varies as the distance of the point from the x-axis. (6 marks) 6(b) Using Duplication Formula prove:

(6 marks) 6(c) Solve: (D² - 1)y = x²sin(3x).(8 marks) 7(a) Evaluate ∫ ∫ (x² + y²)dxdy over the area of a triangle whose vertices are (0,1) (1,1) and (1,2).(6 marks) 7(b) Solve the following:

(6 marks) 7(c) Evaluate the following:

and hence deduce that:

(8 marks)

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