Applied Mathematics 2 - May 2014

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 $\displaystyle \int_{0}^{1}\sqrt{(x-x)}dx$(3 marks) 1(b) Solve D⁴-4D³-8D+4]y=0(3 marks) 1(c) Prove that (1+Δ)(1-∇)=1(3 marks) 1(d) Change to polar co-ordinates and evaluate $\displaystyle \int_{0}^{a}\int_{0}^{\sqrt{x^{2}-x^{2}}}dydx$(3 marks) 1(e) Solve(X⁴-4xy -2y²)dx+(y⁴-4xy -2x²)dy =0 (4 marks) 1(f) Evaluate $$\int_{0}^{2}\int_{0}^{\sqrt{1+x^{2}}}\dfrac{1}{1-x^{2}-y^{2}}dy\ dx$$(4 marks) 2(a) Solve $$xy \left(1+xy^{2}\dfrac{dy}{dx}\right)=1$$(6 marks) 2(b) Change the order of integration and evaluate $$\int_{0}^{\alpha}xe^{-x^{2}/2}dy\ dx $$(6 marks) 2(c) Evaluate $$\int_{0}^{\pi/2}\dfrac{dx}{a^{2}sin^{2}x+b^{2}cos^{2}x}$$ and show that
$$\int_{0}^{\pi/2}\dfrac{dx}{(a^{2}sin^{2}x+b^{2}cos^{2}x)^{2}}=\dfrac{pi}{4ab}\left(\dfrac{1}{a^{2}+\dfrac{1}{b^{2}}}\right)$$(8 marks) 3(a) Evaluate $$\iiint x^{2}yz\ dx\ dx\ dz$$ through the volume bounded by x=0, y=0, z=0. x+y+z =1(6 marks) 3(b) Find the area bounded by parabola y²=4x and the line y = 2x-4.(6 marks) 3(c) Use the method of variation of parameter to solve
$$\dfrac{d^{2}y}{dx^{2}} + 5\dfrac{dy}{dx}+6y =e^{-2x}\sec^{2} x+1 (1+2\tan x)$$(8 marks) 4(a) find the total length of the loop of the curve $$9y^{2}=(x+7)(x+4)^{2}$$(6 marks) 4(b) Solve $$\dfrac{d^{2}y}{dx^{2}}+2y -x^{2}e^{3x}+e^{x}-\cos 2x$$(6 marks) 4(c) Apply Runge-kutta method of fourth order to find an approximate value of y at x=0.2 if $$\dfrac{dy}{dx}=x+y^{2} $$ given that y=1 when x=0 in step of h=0.1(8 marks) 5(a) Solve y(X²y + e^x)dx-e^xdy =0.(6 marks) 5(b) using taylor's series method solve $$\dfrac{dy}{dx}=1-2xy$$ given that y(0)=0 and hence y=(0,2) and y(0,4).(6 marks) 5(c) Compute the value of the definite integral $$\int_{0.2}^{1.4}(\sin x-log_{e}x+e^{x})dx$$, by
Trapezoidal Rule
Simpson's one third Rule
Simpson's three-eighth Rule.(8 marks) 6(a) The Motion of a particle is given by $$\dfrac{d^{2}x}{dt^{2}}=-k^{2}x-2h\dfrac{dx}{dt}$$solve the equation when h=5, k=4 taking x=0, v=v₀ at t =0. Show that the the time of maximum displacement is independent of the initial velocity.(6 marks) 6(b) Evaluate$$ \iint(x^{2}+y^{2})dx\ dy $$ over the area of triangle whose vertices are (0,0),(1,0)(1,2).(6 marks) 6(c) Find the volume bounded by y² =x, x² =y and the planes z=0 and x+y+z=1.(8 marks)