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MU First Year Engineering (Semester 2)
Applied Mathematics 2
May 2014
Total marks: --
Total time: --
INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

1(a) Evaluate $$\displaystyle \int_{0}^{1}\sqrt{(x-x)}dx$$
3 M
1(b) Solve D4-4D3-8D+4]y=0
3 M
1(c) Prove that (1+Δ)(1-∇)=1
3 M
1(d) Change to polar co-ordinates and evaluate $$\displaystyle \int_{0}^{a}\int_{0}^{\sqrt{x^{2}-x^{2}}}dydx$$
3 M
1(e) Solve(X4-4xy -2y2)dx+(y4-4xy -2x2)dy =0
4 M
1(f) Evaluate $\int_{0}^{2}\int_{0}^{\sqrt{1+x^{2}}}\dfrac{1}{1-x^{2}-y^{2}}dy\ dx$
4 M

2(a) Solve $xy \left(1+xy^{2}\dfrac{dy}{dx}\right)=1$
6 M
2(b) Change the order of integration and evaluate $\int_{0}^{\alpha}xe^{-x^{2}/2}dy\ dx$
6 M
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 M

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 M
3(b) Find the area bounded by parabola y2=4x and the line y = 2x-4.
6 M
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 M

4(a) find the total length of the loop of the curve $9y^{2}=(x+7)(x+4)^{2}$
6 M
4(b) Solve $\dfrac{d^{2}y}{dx^{2}}+2y -x^{2}e^{3x}+e^{x}-\cos 2x$
6 M
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 M

5(a) Solve y(X2y + ex)dx-exdy =0.
6 M
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 M
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 M

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 M
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 M
6(c) Find the volume bounded by y2 =x, x2 =y and the planes z=0 and x+y+z=1.
8 M

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