STUPIDSID
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
\[\int_{0}^{\pi/2}\dfrac{dx}{(a^{2}sin^{2}x+b^{2}cos^{2}x)^{2}}=\dfrac{pi}{4ab}\left(\dfrac{1}{a^{2}+\dfrac{1}{b^{2}}}\right)\]
\[\int_{0}^{\pi/2}\dfrac{dx}{(a^{2}sin^{2}x+b^{2}cos^{2}x)^{2}}=\dfrac{pi}{4ab}\left(\dfrac{1}{a^{2}+\dfrac{1}{b^{2}}}\right)\]
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
\[\dfrac{d^{2}y}{dx^{2}} + 5\dfrac{dy}{dx}+6y =e^{-2x}\sec^{2} x+1 (1+2\tan x)\]
\[\dfrac{d^{2}y}{dx^{2}} + 5\dfrac{dy}{dx}+6y =e^{-2x}\sec^{2} x+1 (1+2\tan x)\]
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.
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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