STUPIDSID
1(a)
Evaluate \[\displaystyle \int_{0}^{2} x^{4}(8-x^{3})^{-1/3}dx\]
3 M
1(b)
Solve \(\dfrac{d^{4}y}{dx^{4}}+2\dfrac{d^{2}y}{dx^{2}}+y =0\)
3 M
1(c)
Prove that E = 1+Δ = e4D
3 M
1(d)
Solve \([x\sqrt{x^{2}+y^{2}}-y]dx +[y\sqrt{x^{2}+y^{2}}-x]dy =0\)
3 M
1(e)
Change to polar coordinates and evaluate \(\displaystyle \int_{0}^{2a} \int_{0}^{\sqrt{2ax-x^{3}}} \dfrac{x}{\sqrt{x^{2}+y^{2}}}dy\ dx\)
4 M
1(f)
Evaluate\( \displaystyle \int_{a}^{1}\int_{a}^{x} e^{x+y} dydx\)
4 M
2(a)
Solve \(\dfrac{dy}{dx} +x\sin 2y =x^{3}\cos^{2}y \)
6 M
2(b)
Change the order of integration and evaluate \(\displaystyle \int_{0}^{a}\int_{\frac{y^{2}}{a}}^{y} \dfrac{y}{(a-x)\sqrt{ax-y^{2}}}dxdy\)
6 M
2(c)
prove that using
\(\displaystyle \int_{0}^{\infty}cis\lambda(e^{-ax}-e^{-bx})dx=\dfrac{1}{2} log\left(\dfrac{b^{2}+\lambda^{2}}{a^{2}+\lambda^{2}}\right), a>o ,b>0 \)
DUIS rule
8 M
3(a)
Evaluate \(\displaystyle \iiint\dfrac{dx\ dy\ dy}{x^{2}+y^{2}+z^{2}}\) throughout the volume of the sphere \(x^{2}-y^{2} +z^{2} =a^{2}\)
6 M
3(b)
Find the area common to the cardiods r=a(1+cosθ) and r = a(a-cosθ).
6 M
3(c)
Apply the method of variation of parameters to solve \(\dfrac{d^{2}y}{dx^{2}}-4\dfrac{dy}{dx}+4y =e^{2x}sec^{2}x\)
8 M
4(a)
Find the length of one are of the cycloid x =a (θ -sinθ)and y = a(a+cos θ)
6 M
4(b)
Solve \(\dfrac{d^{2}y}{dx^{2}} +2y =x^{2}e^{3x} + e^{x} \cos x \)
6 M
4(c)
Appply Runge-Kutla method of fourth order to find an approximate value of y at x =1.2 if \(\dfrac{dy}{dx} =x^{2} +y^{2}\), given that y = 1.5 when x=1 choosing h= 0.1
8 M
5(a)
Solve \(\left [xy^{2}-e^{\frac{1}{x^{3}}}\right]dx-yx^{2}dy =0\)
6 M
5(b)
If y satisfies the equation \(\dfrac{dy}{dx} =x^{2}y-1\) and with y=1 when x=0, using Taylor's series method for y about x=0, find y when x =0.1 and x=0.2
6 M
5(c)
Compute the value of the definite integral\( \displaystyle \int_{-1}^{1}\dfrac{dx}{1+x^{2}}\),by using
Trapezoidal Rule
Simpson's (1/3)rd Rule
Simpson's (3/8)th Rule.
Trapezoidal Rule
Simpson's (1/3)rd Rule
Simpson's (3/8)th Rule.
8 M
6(a)
A radial displacement 'u' in rotating a disc at a distance 'r' from the axis in given by \(\dfrac{d^{2}u}{dr^{2}}+\dfrac{1}{r}\dfrac{du}{dr}-\dfrac{u}{r^{2}}+kr=0.\) find the displacement given u=0 when r=0 and r=a
6 M
6(b)
Evaluate\(\displaystyle \iint x^{2} dsdy \) over the region bounded by xy = a2, x=2a, y=0 and y=x in the first quadrant.
6 M
6(c)
Find the volume of the tetrahedron bounded by the co-ordinates plane and the plane \(\dfrac{x}{2}+\dfrac{y}{3}+\dfrac{z}{4} =1\)
8 M
More question papers from Applied Mathematics 2