STUPIDSID
1(a)
Prove that \[\int ^1_0\frac{dx}{\sqrt{-\log x}}=\sqrt{\pi}\]
3 M
1(b)
Solve \[\dfrac{d^3y}{dx^3}-5\dfrac{d^2y}{dx^2}+8\dfrac{dy}{dx}-4y=0\]
3 M
1(c)
Prove that Δ∇=Δ&nabla
3 M
1(d)
Solve [xy sin(xy) + cos (xy)] y dx+[x ysin(xy) - cos (xy)]x dy=0
3 M
1(e)
Change to polar coordinates and evaluate \[\displaystyle \int ^1_0\int ^{\sqrt{2x-x^2}}_x \left ( x^2+y^2 \right )dx\ dy\]
4 M
1(f)
Evaluate \[\int ^1_0\int ^x_0\left ( x^2+y^2 \right )xdy\ dx\]
4 M
2(a)
Solve \[\left ( 1+y^2 \right )dx=\left ( e^{\tan^{-1}y}-x \right )dy\]
6 M
2(b)
Change the order of integration and evaluate \[\int ^1_0\int ^{\sqrt{1-x^2}}_0\frac{e^y}{\left ( e^y+1 \right )\sqrt{1-x^2-y^2}}dy\ dx\]
6 M
2(c)
Prove that \[\int ^{\infty}_0\dfrac{e^{-x}-e^{-ax}}{xsecx}dx=\dfrac{1}{2}\log\left ( \dfrac{a^2+1}{2} \right )\]
8 M
3(a)
Evaluate \[\int ^e_1\int ^{\log y}_1\int ^{e^x}_1\log z\ dz\ dy\ dx\]
6 M
3(b)
Find the total area od the curve r = a sin2?
6 M
3(c)
Solve \[x^2\dfrac{d^3y}{dx^3}+3x\dfrac{d^2y}{dx^2}+\dfrac{dy}{dx}=x^2\log x\]
8 M
4(a)
Show that thw length of the arc of the curve ay2 = x3 from the origin to the point whose abscissa is b is \[\dfrac{8a}{27}\left [ \left ( 1+\dfrac{9b}{4a} \right )^{3/2} -1\right ]\]
6 M
4(b)
Solve \( \left ( D^2-D-2 \right )y=2 \log x+\dfrac{1}{x}+\dfrac{1}{x^2} \)
6 M
4(c)
Apply Runge-Kutta Method of fourth order to find an approximate value of y for \( \dfrac{dy}{dx}=x\ y \) with x0=1, y0 =1 at x=1.2 taking h=0.1
8 M
5(a)
Solve \( \left ( x^2y-2xy^2 \right )dx-(x^3-3x^2y)dy=0 \)
6 M
5(b)
Using Taylor series Method obtain the solution of following differential equation \( \dfrac{dy}{dx}=2y+3e^x \) with y0 = 0 when x0 = 0 for x=0.1, 0.2
6 M
5(c)
Find the approximate value of \( \displaystyle \int ^4_0 e^x dx \)
by i) Trapezoidal Rule, ii) Simpson's 1/3rd Rule
by i) Trapezoidal Rule, ii) Simpson's 1/3rd Rule
8 M
6(a)
In a circuit containing inductance L, resistance R, and voltage E, the current I is given by \( L\dfrac{di}{dt}+Ri=E. \) Find the current i at time t if at t=0, i=0 and L, R, E are constants.
6 M
6(b)
Evaluate \( \iint _R\dfrac{dxdy}{\left ( 1+x^2+y^2 \right )^2} \) over one loop of the lemniscate \( \left ( x^2+y^2 \right )^2=x^2-y^2 \)
6 M
6(c)
Find the volume bounded by the cylinder x2 + y2 = 4 and the planes z=0 and y+z=4
8 M
More question papers from Applied Mathematics 2