STUPIDSID
1 (a)
Evaluate \( \int^2_0 x^2 (2-x)^3 dx \)
3 M
1 (b)
Solve \( \dfrac {d^3y}{dx^3}- 6 \dfrac {d^2y}{dx^2}+11 \dfrac {dy}{dx} - 6y = 0 \)
3 M
1 (c)
Prove that E=1+Δ
3 M
1 (d)
Solve \( \left [ y \left ( 1+ \dfrac {1}{x} \right )+ \text{cosy} \right ] dx + (x+ {\log x} - x\sin y)dy=0 \)
3 M
1 (e)
Change to polar coordinates and evaluate \( \int^a_0 \int^{\sqrt{a^2 - x^2}}_0 (x^2 + y^2) dy \ dx \)
4 M
1 (f)
Evaluate \( \int^1_0 \int^x_0 x\ y\ dy \ dx \)
4 M
2 (a)
Solve \( \dfrac {dy}{dx} + \dfrac {4x}{x^2 +1} y = \dfrac {1}{(x^2 +1)^3} \)
6 M
2 (b)
Change the order of integration and evaluate \[ \int^2_0 \int^2_{\sqrt{2x}} \dfrac {y^2 \ dx \ dy}{\sqrt{y^2 - 4x^2}} \]
6 M
2 (c)
Prove that \( \int^{\pi / 2}_0 \dfrac {\log (1+ a \sin ^2 x) }{\sin^2 x}dx = \pi \big [ \sqrt{a+1}-1 \big ] a>-1 \)
8 M
3 (a)
Evaluate \( \int^1_0 \int^{1-y}_0 \int^{1-x-y}_0 \dfrac {1}{(x+y+z+1)^3} dz \ dy \ dx \)
6 M
3 (b)
Find by double integration the area enclosed by the curve
9xy=4 and the line 2x+y=2.
9xy=4 and the line 2x+y=2.
6 M
3 (c)
Using method of Variation of Parameter solve \( \dfrac {d^2 y}{d x^2} + a^2 y = \sec ax \)
8 M
4 (a)
Find the perimeter of the cardioide r=a (1+ cos θ).
6 M
4 (b)
Solve (D2+4)y=cos 2x
6 M
4 (c)
Apply Runge-kutta Method of fourth order to find an approximate value of y for \( \dfrac {dy}{dx} = \dfrac {1}{x+y} \) with x0 = 0, y0=1 at x=1 taking h=0.5.
8 M
5 (a)
Solve (y-x y2) dx - (x+x2y) dy = 0.
6 M
5 (b)
Using Taylor Series Method obtain the solution of following differential equation \( \dfrac {dy}{dx}= 1+ y^2 \) with y0=0 when x0=0 for x=0.2.
6 M
5 (c)
Find the approximate value of \( \int^6_0 e^x dx \) by i) Trapezoidal Rule, ii) Simpson's 1/3rd Rule, iii) Simpson's 3/8th Rule.
8 M
6 (a)
A resistance of 100 Ohm and inductance of 0.5 Henry are connected in series with a battery of 20 Volt. Find the current at any instant if the relation between L. \( R, \ E \ is \ L\dfrac {di}{dt} + Ri = E \)
6 M
6 (b)
\( \int \int y \ dx \ dy \) over the area bounded by the x=0, y=x2, x+y=2.
6 M
6 (c)
Find the volume bounded by the paraboloid x2+y2=az and the cylinder x2+y2=a2.
8 M
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