1(a)
Evaluate the following:

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1(b)
Solve the following:

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1(c)
Solve the following:

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1(d)
Find by double integration the area enclosed by y

^{2}= x^{3}, y = x.
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2(a)
Solve (4xy + 3y

^{2}- x) dx +x(x+2y)dy
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2(b)
Change the order of integration:

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2(c)
Prove that:

Hence evaluate:

Hence evaluate:

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3(a)
Using Euler's method find approximate value of y at x=1 in five steps taking h=0.2 given dy/dx = x + y, and y(0) = 1.

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3(b)
Evaluate the following

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3(c)
Solve the following:

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4(a)
Show that the following holds true: :

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4(b)
Evaluate the following, where R is the region bounded by y

\[\displaystyle\int\limits_R\displaystyle\int \dfrac{y\ dx\ dy}{(a-x)\sqrt{ax-y^2}}\]

^{2}=ax and y = x.\[\displaystyle\int\limits_R\displaystyle\int \dfrac{y\ dx\ dy}{(a-x)\sqrt{ax-y^2}}\]

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4(c)
Solve by method of variation of parameters (D

^{2}- 2D + 2)y = e^{x}tan(x)
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5(a)
Solve the following: (D

^{2}+ 2)y = e^{x}cos(x) + x^{2}e^{3x}
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5(b)
Using Taylor's Method solve the following: dy/dx = x

^{2}- y with y(0) = 1. Also find y at x - 0.1.
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5(c)
Find the Volume of the Tetrahedron bounded by the planes: x = 0, y = 0, z = 0 and x + y + z = a

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6(a)
In a single closed circuit, the current i at any time t, is given by: R i + L (di/dt) = E.

Find the current i at a time t if at t=0, i=0 and L, R, E are constants.

Find the current i at a time t if at t=0, i=0 and L, R, E are constants.

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6(b)
Find the mass of the octant of the ellipsoid x

^{2}/a^{2}+ y^{2}/b^{2}+ z^{2}/c^{2}=1, the density at any point being kxyz.
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6(c)
Using Runge-Kutta's Fourth order method find y at x = 0.2 if dy/dx = x + y

^{2}given that y = 1, when x = 0 in steps of h = 0.1.
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7(a)
State and prove Duplication formula for gamma functions.

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7(b)
Find the length of the cardiode r = a(1 + cosθ) which lies outside the circle r + acosθ = 0

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7(c)
Solve the following:

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