1 (a)
(i) Find the intervals on which the function f(x) = x

(ii) Graph the set of points whose polar coordinates satisfy the conditions \[ 1 \le r \le 2 \ and \ 0 \le \theta \le \dfrac {\pi}{2} \]

^{3}- 27x is increasing and decreasing.(ii) Graph the set of points whose polar coordinates satisfy the conditions \[ 1 \le r \le 2 \ and \ 0 \le \theta \le \dfrac {\pi}{2} \]

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1 (b)
Which of following series converge and which diverge? \[ (i) \ \sum^{\infty}_{n=1}\dfrac {2n+1}{n^2+2n+1} \\ (ii) \ \sum^{\infty}_{n=1}e^{-n}\]

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1 (c)
For what values of x do the power series \[ \sum^\infty_{n-1}\dfrac {(-1)^{n-1}x^{2n-1}}{2n-1} \ converge? \]

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2 (a)
\[ (i) \ Evaluate \ \lim_{x\rightarrow \frac{\pi}{2}}\dfrac {2x-\pi}{\cos x} \]

(ii) Find the Maclaurin's series for e

(ii) Find the Maclaurin's series for e

^{x}.
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2 (b)
Find the Taylor polynomial of order 3 generated by \[ f(x)=\sqrt{x} \ at a=4 \]

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2 (c)
Sketch the curve y=|x

^{2}-1|
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3 (a)
(i) Obtain reduction formula that expresses the integral ∫(ln x)

\[ (ii) \ Find \ \dfrac {dy}{dx} \ if \ y=\int^5_x 3t\sin tdt \]

^{x}dx in terms of an integral of a lower power of (ln x).\[ (ii) \ Find \ \dfrac {dy}{dx} \ if \ y=\int^5_x 3t\sin tdt \]

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3 (b)
State Leibniz's Rule. Use Leibniz's Rule to find the derivative of \[ g(y)=\int^{2\sqrt{y}}_{\sqrt{y}}\sin^2 tdt. \]

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3 (c)
\[ Evaluate \\ (i) \int^{\frac{\pi}{4}}_{0}\sin^7 2\theta d\theta \\(ii) \ \int^{1}_{0}\dfrac {x^7}{\sqrt{1-x^4}}dx \]

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4 (a)
\[ Evaluate \ \int^{3}_0\dfrac {dx}{x-1} \ if \ possible. \\Evaluate \ \int^{\infty}_{-\infty}\dfrac {dx}{1+x^2} \]

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4 (b)
Find the volume of the solid generated by revolving the region between the parabola x = y

^{2}and the line x = 1 about line x = 1.
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4 (c)
Use the shell method to find volume of solid generated by revolving the region bounded by y = x , x = 0 and y =1 about x-axis.

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5 (a)
\[ (i) \ Show \ that \ \lim_{(x,y)\rightarrow (0,0)}\dfrac {xy}{x^2+y^2} \ does \ not \ exist. \]

(ii) Determine set of all points at which the function \[ f(x,y)= \dfrac {x^2+Y^2}{x-y} \ is \ continuous. \]

(ii) Determine set of all points at which the function \[ f(x,y)= \dfrac {x^2+Y^2}{x-y} \ is \ continuous. \]

4 M

5 (b)
Determine whether \[ u(x,y)=ln \sqrt{x^2+y^2} \] is a solution of Laplace's equation.

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5 (c)
\[ if \ f(x,y)=\dfrac {x^{\frac {1}{4}}+y^{\frac {1}{4}}}{x^{\frac {1}{5}}+y^{\frac {1}{5}}} \ then \ find \ x\dfrac {\partial f}{\partial x}+y\dfrac {\partial f}{\partial y} \ and \\x^2\dfrac {\partial^2f}{\partial x^2}+2xy\dfrac {\partial^2f}{\partial x \partial y}+ y^2 \dfrac {\partial^2f}{\partial y^2} \]

5 M

6 (a)
Find equation for the tangent plane and normal line at point (2,0,2) on the surface 2z - x

^{2}= 0
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6 (b)
Find all local maxima, local minima and saddle point off(x,y) = x

^{2}+ y^{2}+ 4x + 6y + 13.
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6 (c)
Suppose that the Celsius temperature at the point (x,y,z) on the sphere x

^{2}+y^{2}+ z^{2}=1 is T = 400xyz^{2}. Locate the highest and the lowest temperature on the sphere.
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7 (a)
\[ (i) \ Evaluate \ \int^2_1\int^1_0 (1+3xy)dxdy \\(ii) \ \int^1_{-1}\int^{2}_{0}\int^{1}_{0}xz-y^3 \ dzdydx \]

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7 (b)
Find the volume of the solid under the cone \[ z=\sqrt{x^2+y^2} \] and above the disk x

^{2}+ y^{2}≤ 4.
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7 (c)
A solid E lies within the cylinder x

^{2}+ y^{2}= 1, below the plane z = 4 and above the paraboloid z = 1 - x^{2}- y^{2}. The density at any point is proportional to its distance from the axis of the cylinder. Find mass of E.
5 M

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