DEPARTMENT OF MATHEMATICS & STATISTICS

MATH 2513

DECEMBER 1998 TIME: 3 HOURS
 

NO CALCULATORS PERMITTED

MARKS
(9) 1. (a) Let P,Q,R be the points (1,3,4),(0,-1,2),(-2,1,8).
(i) Find the area of the triangle PQR.
(ii) Find the equation of the plane through P, Qand R.
(b) Find the distance of the point (2,5,-2) from the plane 3x + 2y - 6z = 1.
(15)2. (a) Find the equation of the plane containing the line x = 2 + 3t; y = -1 + t; z = t and parallel to the line ${\displaystyle \frac{x-1}{2} = \frac{y+1}{-2} = \frac{z+3}{-1}}$.
(b) Find the line of intersection of the planes x-y-2z = 4 and 2x + y - z = 2.
(c) Find the distance between the lines (x,y,z) = (1,-2,1) + t(1,0,-3) and ${\displaystyle \frac{x+1}{2} = \frac{y}{3}\;;\;z = 1}$.
(6)3. Find the equation of the surface consisting of all the points that are equidistant from the point (1,0,0) and the plane x = -1. Identify and sketch the surface.
(6)4. A conic is given by the polar equation
\begin{displaymath}r = \frac{10}{3 - 2 \cos \theta}.
\end{displaymath}
Identify and sketch the conic. Give the equation of the directrix.
(6)5. Let ${\displaystyle w = \frac{x^2y}{z},\;x =
r \cos \theta,\;y =
r \sin \theta}$ and $z = e^{r \theta}$. Find the value of ${\displaystyle
\frac{\partial w}{\partial \theta}}$.
(6)6. The radius and height of a cylinder are 20 cm. and 30 cm. respectively. If they are increased by .04 cm (both radius and height), what is the volume of the cylinder after the increase?
(6)7. Find the tangent plane and normal line to the surface x sinz + yez = 4 at (3,4,0).
(6)8. Find the rate of change of f = x2 + 2y2 - z2 + 4xyz in the direction of $2 \vec{i} - \vec{j} + 2 \vec{k}$ at the point (-1,2,1).
(8)9. Classify the critical points of the function f = 2x2 + y2 - xy2 + 4.
(8)10. Using the method of Lagrange multipliers, find the minimum of f = x2 + y2 + (z-5)2 subject to xy-z = 0.
(8)11. Reverse the order of integration and evaluate ${\displaystyle \int_0^1 \int_0^{\sqrt{1-x}} xy^2\;dy\;dx}$.
(8)12. Change the integral to polar coordinates and evaluate ${\displaystyle \int_0^2 \int_0^{\sqrt{4-x^{2}}}
\sqrt{5-x^2-y^2}\;dy\;dx}$.
(8)13. Using spherical coordinates, find the volume of the solid bounded by the cone $z = \sqrt{x^2+y^2}$ and the plane z = 2.

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