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).
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(i) | Find the area of the triangle PQR.
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(ii) | Find the equation of the plane through
P, Qand R.
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(b) | Find the distance of the point (2,5,-2) from the plane
3x + 2y - 6z = 1.
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(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
.
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(b) | Find the line of intersection of the planes
x-y-2z = 4 and
2x
+ y - z = 2.
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(c) | Find the distance between the lines
(x,y,z) = (1,-2,1) +
t(1,0,-3) and
.
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(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.
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(6) | 4. | A conic is given by the polar equation
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Let and . Find the value of . | ||||

(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? | ||

Find the tangent plane and normal line to the
surface
x sinz + ye = 4
at (3,4,0).
^{z} | ||||

Find the rate of change of
f = x^{2} + 2y^{2} - z^{2}
+ 4xyz in the direction of
at the
point (-1,2,1).
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Classify the critical points of the function
f
= 2x^{2} + y^{2} - xy^{2} + 4.
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Using the method of Lagrange multipliers, find
the minimum of
f = x^{2} + y^{2} + (z-5)^{2} subject to xy-z = 0.
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Reverse the order of integration and evaluate . | ||||

Change the integral to polar coordinates and evaluate . | ||||

(8) | 13. | Using spherical coordinates, find the volume of
the solid bounded by the cone
and the plane z =
2.
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(100) |