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Unformatted text preview: MA 162 FINAL EXAM PRACTICE PROBLEMS Spring 2011 1. Find the angle between the vectors v = 2 i + 2 j + k and w = 2 i + 2 j k . A. cos 1 8 9 B. cos 1 5 9 C. cos 1 2 3 D. cos 1 7 9 E. cos 1 1 3 2. Find a such that u = 2 i j + a k and v = i + 4 j + 2 k are perpendicular. A. 3 B. 2 C. 1 D. 1 E. 2 3. If w = w 1 i + w 2 j + w 3 k is perpendicular to u = i + j k and v = 2 i + j + k , and if w 3 = 2, then w 1 = A. 4 B. 2 C. 2 D. 4 E. 1 4. If v = i + j + k and w = 2 i k , find  proj v ( w )  . A. 1 / √ 3 B. √ 3 C. √ 3 / 5 D. 2 √ 3 E. √ 3 / 2 5. Find the area of the triangle with vertices P = (0 , , 0) , Q = (1 , 2 , 1), and R = (2 , 1 , 1). A. √ 27 B. √ 27 2 C. √ 11 2 D. √ 19 E. √ 3 2 6. The radius of the sphere x 2 + y 2 + z 2 + 2 x + 4 y 6 z = 3 is A. 3 + √ 13 B. √ 13 C. √ 65 D. 3 + √ 56 E. √ 17 7. The area of the region enclosed by the curves y = x 2 + 1 and y = 2 x + 9 is given by A. Z 4 2 ( x 2 + 1 2 x 9) dx B. Z 4 2 (2 x + 9 x 2 1) dx C. Z 2 2 (2 x + 9 x 2 1) dx D. Z 2 4 (2 x + 9 x 2 1) dx E. Z 2 4 ( x 2 + 1 2 x 9) dx 8. Let R be the region between the graphs of y = x 2 and y = x . Find the volume of the solid generated by revolving R about the x –axis. A. π 6 B. π 12 C. π 4 D. π 15 E. 2 π 15 9. If the region in problem 8 is revolved about the y –axis, then the volume of the solid is A. π 6 B. π 12 C. π 24 D. 2 π 15 E. π 15 10. If R is the region bounded by the curves x = 0 and x = y y 2 , and if R is revolved around the y –axis, then the volume of the solid is A. π 15 B. π 30 C. π 12 D. π 3 E. π 6 1 11. A force of 4 lb. is required to stretch a spring 1 / 2 ft. beyond its natural length. How much work is required to stretch the spring from its natural length to 2 ft. A. 8 ft–lbs. B. 12 ft–lbs. C. 16 ft–lbs. D. 24 ft–lbs. E. 32 ft–lbs. 12. A cylindrical tank of height 4 feet and radius 1 foot is filled with water. How much work is required to pump all but 1 foot of water out of the tank. (Density = 62.5 lbs./ft 3 ) A. 9 π (62 . 5) ftlbs. B. 3 π (62 . 5) ftlbs. C....
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This document was uploaded on 01/21/2012.
 Spring '09
 Vectors

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