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}{CSTYLE "" -1 574 "" 1 18 128 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 575 "" 1 18 128 0 128 1 0 1 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier " 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 " Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 1" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 259 1 {CSTYLE "" -1 -1 "T imes" 1 14 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 260 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 261 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 472 29 " Math 2331 Spring 2004 E xam 1" }}{PARA 0 "" 0 "" {TEXT 258 2 " " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 263 1 "1" }{TEXT 336 2 ". " }{TEXT 308 5 "If " }{XPPEDIT 334 1 "u" "6#%\"uG" }{TEXT -1 1 " " }{TEXT 298 6 " and " }{TEXT -1 1 " " }{XPPEDIT 335 1 "v" "6#%\"vG" }{TEXT 299 19 " are vectors and " } {XPPEDIT 332 1 "u=lambda*v" "6#/%\"uG*&%'lambdaG\"\"\"%\"vGF'" }{TEXT 333 21 " for some scalar " }{XPPEDIT 302 1 "lambda" "6#%'lambdaG" }{TEXT 301 31 " then which of the following " }{TEXT 303 4 "must" } {TEXT 304 11 " be true? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 8 "I) " }{XPPEDIT 328 1 "u" "6#%\"uG" } {TEXT -1 30 " has the same direction as " }{XPPEDIT 300 1 "v" "6#% \"vG" }{TEXT -1 14 " \nII) " }{XPPEDIT 329 1 "u" "6#%\"uG" } {TEXT -1 34 " is opposite in direction to " }{XPPEDIT 305 1 "v" " 6#%\"vG" }{TEXT -1 16 " \nIII) " }{XPPEDIT 330 1 "u" "6#%\"uG " }{TEXT -1 25 " is perpendicular to " }{XPPEDIT 306 1 "v" "6#%\"v G" }{TEXT -1 18 " \nIV) " }{XPPEDIT 331 1 "u" "6#%\"uG" } {TEXT -1 28 " has the same length as " }{XPPEDIT 307 1 "v" "6#%\"v G" }{TEXT -1 15 " \nV) " }{XPPEDIT 257 1 "u;" "6#%\"uG" } {TEXT -1 2 " " }{TEXT 473 1 "x" }{TEXT -1 2 " " }{XPPEDIT 256 1 "v*` ` = ` `*0;" "6#/*&%\"vG\"\"\"%\"~GF&*&F'F&\"\"!F&" }{TEXT -1 5 " \n \n" }}{PARA 260 "" 0 "" {TEXT -1 192 "a) I only b) II o nly c) III only d) IV only e) V only\nf) \+ I, IV only g) I, V only h) II, IV only i) II, V only \+ j) III, V only" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 554 14 "Answer: (e)" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 261 "" 0 "" {TEXT 277 4 "2. " }{TEXT -1 29 "If the magnitude of vector " }{XPPEDIT 19 1 " v = `<`*1,-2,c*`>`;" "6%/%\"vG*&%\"GF'" }{TEXT -1 20 " is 3 and if " }{XPPEDIT 19 1 "v " "6#%\" vG" }{TEXT -1 24 " is parallel to\n " }{XPPEDIT 19 1 "`<`*alpha ,4,gamma*`>`;" "6%*&%\"GF%" } {TEXT -1 18 " then what is " }{XPPEDIT 19 1 "c;" "6#%\"cG" }{TEXT -1 2 "? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 309 4 "a) " }{XPPEDIT 310 1 "0" "6#\"\"!" }{TEXT 311 9 " b) " } {XPPEDIT 312 1 "1" "6#\"\"\"" }{TEXT 313 10 " c) " }{XPPEDIT 314 1 "-1" "6#,$\"\"\"!\"\"" }{TEXT 315 7 " d) " }{XPPEDIT 316 1 "2 " "6#\"\"#" }{TEXT 317 10 " e) " }{XPPEDIT 318 1 "-2" "6#,$\"\"# !\"\"" }{TEXT 319 11 " f) " }{XPPEDIT 320 1 "3" "6#\"\"$" } {TEXT 321 11 " g) " }{XPPEDIT 322 1 "-3" "6#,$\"\"$!\"\"" } {TEXT 323 8 " h) " }{XPPEDIT 324 1 "4" "6#\"\"%" }{TEXT 325 8 " \+ i) " }{XPPEDIT 326 1 "-4" "6#,$\"\"%!\"\"" }{TEXT 327 80 " \+ \nj) Insufficient information to determine the answer \+ " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 555 16 "Answer: ( j )" } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "solve(sqrt(1^2 +(- 2)^2+c^2) = 3 , c);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"#!\"#" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "No inform ation is given to determine which of these two values c has." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 260 18 "3. If the vectors " }{TEXT 340 1 " " }{XPPEDIT 542 0 "`<`;" "6#%\"`;" "6#%\">G " }{TEXT 541 3 " " }{TEXT 547 3 "and" }{TEXT 549 1 " " }{TEXT 548 3 " " }{XPPEDIT 544 0 "`<`;" "6#%\"`;" "6#%\">G" }{TEXT 540 38 " are perpendicul ar, then what might" }{TEXT 342 1 " " }{XPPEDIT 343 1 "c;" "6#%\"cG" } {TEXT 339 2 " " }{TEXT 344 4 "be? " }{TEXT 345 2 " " }}{PARA 3 "" 0 "" {TEXT 276 182 "\na) -2/3 b) -1/2 c) -1/3 \+ d) -1/4 e) -1/6 \n f) 0 g) 1/ 6 h) 1/4 i) 1/3 j) 1/2 " } {TEXT 346 7 " \n" }}{PARA 0 "" 0 "" {TEXT 556 16 "Answer: ( e \+ )" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 44 "solve( (1/3)*(-1/3)+(1/2)*(-c)+c*c = 0 , c);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$#\"\"#\"\"$#!\"\"\"\"'" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 261 1 " " }{TEXT 353 36 "4. What is the angle between \+ vectors" }{TEXT 354 3 " " }{XPPEDIT 351 1 "`<`*2,2,1*`>`;" "6%*&%\"< G\"\"\"\"\"#F%F&*&F%F%%\">GF%" }{TEXT 347 1 " " }{TEXT 349 3 "and" } {TEXT 350 2 " " }{XPPEDIT 352 1 "`<`*2,3,6*`>`;" "6%*&%\"GF%" }{TEXT 348 3 " ?" }}{PARA 0 "" 0 "" {TEXT 267 4 "a) " }{XPPEDIT 256 0 "arccos(3/7);" "6#-%'arccosG6#*&\"\"$\"\" \"\"\"(!\"\"" }{TEXT 257 4 " " }{TEXT 355 4 "b) " }{XPPEDIT 256 0 "arccos(5/7);" "6#-%'arccosG6#*&\"\"&\"\"\"\"\"(!\"\"" }{TEXT 474 1 " \+ " }{TEXT 356 7 " c) " }{XPPEDIT 256 0 "arccos(3/11);" "6#-%'arccosG 6#*&\"\"$\"\"\"\"#6!\"\"" }{TEXT 357 10 " d) " }{XPPEDIT 256 0 " arccos(7/11);" "6#-%'arccosG6#*&\"\"(\"\"\"\"#6!\"\"" }{TEXT 476 9 " \+ e) " }{XPPEDIT 256 0 "arccos(4/15);" "6#-%'arccosG6#*&\"\"%\"\"\" \"#:!\"\"" }{TEXT 475 10 " \nf) " }{XPPEDIT 256 0 "arccos(8/15); " "6#-%'arccosG6#*&\"\")\"\"\"\"#:!\"\"" }{TEXT 477 9 " g) " } {XPPEDIT 256 0 "arccos(8/21);" "6#-%'arccosG6#*&\"\")\"\"\"\"#@!\"\"" }{TEXT 478 9 " h) " }{XPPEDIT 256 0 "arccos(16/21);" "6#-%'arccos G6#*&\"#;\"\"\"\"#@!\"\"" }{TEXT 479 9 " i) " }{XPPEDIT 256 0 "ar ccos(24/35);" "6#-%'arccosG6#*&\"#C\"\"\"\"#N!\"\"" }{TEXT 481 10 " \+ j) " }{XPPEDIT 256 0 "arccos(32/35);" "6#-%'arccosG6#*&\"#K\"\"\" \"#N!\"\"" }{TEXT 482 6 " " }{TEXT 480 2 " " }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 557 16 "Answer: ( h )" } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "arccos( (2*2+2*3+1 *6)/(sqrt(2^2+2^2+1^2)* sqrt(2^2+3^2+6^2)) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'arccosG6##\"#;\"#@" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT 268 2 " 5" }{TEXT -1 2 ". " }{TEXT 358 34 "What is the radius of the sphere " }{TEXT 361 1 " " }{XPPEDIT 359 1 "x^2+y^2+z^2-10*x+8*y-6*z+1 = 0;" "6#/,0*$%\"xG\"\"#\"\"\"*$%\"y GF'F(*$%\"zGF'F(*&\"#5F(F&F(!\"\"*&\"\")F(F*F(F(*&\"\"'F(F,F(F/F(F(\" \"!" }{TEXT 360 2 " ?" }}{PARA 261 "" 0 "" {TEXT -1 50 "a) 1 b) \+ 2 c) 3 d) 4 e) 5" }}{PARA 261 "" 0 "" {TEXT -1 53 "f) 6 g) 7 h) 8 i) 9 j) 10" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 558 16 "Answer: ( g )" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "x^2 -10*x +25 + y^2 +8*y +16 + z^2 - 6*z + 9 = 25 +16+9 - 1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,0*$)%\"xG\"\"#\"\"\"F) *&\"#5F)F'F)!\"\"\"#]F)*$)%\"yGF(F)F)*&\"\")F)F0F)F)*$)%\"zGF(F)F)*&\" \"'F)F5F)F,\"#\\" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "(x-5)^2 + (y+4)^2 + (z-3)^2 = 7^2" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " } {TEXT 264 6 "6. If " }{TEXT 370 2 " " }{XPPEDIT 259 1 "`<`*a,b,c*`>` " "6%*&%\"GF%" }{TEXT 362 1 " " } {TEXT 371 29 " is the vector projection of " }{TEXT 372 2 " " } {XPPEDIT 363 1 "`<`*7,2,1*`>`;" "6%*&%\" GF%" }{TEXT 364 2 " " }{TEXT 373 3 " on" }{TEXT 374 3 " " } {XPPEDIT 365 1 "`<`*2,3,1*`>`;" "6%*&%\" GF%" }{TEXT 366 1 " " }{TEXT 375 13 " then what is" }{TEXT 376 1 " " } {TEXT 368 1 " " }{XPPEDIT 369 0 "a" "6#%\"aG" }{TEXT 367 2 " ?" }} {PARA 0 "" 0 "" {TEXT 265 128 "a) -5 b) -4 c) -3 \+ d) -2 e) -1 \n\n f) 0 g) 1 h) 2 \+ i) 3 j) 4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 559 16 "Answer: ( i )" } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning , the protected names norm and trace have been redefined and unprotect ed\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "u := vector([7,2,1] ); w := vector([2,3,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG-% 'vectorG6#7%\"\"(\"\"#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG -%'vectorG6#7%\"\"#\"\"$\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "p := dotprod(u,w)/dotprod(w,w); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG#\"\"$\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "a := p*2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG\"\"$" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 266 170 "7. Mr. Woodman is pulling a w agon with a force of 100 N. With what force, measured in newtons,\n \+ must Mrs. Woodman pull if the wagon is to move along the dotted line ? 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Let " }{XPPEDIT 388 1 "`<`*a,b,c*`>`" "6%*&%\"GF%" } {TEXT 389 1 " " }{TEXT 384 28 " be the vector cross product" }{TEXT 394 1 " " }{XPPEDIT 392 1 "`<`*2,1,1*`>` " "6%*&%\"GF%" }{TEXT 393 1 " " }{TEXT 395 2 " " }{TEXT 390 1 "X" } {TEXT 391 1 " " }{TEXT 385 1 " " }{XPPEDIT 379 1 "`<`*2,-1,2*`>`;" "6% *&%\"GF%" }{TEXT 380 1 " " }{TEXT 386 10 ". What is" }{TEXT 387 1 " " }{TEXT 382 1 " " }{XPPEDIT 383 0 "a+b+c" "6#,(%\"aG\"\"\"%\"bGF%%\"cGF%" }{TEXT 381 2 " ?" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 397 128 "a ) -5 b) -4 c) -3 d) -2 e) -1 \n\n f) \+ 0 g) 1 h) 2 i) 3 j) 4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 560 16 "Answer: ( c )" } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "crossprod(vector([2,1,1]), vector([2,-1,2]));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%\"\"$!\"#!\"%" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "3+(-2)+(-4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 269 29 "9. The magnitude of vector " }{XPPEDIT 501 0 "u;" "6#%\"uG" }{TEXT 483 37 " is 3. The magnitude of vector \+ " }{XPPEDIT 502 0 "v;" "6#%\"vG" }{TEXT 484 36 " is 4. The magnitu de of vector " }{XPPEDIT 503 0 "u" "6#%\"uG" }{TEXT 487 2 " " } {TEXT 485 3 "x " }{XPPEDIT 506 0 "v;" "6#%\"vG" }{TEXT 486 52 " is \+ 6. It is known that the angle between " }{XPPEDIT 507 0 "u" "6 #%\"uG" }{TEXT 488 1 " " }{TEXT 508 4 " and" }{TEXT 490 1 " " }{TEXT 509 1 " " }{XPPEDIT 510 0 "v;" "6#%\"vG" }{TEXT 489 36 " is acute. \+ What is that angle? " }}{PARA 258 "" 0 "" {TEXT 398 5 "\na) " } {XPPEDIT 399 1 "Pi/6;" "6#*&%#PiG\"\"\"\"\"'!\"\"" }{TEXT 400 9 " \+ b) " }{XPPEDIT 401 1 "Pi/3;" "6#*&%#PiG\"\"\"\"\"$!\"\"" }{TEXT 402 10 " c) " }{XPPEDIT 403 1 "Pi/4;" "6#*&%#PiG\"\"\"\"\"%!\"\"" } {TEXT 404 7 " d) " }{XPPEDIT 405 1 "Pi/3;" "6#*&%#PiG\"\"\"\"\"$!\" \"" }{TEXT 406 10 " e) " }{XPPEDIT 407 1 "Pi/12;" "6#*&%#PiG\"\" \"\"#7!\"\"" }{TEXT 408 7 " f) " }{XPPEDIT 409 1 "arccos(6/7);" "6# -%'arccosG6#*&\"\"'\"\"\"\"\"(!\"\"" }{TEXT 410 9 " g) " } {XPPEDIT 411 1 "arcsin(6/7);" "6#-%'arcsinG6#*&\"\"'\"\"\"\"\"(!\"\"" }{TEXT 412 7 " h) " }{XPPEDIT 413 1 "arcsin(1/6);" "6#-%'arcsinG6#* &\"\"\"F'\"\"'!\"\"" }{TEXT 414 8 " i) " }{XPPEDIT 415 1 "arcsin(1 /72);" "6#-%'arcsinG6#*&\"\"\"F'\"#s!\"\"" }{TEXT 416 76 " \+ \nj) Insufficient information to determine answer " } {TEXT -1 1 " " }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 573 16 "Answer: ( a )" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "eqn := 6 = 3*4*sin(theta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/\"\"',$*&\"#7\"\"\"-%$sinG6#%&thetaGF*F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "solve(eqn, theta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"'!\"\"%#PiG\"\"\"F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 270 38 "10. What is the area of parallelogram " }{TEXT 417 4 "PQRS" } {TEXT 418 4 " if " }{TEXT 419 1 "P" }{TEXT 420 12 " = (4,3,5), " } {TEXT 421 1 "Q" }{TEXT 422 13 " = (5,4,6), " }{TEXT 423 1 "R" }{TEXT 424 27 " = (2,3,5), and S =(1,2,4)?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT 279 21 "a) 4 b) " }{XPPEDIT 280 0 "3*sqrt(2)" "6#*&\"\"$\"\"\"-%%sqrtG6#\"\"#F%" }{TEXT 281 16 " \+ c) " }{XPPEDIT 282 0 "2*sqrt(6)" "6#*&\"\"#\"\"\"-%%sqrtG6#\"\"'F %" }{TEXT 283 19 " d) " }{XPPEDIT 284 0 "8;" "6#\"\")" }{TEXT 285 21 " e) " }{XPPEDIT 286 0 "4*sqrt(2)" "6#* &\"\"%\"\"\"-%%sqrtG6#\"\"#F%" }{TEXT 287 21 " \nf) " }{XPPEDIT 288 0 "6" "6#\"\"'" }{TEXT 289 14 " g) " } {XPPEDIT 290 0 "2*sqrt(3);" "6#*&\"\"#\"\"\"-%%sqrtG6#\"\"$F%" }{TEXT 291 18 " h) " }{XPPEDIT 292 0 "4*sqrt(3)" "6#*&\"\"%\"\" \"-%%sqrtG6#\"\"$F%" }{TEXT 293 19 " i) " }{XPPEDIT 294 0 "2*sqrt(2);" "6#*&\"\"#\"\"\"-%%sqrtG6#F$F%" }{TEXT 295 16 " \+ j) " }{XPPEDIT 296 0 "sqrt(6);" "6#-%%sqrtG6#\"\"'" }}{PARA 0 "" 0 "" {TEXT 297 1 " " }}{PARA 0 "" 0 "" {TEXT 574 16 "Answer: ( i )" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "u := vecto r([5-4,4-3,6-5]); #PQ" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG-%'ve ctorG6#7%\"\"\"F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "v := vector([1-4,2-3,4-5]); #PS" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"v G-%'vectorG6#7%!\"$!\"\"F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "crossprod(u,v);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7% \"\"!!\"#\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "sqrt(0^2+ 2^2+2^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"#\"\"\"F%#F&F%F& " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 271 76 "11. What is the volume of the parallelepiped formed from the three vectors" }{TEXT 431 1 " " }{XPPEDIT 425 1 "`<`*2,2,1*`>` " "6%* &%\"GF%" }{TEXT 426 3 ", " }{XPPEDIT 427 1 "`<`*2,1,1*`>` " "6%*&%\"GF%" }{TEXT 428 3 ", " }{TEXT 433 3 "and" }{TEXT 434 2 " " }{XPPEDIT 429 1 "`<`*1,2, 3*`>` " "6%*&%\"GF%" }{TEXT 430 1 "?" } {TEXT 432 3 " \n\n" }{TEXT 278 70 "a) 1 b) 2 c ) 3 d) 4 e) 5" }{TEXT 435 21 " \+ " }}{PARA 0 "" 0 "" {TEXT 437 34 "f) 6 g) 7 \+ " }{TEXT -1 3 " h)" }{TEXT 436 36 " 8 i) 9 j) \+ 12" }{TEXT -1 2 " " }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 571 16 "Answer: ( e )" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "u := vector([2,2,1]);\nv := vector([2,1,1]);\nw := vector([1,2,3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG-%'vectorG6 #7%\"\"#F)\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG-%'vectorG6# 7%\"\"#\"\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG-%'vectorG6#7 %\"\"\"\"\"#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "abs(do tprod(u,crossprod(v,w)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " \+ " }{TEXT 272 105 "12. The line through the points ( 4, -1, -2) and (6, -2, -1) intersects the plane y = 2 at the point (" }{TEXT 444 1 "a " }{TEXT 445 2 ", " }{TEXT 446 1 "b" }{TEXT 447 24 ", c). \n Wh at is " }{TEXT 442 1 "a" }{TEXT 443 79 " ? \n\na) -4 b) - 3 c) -2 d) -1 e) 0" }{TEXT 439 21 " \+ " }}{PARA 0 "" 0 "" {TEXT 440 36 "f) 1 \+ g) 2 " }{TEXT -1 1 " " }{TEXT 441 39 "h) 3 \+ i) 4 j) " }{TEXT -1 1 " " }{TEXT 448 1 "5" }{TEXT -1 2 " " }}{PARA 3 "" 0 "" {TEXT 438 1 " " }}{PARA 0 "" 0 "" {TEXT 570 16 " Answer: ( c )" }{TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "eqn1 := x = 4 + t*(4-6); \ne qn2 := y = -1 + t*(-1-(-2));\neqn3 := z = -2 + t*(-2-(-1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/%\"xG,&\"\"%\"\"\"*&\"\"#F)%\"tGF)! \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/%\"yG,&\"\"\"!\"\"%\" tGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/%\"zG,&\"\"#!\"\"%\"t GF)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "eqn3 := subs(y=2,eqn 2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/\"\"#,&\"\"\"!\"\"%\"t GF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "eqn4 := t = solve(eq n3, t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn4G/%\"tG\"\"$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "subs(eqn4,eqn1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"xG!\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 273 1 " " }{TEXT 453 17 "13. The vector " }{TEXT 449 1 "<" }{TEXT 450 11 " a , b , 3 " }{TEXT 451 1 ">" }{TEXT 452 69 " is perpendicula r to the plane z = x - 2y + 1. What is a ? " }{TEXT 454 3 " \+ \n\n" }{TEXT 455 132 "a) -4 b) - 3 c) - 2 d) - 1 e) 0 \nf) 1 g) 2 h) 3 i) 4 j) 5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 569 16 "Answer: ( b )" } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "a = k*1; b = k*(- 2); 3 = k*(-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"$%\"kG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"bG,$*&\"\"#\"\"\"%\"kGF(!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"$,$%\"kG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "b = 6; a = -3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"bG\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"$! \"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 261 "" 0 "" {TEXT -1 84 "14. A plane passes through the three points (1, 0, 0), (0, 4, 0) , and (2, 3, 2). " }}{PARA 3 "" 0 "" {TEXT 456 51 "Its equation is \+ Ax + 2y + Cz = D. What is C? " }{TEXT 458 1 "\n" }}{PARA 3 "" 0 " " {TEXT 457 117 "a) -1 b) -2 c) -3 d) -4 e) -5 f) -6 g) -7 h) -8 i) -9 j) -10 " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 568 16 "Answer: \+ ( g )" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Method 1:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "u := vector([1-0,0-4,0-0]); v := vector([0-2,4-3,0-2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG-%'vectorG6#7%\"\"\"!\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG-%'vectorG6#7%!\"#\"\"\"F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "crossprod(u,v);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%\"\")\"\"#!\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "C = -7;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"CG! \"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Me thod 2:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "eqn1 := subs( \{x=1,y=0,z=0\} , A*x + 2*y + C*z = D ) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/%\"AG%\"DG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "eqn2 := subs( \{x=0,y=4,z=0\} , A*x + 2*y + C*z = D );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/\"\")% \"DG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "eqn3 := subs( \{x=2 ,y=3,z=2\} , A*x + 2*y + C*z = D );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%%eqn3G/,(*&\"\"#\"\"\"%\"AGF)F)\"\"'F)*&F(F)%\"CGF)F)%\"DG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "solve(\{eqn1,eqn2,eqn3\},\{A ,C,D\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/%\"DG\"\")/%\"AGF&/%\"C G!\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 461 167 "15. A plane is perpendicul ar to the line 2 - x = (y + 1)/3 = 2z . The plane passes through the point (1, -2,- 1) as well as the point ( a, -4, 3 ). What is a \+ ?" }{TEXT 462 3 " \n" }{TEXT 460 133 "\na) -4 b) - 3 \+ c) - 2 d) - 1 e) 0 \nf) 1 g) 2 h ) 3 i) 4 j) 5" }{TEXT 459 1 "\n" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 566 16 "Answer: ( b )" } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "eqn := (-1)*(x-1) + 3*(y-(-2)) + (1/2)*(z-(-1)) = 0; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,*%\"xG!\"\"#\"#:\"\"#\"\" \"*&\"\"$F,%\"yGF,F,*&F+F(%\"zGF,F,\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "eqn2 := subs( \{ y=-4, z=3 \} , eqn );" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%%eqn2G/,&%\"xG!\"\"\"\"$F(\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "solve(eqn2, x);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 262 108 "16. The two planes x + y + z = 1 and x - y + 3z = 5 intersect in a line that is parallel to the \nvector " }{TEXT 470 2 " " }{TEXT 465 1 "<" }{TEXT 466 11 " a , b , 1 " } {TEXT 467 1 ">" }{TEXT 468 16 ". What is b ? " }{TEXT 469 3 " \n\n" }{TEXT 464 132 "a) -4 b) - 3 c) - 2 d) - 1 \+ e) 0 \nf) 1 g) 2 h) 3 i) 4 \+ j) 5" }{TEXT 463 1 "\n" }}{PARA 0 "" 0 "" {TEXT 565 16 "Answer: \+ ( f )" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "crosspr od(vector([1,1,1]),vector([1,-1,3]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%\"\"%!\"#F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "scalarmul(vector([4, -2, -2]), (-1/2) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%!\"#\"\"\"F(" }}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 261 "" 0 "" {TEXT -1 163 "17. A plane is parallel to the plane x + 2y + 3z = 0 and also passes through the point (4, 1, 2). What is the sum of its three intercepts with the coordinate axes?" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 260 "" 0 "" {TEXT -1 161 "a) 4 b) 6 c) 8 d) 9 e) 12 \nf) 16 \+ g) 18 h) 20 i) 22 j) 24" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 564 16 "Answer: ( i ) " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "eqn := x + 2*y + 3*z = D;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,(%\"xG\"\"\"*&\"\"#F(%\"yGF(F(*&\"\"$F(%\"zGF( F(%\"DG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "subs(\{x=4,y=1,z =2\},eqn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"#7%\"DG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "eqn := subs(D=12,eqn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,(%\"xG\"\"\"*&\"\"#F(%\"yGF(F(*&\"\"$F (%\"zGF(F(\"#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "12+(12/2) +(12/3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#A" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 261 "" 0 "" {TEXT -1 66 "18. Let S denote a particular quadric surface. When w e slice " }{TEXT 536 1 "S" }{TEXT -1 304 "\nwith each plane that is p arallel to one of the three coordinate planes, the intersection\nis ei ther (1) an ellipse, (2) a single point, or (3) empty. Each of\nthese \+ three types of intersections does occur with at least one such plane\n (and no other type of intersection occurs). What might the equation of " }{TEXT 537 1 "S" }{TEXT -1 11 " be?\n\na) " }{XPPEDIT 19 1 "z = x ^2/4+y^2/9;" "6#/%\"zG,&*&%\"xG\"\"#\"\"%!\"\"\"\"\"*&%\"yGF(\"\"*F*F+ " }{TEXT -1 38 " b) " }{XPPEDIT 19 1 "z^2 = x^2/4+y^2/9;" "6#/*$%\"zG\"\"#,&*&%\"xGF&\"\"%!\"\"\"\"\"*&% \"yGF&\"\"*F+F," }{TEXT -1 11 " \nc) " }{XPPEDIT 19 1 "z = x^2/4 -y^2/9;" "6#/%\"zG,&*&%\"xG\"\"#\"\"%!\"\"\"\"\"*&%\"yGF(\"\"*F*F*" } {TEXT -1 37 " d) " }{XPPEDIT 19 1 "z^ 2 = x^2/4-y^2/9;" "6#/*$%\"zG\"\"#,&*&%\"xGF&\"\"%!\"\"\"\"\"*&%\"yGF& \"\"*F+F+" }{TEXT -1 15 " \ne) " }{XPPEDIT 19 1 "z = x/4-y/9 ;" "6#/%\"zG,&*&%\"xG\"\"\"\"\"%!\"\"F(*&%\"yGF(\"\"*F*F*" }{TEXT -1 42 " f) " }{XPPEDIT 19 1 "z^2 = \+ 1-x^2/4-y^2/9;" "6#/*$%\"zG\"\"#,(\"\"\"F(*&%\"xGF&\"\"%!\"\"F,*&%\"yG F&\"\"*F,F," }}{PARA 0 "" 0 "" {TEXT 511 4 "g) " }{TEXT -1 2 " " } {XPPEDIT 513 1 "z^2 = 1-x/4-y/9;" "6#/*$%\"zG\"\"#,(\"\"\"F(*&%\"xGF( \"\"%!\"\"F,*&%\"yGF(\"\"*F,F," }{TEXT 512 32 " \+ h) " }{XPPEDIT 515 1 "x^2+y^2-z^2 = 0;" "6#/,(*$%\"xG\"\"#\"\"\" *$%\"yGF'F(*$%\"zGF'!\"\"\"\"!" }{TEXT 514 7 "\n i) " }{XPPEDIT 257 1 "x^2+2*x+y^2+z^2+2 = 0;" "6#/,,*$%\"xG\"\"#\"\"\"*&F'F(F&F(F(*$%\"yG F'F(*$%\"zGF'F(F'F(\"\"!" }{TEXT 516 1 " " }{TEXT 550 11 " j) \+ " }{XPPEDIT 517 1 "x^2/4+y^2/9-z = 1;" "6#/,(*&%\"xG\"\"#\"\"%!\"\"\" \"\"*&%\"yGF'\"\"*F)F*%\"zGF)F*" }{TEXT 518 6 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 562 16 "Answer: ( f )" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 471 76 "19. Which one of the following equat ions has a graph that is a cylinder in " }{TEXT 552 1 " " }{XPPEDIT 553 0 "xyz;" "6#%$xyzG" }{TEXT 551 13 "-coordinates?" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 524 5 "a) \+ " }{XPPEDIT 525 1 "z = x^2/4+y^2/9;" "6#/%\"zG,&*&%\"xG\"\"#\"\"%!\"\" \"\"\"*&%\"yGF(\"\"*F*F+" }{TEXT 526 35 " \+ b) " }{XPPEDIT 527 1 "z^2 = x^2/4+y^2/9;" "6#/*$%\"zG\"\"#,&*&%\"xG F&\"\"%!\"\"\"\"\"*&%\"yGF&\"\"*F+F," }{TEXT 528 11 " \nc) " } {XPPEDIT 529 1 "z = x^2/4-y^2/9;" "6#/%\"zG,&*&%\"xG\"\"#\"\"%!\"\"\" \"\"*&%\"yGF(\"\"*F*F*" }{TEXT 530 35 " \+ d) " }{XPPEDIT 531 1 "z^2 = x^2/4-y^2/9;" "6#/*$%\"zG\"\"#,&*&%\"xGF& \"\"%!\"\"\"\"\"*&%\"yGF&\"\"*F+F+" }{TEXT 532 15 " \ne) " } {XPPEDIT 533 1 "x^2+y^2/4+z^4-2*z^2 = 0;" "6#/,**$%\"xG\"\"#\"\"\"*&% \"yGF'\"\"%!\"\"F(*$%\"zGF+F(*&F'F(*$F.F'F(F,\"\"!" }{TEXT 534 15 " \+ f) " }{XPPEDIT 535 1 "z^2 = 1-x^2/4-y^2/9;" "6#/*$%\"zG\"\"#, (\"\"\"F(*&%\"xGF&\"\"%!\"\"F,*&%\"yGF&\"\"*F,F," }}{PARA 0 "" 0 "" {TEXT 519 6 "g) " }{XPPEDIT 259 1 "x^2+y^2+z^2 = 1;" "6#/,(*$%\"xG \"\"#\"\"\"*$%\"yGF'F(*$%\"zGF'F(F(" }{TEXT 520 27 " \+ h) " }{XPPEDIT 261 1 "x^2+y^2-z^2 = 0;" "6#/,(*$%\"xG\"\"#\"\"\" *$%\"yGF'F(*$%\"zGF'!\"\"\"\"!" }{TEXT 521 7 "\n i) " }{XPPEDIT 256 1 "x^2+2*x+y = 2;" "6#/,(*$%\"xG\"\"#\"\"\"*&F'F(F&F(F(%\"yGF(F'" } {TEXT 522 28 " j) " }{XPPEDIT 263 1 "x^2/4+y^2 /9-z = 1;" "6#/,(*&%\"xG\"\"#\"\"%!\"\"\"\"\"*&%\"yGF'\"\"*F)F*%\"zGF) F*" }{TEXT 523 6 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 561 16 "Answer: ( i ) " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 274 200 "20. The traces of z = f(x,y) corres ponding to z = -3/2, -1, -1/2, 0, 1/2, 1, 3/2, 2 are graphed in the f igure below. These traces are typical of the traces of f correspondi ng to horizontal slices." }{TEXT -1 11 " " }}{PARA 0 "" 0 " " {OLE 1 492048 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:J>j:J:ZJJ;j:JyyyyyI<:^: f:n:v:>;F;N;vyyyyy=::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::uyyyyyyw:nYnyyyYuA::::S:UJ:n;v;;JBB:]:_:aZ:^=F= N=V=^=f=n=v=>>AFANA VA^AfAnA;jYJZB:=EFENEV E^EfEnEvE>F:::::::JEf:yyyxIN::;`:Z@[::JjuO;MMu@ y[Aj;J:@:<:=j[vGUMrvC?MoJ::::::::JCNZ;^:vYxI>:<::::::j`J:j:vCSmlF@[KaF FcmnnHEM:>:::::::oJ;@j:j;B:yayA:<::::::Q:CuSIZ:>:=:jRj^^HEmpnCfGEM:>:: ::::n=?R:yyyyyy:>:<::::::?:EJ:>:F:;JyKy;vY:::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::F:wyyAbR<:Tn=JbDNsPqs\\`A:::ZnIalHH>::::::::::j:J:::::::::::=Zk>K?C:Q>S> U>W>Y>[^:FLNLVL^LfLnLvL>M;jFKGkGKHkHKIkIKJ=Z:FNNNVN^NfNnNvN>OFONOVO^Of OnOvO>PUFUNUVU^UfUnUvu:Jj=Z:FVNVVV^VfVnVvV>WFWNWVW^WfWnWv w:JrC:]Q:NXVX^XfXnXvX>YFY;Jw?:qAsAuAwI:vy:J:@:=R:NZ;>:AR:[;j>@:OR:V[;J@@:UR:n[;jA@:[R:_;jN@:OS:V_;JP@:US:n_;jQ@: [S:F`;JS@:aS:^`;jT@:gS:v`;JV@:mS:Na;jW@:sS:fa;JY@:yS:>b;jZ@:?T:Vb;J\\@ :ET:nb;j]@:KT:Fc;J_@:QT:^c;j`@:WT:vc;Jb@:]T:Nd;jc@:cT:fd;Je@:iT:>e;jf@ :oT:Ve;Jh@:uT:ne;ji@:;U:Ff;Jk@:AU:^f;jl@:GU:vf;Jn@:MU:Ng;jo@:SU:fg;Jq@ :YU:>h;jr@:_U:Vh;Jt@:eU:nh;ju@:kU:Fi;Jw@:qU:^i;jx@:wU:vi;J:A:=V:Nj;j;A :CV:fj;J=A:IV:>k;j>A:OV:Vk;J@A:UV:nk;jAA:[V:Fl;JCA:aV:^l;jDA:gV:vl;JFA :mV:Nm;jGA:sV:fm;JIA:yV:>n;jJA:?W:Vn;JLA:EW:nn;jMA:KW:Fo;JOA:QW:^o;>:U W:no;jQA:[W:Fp;JSA:aW:^p;jTA:gW:vp;JVA:mW:Nq;jWA:sW:fq;JYA:yW:>r;jZA:? 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