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vY::::::::::::::::::::::::::::::::::::::::::::::::::::Z::yA:::B:<:xI:: ::;jysy::::::::::::::::::::::::::::::::::::::::::::::::::::;:xI:::::yay=:::::::::::::::::::::::::::::::::::::::::::::::::::>: ry::::B:<:xI::::B:xIyA:::::::::::::::::::::::::::::::::::::::::::::::: ::B:jy;:::::yay=:::::::::::::::::::::::::::::::::::::: :::::::::::>:ry:::::B:<:xI:::::;jysy:::::::::::::::::::::::::::::::::: ::::::::::::::::;:xI::::::yay=::::::::: ::::::::::::::::::::::::::::::::::::::>:ry::::::B:<:xI:::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::B:Zy=::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::<:ry::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::Z::xI:::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::B:3:" }{TEXT -1 24 " \nand the mag nitude of " }{BITMAP 25 35 35 1 "?TMof?F=V>F:k:;::::yyyyyy::::: ::::::Z:vyyuy:::::::::::Z:Zy=:::::::::::<:xI::;B:yI:::Zyuy:::::B:yay=: :J:Z:B:Zyuy::::::B:yay=: :J::ry::::B:yay=::J::ry::::B:yay=::J::ry::: 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::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::Z::xI::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::B:Zy=:::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::<:ry::::::::::::::::::::::::::::::::::::::::::::::::::::::::Ry wyyuiyvyyiyyuyyA:::::::::::::::::::::::::::::::::::::::::::::::::B:Zy= :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::<:ry:::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::Z:J:::::::::::::::::::::::::::::::::::::::::::::::::::::::FZ :vc:^g:ff:J::::::::::OB:?B:yyyxI:;Z::::::j;:<:::::::::::::::::::vYxI:;Z::::::::::1: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 302 4 "a) \+ " }{XPPEDIT 303 1 "1;" "6#\"\"\"" }{TEXT 304 9 " b) " }{XPPEDIT 305 1 "2;" "6#\"\"#" }{TEXT 306 10 " c) " }{XPPEDIT 307 1 "3;" " 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" }{TEXT 333 2 " " }}{PARA 3 "" 0 "" {TEXT 274 211 "\n a) -4 b) -3 c) -2 d) -1 \+ e) 0 \n f) 1 g) 2 h) 3 \+ i) 4 j) insufficient information to answer question " }{TEXT 334 7 " \n" }}{PARA 0 "" 0 "" {TEXT 497 16 "Answer: \+ ( g )" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 10 "Whatever " }{TEXT 553 1 "w" }{TEXT -1 7 " is, " } {TEXT 554 1 "v" }{TEXT -1 24 " is perpendicular to " }{TEXT 556 2 " v " }{TEXT 555 1 "x" }{TEXT 557 2 " w" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "eqn := do tprod( [3,1,c],[-1,-5,4]) = 0 ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ eqnG/,&\"\")!\"\"*&\"\"%\"\"\"%\"cGF+F+\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "c = solve(eqn, c);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"cG\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 261 1 " " }{TEXT 341 36 "4. What is the angle between vectors" }{TEXT 342 3 " " } {XPPEDIT 339 1 "`<`*2,3,1*`>`;" "6%*&%\" GF%" }{TEXT 335 1 " " }{TEXT 337 3 "and" }{TEXT 338 2 " " }{XPPEDIT 340 1 "`<`*3,1,(-2)*`>`;" "6%*&%\"GF%" }{TEXT 336 3 " ?" }}{PARA 0 "" 0 "" {TEXT 266 4 "a) " } {XPPEDIT 256 0 "Pi/6;" "6#*&%#PiG\"\"\"\"\"'!\"\"" }{TEXT 257 4 " \+ " }{TEXT 343 16 " " }{TEXT 522 2 "b)" }{TEXT 523 2 " \+ " }{XPPEDIT 256 0 "Pi/4;" "6#*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT 458 1 " \+ " }{TEXT 344 21 " c) " }{XPPEDIT 256 0 "Pi/3;" "6#*&% #PiG\"\"\"\"\"$!\"\"" }{TEXT 345 22 " " }{TEXT 528 3 "d) " }{TEXT 529 1 " " }{XPPEDIT 256 0 "Pi/2;" "6#*&%#PiG\"\"\" \"\"#!\"\"" }{TEXT 460 23 " " }{TEXT 536 2 "e)" }{TEXT 537 3 " " }{XPPEDIT 256 0 "3*Pi/4;" "6#*(\"\"$\"\"\"%#PiGF%\" \"%!\"\"" }{TEXT 459 6 " \n" }{TEXT 526 2 "f)" }{TEXT 527 2 " " } {XPPEDIT 256 0 "arccos(1/7);" "6#-%'arccosG6#*&\"\"\"F'\"\"(!\"\"" } {TEXT 461 7 " " }{TEXT 524 2 "g)" }{TEXT 525 2 " " }{XPPEDIT 256 0 "arccos(2/7);" "6#-%'arccosG6#*&\"\"#\"\"\"\"\"(!\"\"" }{TEXT 462 8 " " }{TEXT 530 2 "h)" }{TEXT 531 2 " " }{XPPEDIT 256 0 " arccos(3/7);" "6#-%'arccosG6#*&\"\"$\"\"\"\"\"(!\"\"" }{TEXT 463 6 " \+ " }{TEXT 532 3 " i)" }{TEXT 533 2 " " }{XPPEDIT 256 0 "arccos(4/7 );" "6#-%'arccosG6#*&\"\"%\"\"\"\"\"(!\"\"" }{TEXT 465 7 " " } {TEXT 534 2 "j)" }{TEXT 535 2 " " }{XPPEDIT 256 0 "arccos(5/7);" "6#- %'arccosG6#*&\"\"&\"\"\"\"\"(!\"\"" }{TEXT 466 6 " " }{TEXT 464 2 " " }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 498 16 "Answer: ( c )" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "v := [2,3,1]; w := [3,1,-2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"vG7%\"\"#\"\"$\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG7%\"\" $\"\"\"!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "eqn := cos(t heta) = dotprod(v,w)/(vectorLength(v)*vectorLength(w)); #cosine of ang le between v,w" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/-%$cosG6#%&t hetaG#\"\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "theta \+ = map(arccos, rhs(eqn));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&thetaG, $*&\"\"$!\"\"%#PiG\"\"\"F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 267 2 " 5" }{TEXT -1 2 ". " }{TEXT 346 34 "What \+ is the radius of the sphere " }{TEXT 349 1 " " }{XPPEDIT 347 1 "x^2+y ^2+z^2-6*x+5*y-7*z = 17/2;" "6#/,.*$%\"xG\"\"#\"\"\"*$%\"yGF'F(*$%\"zG F'F(*&\"\"'F(F&F(!\"\"*&\"\"&F(F*F(F(*&\"\"(F(F,F(F/*&\"# " 0 "" {MPLTEXT 1 0 38 "eqn := x^2+y^2+z^2-6*x+5*y-7*z = 17/2;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%$eqnG/,.*$)%\"xG\"\"#\"\"\"F+*$)%\"yGF*F+F+*$) %\"zGF*F+F+*&\"\"'F+F)F+!\"\"*&\"\"&F+F.F+F+*&\"\"(F+F1F+F4#\"# " 0 "" {MPLTEXT 1 0 38 "eqn := student[completesqua re](eqn,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,.*$),&%\"xG\" \"\"\"\"$!\"\"\"\"#F+F+\"\"*F-*$)%\"yGF.F+F+*$)%\"zGF.F+F+*&\"\"&F+F2F +F+*&\"\"(F+F5F+F-#\"# " 0 "" {MPLTEXT 1 0 38 "eqn := student[completesquare](eqn,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,,*$),&%\"yG\"\"\"#\"\"&\"\"#F+F.F+F+#\"#h\"\"% !\"\"*$),&%\"xGF+\"\"$F2F.F+F+*$)%\"zGF.F+F+*&\"\"(F+F:F+F2#\"# " 0 "" {MPLTEXT 1 0 38 "eqn := student[completesqua re](eqn,z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,**$),&%\"zG\" \"\"#\"\"(\"\"#!\"\"F.F+F+#\"#bF.F/*$),&%\"yGF+#\"\"&F.F+F.F+F+*$),&% \"xGF+\"\"$F/F.F+F+#\"# " 0 "" {MPLTEXT 1 0 22 "map(z -> z+53/2, eqn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*$),& %\"zG\"\"\"#\"\"(\"\"#!\"\"F,F)F)*$),&%\"yGF)#\"\"&F,F)F,F)F)*$),&%\"x GF)\"\"$F-F,F)F)\"#O" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 264 18 "6. Suppose that " }{XPPEDIT 539 0 "w = lambda*v;" "6#/%\"wG* &%'lambdaG\"\"\"%\"vGF'" }{TEXT 538 7 " and " }{TEXT 358 2 " " } {XPPEDIT 259 1 "v;" "6#%\"vG" }{TEXT 350 1 " " }{TEXT 359 29 " is the \+ vector projection of " }{TEXT 360 2 " " }{XPPEDIT 351 1 "`<`*3,4,1*`> `;" "6%*&%\"GF%" }{TEXT 352 2 " " } {TEXT 361 3 " on" }{TEXT 362 3 " " }{XPPEDIT 353 1 "w = `<`*1,4,8*`> `;" "6%/%\"wG*&%\"GF'" }{TEXT 354 1 " " }{TEXT 363 10 ". \nWhat is" }{TEXT 364 1 " " }{TEXT 356 1 " " } {XPPEDIT 357 0 "lambda;" "6#%'lambdaG" }{TEXT 355 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 500 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 "w := [1,4,8];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG7%\"\"\"\" \"%\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "v = vectorProje ction([3,4,1],w);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"vG7%#\"\"\"\" \"$#\"\"%F(#\"\")F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 381 8 "7. If " }{XPPEDIT 373 1 "`<`*a,b,c*`>`" "6%*&%\"GF%" }{TEXT 374 1 " " }{TEXT 566 20 "is the cross pro duct" }{TEXT 379 1 " " }{XPPEDIT 377 1 "`<`*2,2,1*`>`;" "6%*&%\"GF%" }{TEXT 378 1 " " }{TEXT 380 2 " " }{TEXT 375 1 "X" }{TEXT 376 1 " " }{TEXT 370 1 " " }{XPPEDIT 365 1 "`<`*2,1,- 2*`>`;" "6%*&%\"GF%!\"\"" }{TEXT 366 1 " \+ " }{TEXT 371 15 ", then what is" }{TEXT 372 1 " " }{TEXT 368 1 " " } {XPPEDIT 369 0 "a+b+c" "6#,(%\"aG\"\"\"%\"bGF%%\"cGF%" }{TEXT 367 2 " \+ ?" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 382 128 "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 501 16 "Answ er: ( e )" }{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 61 "crossProduct := crossprod(vector([2 ,2,1]), vector([2,1,-2]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-cross ProductG-%'vectorG6#7%!\"&\"\"'!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "crossProduct[1]+crossProduct[2]+crossProduct[3];" }} {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 268 17 "8. Suppose that " }{XPPEDIT 472 0 "u;" "6#%\"uG" }{TEXT 467 7 " and " }{XPPEDIT 473 0 "v;" "6#% \"vG" }{TEXT 468 36 " are vectors. The magnitudes of " }{TEXT 558 1 "u" }{TEXT 559 8 " and " }{TEXT 560 1 "v" }{TEXT 561 75 " are \+ 2 and 5 respectively. \nThe dot product of u and v is " }{TEXT 562 1 "-" }{TEXT 563 37 " 6. What is the magnitude of \+ " }{XPPEDIT 474 0 "u" "6#%\"uG" }{TEXT 471 2 " " }{TEXT 469 3 "x " } {XPPEDIT 475 0 "v;" "6#%\"vG" }{TEXT 470 6 " ? " }}{PARA 258 "" 0 " " {TEXT 383 5 "\na) " }{XPPEDIT 384 1 "1;" "6#\"\"\"" }{TEXT 385 10 " b) " }{XPPEDIT 386 1 "2;" "6#\"\"#" }{TEXT 387 10 " c) " }{XPPEDIT 388 1 "3;" "6#\"\"$" }{TEXT 389 10 " d) " }{XPPEDIT 390 1 "4;" "6#\"\"%" }{TEXT 391 10 " e) " }{XPPEDIT 392 1 "5;" " 6#\"\"&" }{TEXT 393 10 " \n f) " }{XPPEDIT 394 1 "6;" "6#\"\"'" } {TEXT 395 10 " g) " }{XPPEDIT 396 1 "7;" "6#\"\"(" }{TEXT 397 10 " h) " }{XPPEDIT 398 1 "8;" "6#\"\")" }{TEXT 399 9 " i) \+ " }{XPPEDIT 400 1 "9;" "6#\"\"*" }{TEXT 401 57 " j) Insufficien t information to determine answer " }{TEXT -1 1 " " }}{PARA 3 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 510 16 "Answer: ( h )" } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 25 " be t he angle between " }{TEXT 564 1 "u" }{TEXT -1 7 " and " }{TEXT 565 3 " v" }{TEXT -1 12 " . Then " }{XPPEDIT 18 0 "-6 = abs(u)*ab s(v)*cos(theta);" "6#/,$\"\"'!\"\"*(-%$absG6#%\"uG\"\"\"-F)6#%\"vGF,-% $cosG6#%&thetaGF," }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "cos(theta) = solve( -6 = 2*5*cos(theta), cos(theta));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#%&thetaG#!\"$\"\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Since, " }{XPPEDIT 18 0 "sin(theta) = sqrt(1-cos(theta)^2);" "6#/-%$sinG6#%&thetaG-%%sqrt G6#,&\"\"\"F,*$-%$cosG6#F'\"\"#!\"\"" }{TEXT -1 9 ", we have" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "sin(theta) = sqrt(1 - (-3/5)^2);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$sinG6#%&thetaG#\"\"%\"\"&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "2*5*4/5; #Required magnitud e" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 269 37 "9. Wh at is the area of parallelogram " }{TEXT 402 4 "PQRS" }{TEXT 403 4 " i f " }{TEXT 404 1 "P" }{TEXT 405 12 " = (4,3,5), " }{TEXT 406 1 "Q" } {TEXT 407 13 " = (5,5,6), " }{TEXT 408 1 "R" }{TEXT 409 27 " = (6,7,8 ), and S =(5,5,7)?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 277 24 "a) 1 b) " }{XPPEDIT 278 0 "sqrt(2);" "6# -%%sqrtG6#\"\"#" }{TEXT 279 18 " c) " }{XPPEDIT 280 0 "s qrt(3);" "6#-%%sqrtG6#\"\"$" }{TEXT 281 19 " d) " } {XPPEDIT 282 0 "2;" "6#\"\"#" }{TEXT 283 21 " e) " } {XPPEDIT 284 0 "sqrt(5);" "6#-%%sqrtG6#\"\"&" }{TEXT 285 21 " \+ \nf) " }{XPPEDIT 286 0 "sqrt(6);" "6#-%%sqrtG6#\"\"'" }{TEXT 287 14 " g) " }{XPPEDIT 288 0 "2*sqrt(2);" "6#*&\"\"#\"\"\"- %%sqrtG6#F$F%" }{TEXT 289 17 " h) " }{XPPEDIT 290 0 "3;" "6#\"\"$" }{TEXT 291 21 " i) " }{XPPEDIT 292 0 "2*sqr t(3);" "6#*&\"\"#\"\"\"-%%sqrtG6#\"\"$F%" }{TEXT 293 18 " \+ j) " }{XPPEDIT 294 0 "3*sqrt(2);" "6#*&\"\"$\"\"\"-%%sqrtG6#\"\"#F% " }}{PARA 0 "" 0 "" {TEXT 295 1 " " }}{PARA 0 "" 0 "" {TEXT 511 16 "An swer: ( e )" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "u := vector([5-4,5-3,6-5]); #vector from P to Q" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG-%'vectorG6#7%\"\"\"\"\"#F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "v := vector([6-5,7-5,8-6]); #Q to R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG-%'vectorG6#7%\"\"\"\"\"#F *" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "w := crossprod(u,v);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG-%'vectorG6#7%\"\"#!\"\"\"\"! " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "vectorLength(evalm(w)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$\"\"&#\"\"\"\"\"#" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 270 76 "10. Wh at is the volume of the parallelepiped formed from the three vectors" }{TEXT 416 1 " " }{XPPEDIT 410 1 "`<`*2,2,4*`>`;" "6%*&%\"GF%" }{TEXT 411 3 ", " }{XPPEDIT 412 1 "`<`*2,1,1*` >` " "6%*&%\"GF%" }{TEXT 413 3 ", " } {TEXT 418 3 "and" }{TEXT 419 2 " " }{XPPEDIT 414 1 "`<`*1,2,3*`>` " " 6%*&%\"GF%" }{TEXT 415 1 "?" }{TEXT 417 3 " \n\n" }{TEXT 276 70 "a) 1 b) 2 c) 3 \+ d) 4 e) 5" }{TEXT 420 21 " " }} {PARA 0 "" 0 "" {TEXT 422 34 "f) 6 g) 7 " } {TEXT -1 1 " " }{TEXT 567 2 "h)" }{TEXT 421 36 " 8 i) 9 \+ j) 12" }{TEXT -1 2 " " }}{PARA 3 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 509 16 "Answer: ( d )" }{TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "u := vector([2,2,4]);\nv := \+ vector([2,1,1]);\nw := vector([1,2,3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG-%'vectorG6#7%\"\"#F)\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG-%'vectorG6#7%\"\"#\"\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG-%'vectorG6#7%\"\"\"\"\"#\"\"$" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 31 "abs(dotprod(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 569 203 "11. The line with parametric equations x = 1 + 2t, y = 3t, z = 2 - t intersects the line with parametric eq uations x = 9 + 2t, y = 5 - 4t, z = 1 + 2t in a point (a,b,c) . What is b ?" }}{PARA 0 "" 0 "" {TEXT 588 70 "a) 1 b) \+ 2 c) 3 d) 4 e) 5" }{TEXT 589 22 " \+ \n" }{TEXT 591 34 "f) 6 g) 7 \+ " }{TEXT -1 1 " " }{TEXT 592 2 "h)" }{TEXT 590 36 " 8 i) 9 j) 12" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 593 12 "Answer ( i )" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "line[1] := [x = 1 + 2*t, y = 3*t, z = 2 - t];" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%lineG6#\"\"\"7%/%\"xG,&F'F'*&\"\"# F'%\"tGF'F'/%\"yG,$*&\"\"$F'F.F'F'/%\"zG,&F-F'F.!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "line[2] := [x = 9 + 2*s, y = 5 - \+ 4*s, z = 1 + 2*s ];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%lineG6#\" \"#7%/%\"xG,&\"\"*\"\"\"*&F'F-%\"sGF-F-/%\"yG,&\"\"&F-*&\"\"%F-F/F-!\" \"/%\"zG,&F-F-*&F'F-F/F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "for j from 1 to 3 do\neqn[j] := rhs(line[1][j]) = rhs(line[2][j]); \nend do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$eqnG6#\"\"\"/,&F'F'*& \"\"#F'%\"tGF'F',&\"\"*F'*&F+F'%\"sGF'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$eqnG6#\"\"#/,$*&\"\"$\"\"\"%\"tGF,F,,&\"\"&F,*&\"\"%F,%\"sGF ,!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$eqnG6#\"\"$/,&\"\"#\"\" \"%\"tG!\"\",&F+F+*&F*F+%\"sGF+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "parameterSolns := solve(\{eqn[1],eqn[2]\},\{s,t\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/parameterSolnsG<$/%\"sG!\"\"/%\"t G\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "subs(parameterSol ns,eqn[3] ); #Verification that last coordinate works out" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/!\"\"F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "subs(parameterSolns,[seq(rhs(line[1][j]),j=1..3)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"(\"\"*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "subs(parameterSolns,[seq(rhs(line[2][j]), j=1..3)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"(\"\"*!\"\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " } {TEXT 271 105 "12. The line through the points ( 4, -1, -2) and (6, 3, -1) intersects the plane y = 11 at the point (" }{TEXT 429 1 "a" } {TEXT 430 2 ", " }{TEXT 431 1 "b" }{TEXT 432 24 ", c). \n What \+ is " }{TEXT 427 5 "a + c" }{TEXT 428 78 " ? \n\na) 8 b) \+ 9 c) 10 d) 11 e) 12" }{TEXT 424 21 " \+ " }}{PARA 0 "" 0 "" {TEXT 425 33 "f) 13 \+ g) 14 " }{TEXT -1 1 " " }{TEXT 426 37 "h) 15 i) 16 j) " }{TEXT 587 2 "17" }}{PARA 3 "" 0 "" {TEXT 423 1 " \+ " }}{PARA 0 "" 0 "" {TEXT 508 16 "Answer: ( d )" }{TEXT -1 0 "" }} {PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "eq n1 := x = 4 + t*(4-6); \neqn2 := y = -1 + t*(-1-3);\neqn3 := z = -2 + \+ t*(-2-(-1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/%\"xG,&\"\"% \"\"\"*&\"\"#F)%\"tGF)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2 G/%\"yG,&\"\"\"!\"\"*&\"\"%F(%\"tGF(F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/%\"zG,&\"\"#!\"\"%\"tGF)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "eqn4 := subs(y=11,eqn2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn4G/\"#6,&\"\"\"!\"\"*&\"\"%F(%\"tGF(F)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "eqn5 := t = solve(eqn4, t); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn5G/%\"tG!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "rhs(subs(eqn5,eqn1))+rhs(subs(eqn5, eqn3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 272 1 " " }{TEXT 437 17 "13. The vector " }{TEXT 433 1 "<" }{TEXT 434 11 " a , b , 2 " }{TEXT 435 1 ">" }{TEXT 436 69 " is perpe ndicular to the plane z = 2x - 3y + 1. What is a ? " }{TEXT 438 3 " \n\n" }{TEXT 439 132 "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 507 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 60 "eqn1 := a = k*2; \+ eqn2 := b = k*(-3); eqn3 := 2 = k*(-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/%\"aG,$*&\"\"#\"\"\"%\"kGF*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/%\"bG,$*&\"\"$\"\"\"%\"kGF*!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/\"\"#,$%\"kG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "subs(k=-2,eqn1);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%\"aG!\"%" }}}{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 po ints (1, 0, 0), (0, 4, 0), and (2, 3, 2). " }}{PARA 3 "" 0 "" {TEXT 440 50 "Its equation is Ax + y + Cz = D. What is A? " } {TEXT 442 1 "\n" }}{PARA 3 "" 0 "" {TEXT 441 107 "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 506 16 "Answer: ( d )" }{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 20 "w := crossprod (u,v);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG-%'vectorG6#7%\"\")\" \"#!\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "A = (1/2)*w[1]; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Method 2:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "eqn1 := s ubs( \{x=1,y=0,z=0\} , A*x + y + C*z = D );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/%\"AG%\"DG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "eqn2 := subs( \{x=0,y=4,z=0\} , A*x + y + C*z = D ); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/\"\"%%\"DG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "eqn3 := subs( \{x=2,y=3,z=2\} , A*x + y + C*z = D );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/,(*&\"\" #\"\"\"%\"AGF)F)\"\"$F)*&F(F)%\"CGF)F)%\"DG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "solve(\{eqn1,eqn2,eqn3\},\{A,C,D\});" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#<%/%\"AG\"\"%/%\"CG#!\"(\"\"#/%\"DGF&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 445 168 "15. A plane is perpendicular to the lin e 2 - x = (2y + 2)/6 = 2z . The plane passes through the point ( 1, -2,- 1) as well as the point ( a, -4, 3 ). What is a ?" }{TEXT 446 3 " \n" }{TEXT 444 131 "\na) -3 b) - 2 c) - 1 \+ d) 0 e) 1 \nf) 2 g) 3 h) 4 \+ i) 5 j) " }{TEXT 443 2 "6\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 505 16 "Answer: ( a )" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "eqn := (-1)*(x-1) + 3*(y-(-2)) + (1/2)*(z-(-1)) = 0;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%$eqnG/,*%\"xG!\"\"#\"#:\"\"#\"\"\"*&\"\"$F,%\" yGF,F,*&F+F(%\"zGF,F,\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "eqn2 := subs( \{x = a, y=-4, z=3 \} , eqn );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/,&%\"aG!\"\"\"\"$F(\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "solve(eqn2, a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 " " {TEXT 262 109 "16. The two planes x + y + 2z = 1 and x - y + 4z \+ = 5 intersect in a line that is parallel to the \nvector" }{TEXT 454 2 " " }{TEXT 449 1 "<" }{TEXT 450 12 " a , b , -1 " }{TEXT 451 1 ">" }{TEXT 452 18 ". What is a ? " }{TEXT 453 3 " \n\n" }{TEXT 448 132 "a) -4 b) - 3 c) - 2 d) - 1 e) 0 \+ \nf) 1 g) 2 h) 3 i) 4 j) 5" }{TEXT 447 1 "\n" }}{PARA 0 "" 0 "" {TEXT 504 16 "Answer: ( h )" } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "v := crossprod(vec tor([1,1,2]),vector([1,-1,4]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"vG-%'vectorG6#7%\"\"'!\"#F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "scalarmul(v , (1/2) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vec torG6#7%\"\"$!\"\"F(" }}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 261 "" 0 "" {TEXT -1 164 "17. A plan e is parallel to the plane x + 2y - 3z = 0 and also passes through the point (10, 1, 2). What is the sum of its three intercepts with the co ordinate axes?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 161 "a) 6 b) 7 c) 8 d) \+ 9 e) 12 \nf) 16 g) 18 h) 2 0 i) 22 j) 24" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 503 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 25 "eq n := x + 2*y - 3*z = D;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,(% \"xG\"\"\"*&\"\"#F(%\"yGF(F(*&\"\"$F(%\"zGF(!\"\"%\"DG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "eqn2 := subs(\{x=10,y=1,z=2\},eqn); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/\"\"'%\"DG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "eqn := subs(D=lhs(eqn2),eqn);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,(%\"xG\"\"\"*&\"\"#F(%\"yGF(F (*&\"\"$F(%\"zGF(!\"\"\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "6/1+(6/2)+(6/(-3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"(" }}} {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 48 "18. 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