CE GREAT MINDS DIAGNOSTIC EXAM MATHEMATICS, SURVEYING, AND TRANSPORTATION ENGINEERING INSTRUCTION: Select the best answe !" each "! the !"ll"w#n$ %&est#"ns' Ma( "nl) "ne answe !" each #te* b) sha+#n$ the b" c"es-"n+#n$ t" the lette "! )"& ch"#ce "n the answe sheet -".#+e+' STRICT/Y NO ERASURES A//O0ED' Use -enc#l n"' 1 "nl)' 2' 0hat #s the lateal lateal aea aea "! the !&st&* !&st&* "! a #$ht #$ht c#c&la c#c&la c"ne wh"se wh"se base +#a*etes +#a*etes ae 1 an+ 3 *' #! #ts alt#t&+e #s 4 *5 b. 48.06 a' 33'67 *1 c' 41'67 +' 47'67
1' Dete Dete*#n *#ne e the the l#*#t l#*#t "! a'
6' 6'14
lim x→ 0
csc x −cot x x
b' 6
c' Un+e!#ne+
d. 0.5
8' 9) the c"n+#t# c"n+#t#"n "n "! a w#ll s&* s&* "! P 14666 #s #s le!t t" be hel+ hel+ #n t&st t&st b) he $&a+ $&a+ a*"&nts a*"&nts t" P 34666' 0hen w#ll the la+) ece#.e the *"ne) #! the !&n+ #s #n.este+ c"*-"&n+e+ %&atel)5 a. 7 years, 5 months
b' 4 )eas, *"nths
c' 21 )eas
+' 7 )eas
3' C"*-&te C"*-&te the the s#n(#n$ s#n(#n$ !&n+ !&n+ !act" !act" "! a et&n et&n ate ate "! 8; #n < )eas' )eas' a' 6'231 b' 6'221 c' '616 +' <'<=1 4' In the tw" -e$ test test w#th the the le.el at 9, the #nst& #nst&*ent *ent #s t" be 2'36 2'36 * ab".e 9, an+ the the "+ hel+ at '21 * w#th the #nst&*ent at A, the "+ ea+#n$ at A an+ 9 ae 3'8 * an+ 2'7 * es-ect#.el)' es-ect#.el)' Dete*#ne the +#!!eence #n ele.at#"n between A an+ 9' a' 4'6=1 * b' 8'=74 * c' 3'7=3 +' 3'314 7' Us#n$ Us#n$ the !"ll"w# !"ll"w#n$ n$ n"tes, n"tes, +ete* +ete*#ne #ne the the ele.at#" ele.at#"n n "! 9M 235 STATION 9M21 2 1 9M28 8 3 4 7 9M23 a'
2<8'=4
b' 183'1
9S 3'73 4'<6 1'14 7'61 <'=7 <'67 ='34 21'81
>S
E/EVATION 16='74
4'67 4'61 4'<4 3'=3 8'11 8'2 1'61 2'=7 c. 235.35
+' 184'17
' >#n+ >#n+ the the ."l& ."l&*e *e "! the the la$ la$es est t #$h #$ht t c#c c#c&l &la a c)l# c)l#n+ n+e e that that can can be c#c c#c&* &*sc sc# #be be+ + ectan$&la -aalell-#-e+ "! +#*ens#"ns 1 !t b) 8 !t b) 3 !t' a. 121.54 cu in b' 221'34 c' 234'41 +' 243'22 <' The tan$en tan$ent t t" the the c&. c&.e e ) ? e e at ? 6'< #s: c. 4 a' 2'< b' 8'7
ab"& ab"&t t
a
+' 4'1
=' The c"ss@sect#"nal c"ss@sect#"nal aea aea "! a "a+ w#th w#th w#+th "! 26 * #s 31'= s% s% *' The c"ss c"ss sect#"nal aea #s as !"ll"ws:
9.8 0 7.4 2.4 x 1.2
Dete*#ne the .al&e "! ' a' 8'=3 b' 1'34
c' 8'6
+' 8'34
26' A s#$ht #s ta(en w#th a en$#nees en$#nees le.el at "+ hel+ hel+ 266 * awa) an+ an #n#t#al ea+#n$ ea+#n$ "! 2'<8 * was "bse.e+' The b&bble #s then le.el th"&$h s# s-aces "n le.el t&be an+ the "+ ea+#n$ #s 2'=2 *' 0hat #s the sens#t#.#t) "! the b&bble t&be #n sec"n+s "! ac5
a' 1=B
b' 13B
c' 1<'4B
+' 1'4B
SITUATION 2: A *"+el "c(et #s la&nche+ !"* -"#nt A w#th an #n#t#al .el"c#t) "! <4 *s' I! the "c(ets +escent -aach&te +"es n"t +e-l") an+ the "c(et lan+s 266 * !"* A, 22' Dete*#ne the an$le the !"*s w#th the .et#cal' a' 8'=6 b' 1'84 c' 1'14
+' 4'14
21' Dete*#ne the *a#*&* he#$ht eache+ b) the "c(et' a' <24 * b' 877 c' 21=
+' 2=3
28' Dete*#ne the +&at#"n "! the !l#$ht' b. 17.3 a' 28'3 s
+' 26'3
c' 22'7
23' The +#stance !"* the *#+-"#nt "! the c&.e t" the *#+-"#nt "! the ch"+ "! a s#*-le c&.e #s <'21 *' I! the cental an$le #s 36, !#n+ the a+#&s' a' 283'73 * b' 213'78 c' 2<7'32 +' 224'41 24' It #s the +e$ee "! -ea(e+ness "! a stat#st#cal +#st#b&t#"n &s&all) ta(en elat#.e t" n"*al +#st#b&t#"n' a' S(ewness b' H)-"theses c' &t"s#s +' ANOVA 27' The e.en&e R !"* sell#n$ n&*be "! c"*-&te b"a+s #s $#.en b) C ? 46 F 4666' >#n+ h"w *an) b"a+s *&st be s"l+ t" bea( e.en5 a' 866 b' 366 c' 466 +' 766 2' A sw#**#n$ -""l #s c"nst&cte+ #n the sha-e "! tw" ".ela--#n$ #+ent#cal c#cles ha.#n$ a a+#&s "! = *' an+ each c#cle -asses th"&$h cente "! each "the' >#n+ the aea c"**"n t" the tw" c#cles a' 268 b' 261 c' 262 +' 266 2<' T#c(ets t" a bene!#t c"ncet c"st e#the P 21 " P 24' A t"tal "! 866 t#c(ets ae s"l+, an+ the t"tal ece#-ts wee P3236'66' H"w *an) P 21 t#c(ets wee s"l+5 a. 120 b' 2<6 c' 246 +' 166 3
2=' Calc&late the !"ll"w#n$ +e!#n#te #nte$al:
∫ ( x
2
+ 4 x ) dx
0
a' 34
b' 38
c' 3
+' 3=
16' A Ge*an s#l.e all") c"ns#sts "! 76; c"--e, 14; #nc an+ 24; n#c(el' Dete*#ne the *ass "! c"--e #n a 26 ($ sa*-le' c. 6 a' 1'4 b' 2'4 +' N"ne 12' Dete*#ne the +ete*#nant "! th#s *at#:
[ ] 8
2
0
0
2
8
2
0
0
2
8
2
0
0
2
4
a. 1552
b' 2431
c' 2417
+' 247=
11' The c"l" "! c&b *a(#n$s !" -a(#n$ est#ct#"ns shall be "! c".e#n$ the !ace "! the c&b' a' s"l#+ e+ c"l" b' s"l#+ )ell"w c"l" c' s"l#+ wh#te c"l" +' s"l#+ e+ an+ )ell"w c"l" 18' A ceta#n "-eat#"n #s n"w -e!"*e+ b) han+, the lab" c"st -e &n#t #s P 6'43 an+ the ann&al !#e+ cha$e !" t""l &se+ #s P 266 -e )ea' A *ach#ne that #s be#n$ c"ns#+ee+ !" th#s "b w#ll c"st P 1466 , ha.e a sal.a$e .al&e "! P 266 at an) t#*e an+ a !#e+ ann&al c"st "! P 866' 0#th the *ach#ne, the lab" c"st #s P 6'11 -e &n#t' Use sta#$ht l#ne *eth"+ "! +e-ec#at#"n' >" what
n&*be "! &n#ts "! -"+&ct -e )ea at e" #nteest an+ l#!e "! 7 )eas !" the *ach#ne w#ll the ann&al c"st "! the tw" *eth"+s bea( e.en5 a' 24 b' 2<4 c' 2=4 +' 164 13' An e-e#*ent c"ns#sts "! select#n$ thee #te*s !"* a b" "! se.eal #te*s' The thee #te*s ae class#!#e+ +e!ect#.e JDK " n"n@+e!ect#.e JNK as the) ae selecte+' G#.e the sa*-le s-ace !" th#s e-e#*ent' 0h#ch "&tc"*es ae #n the e.ents A ,9, an+ C whee A eactl) "ne +e!ect#.e5 a' JNNN, NND, DNNK b' JDNN, NDN, NNDK c' JDDD, DNN, NDDK +' JNDD, DND, DDNK 14' >#n+ the 14th te* "! an AP 4,<,22,23L a' <2 b' =
c'
+' 4
17' A *#t&e c"*-"&n+ !"* e%&al -ats "! tw", l#%&#+s, "ne wate an+ the "the "#l was -lace+ #n a he*#s-he#cal b"wl' The b"wl +e-th "! the tw" l#%&#+s #s 8 c*' A!te stan+#n$ !" a sh"t t#*e the *#t&e se-aate+ , the "#l settl#n$ ab".e the wate' I! the th#c(ness "! the se$*ent "! the "#l #s 2 c*, !#n+ the a+#&s "! the b"wl' a' 3'81 b' 2'44 c' 4'11 +' 8'7 SITUATION 1: The tan$ents "! a s-#al c&.e !"*s an an$le "! #ntesect#"n "! 14 at stat#"n 1 F 64<' Des#$n s-ee+ #s 76 (*h' >" a a+#&s "! cental c&.e "! 866 * an+ a len$th "! s-#al "! 41'26 *' 1' >#n+ the stat#"n#n$ at the -"#nt whee the s-#al stats' a' 2F=<7'11 b' 2F=<4'84 c' 2F=<<'11
+' 2F=<2'87
1<' >#n+ the stat#"n#n$ "! the stat "! the cental c&.e' a' 1F62'37 b' 1F66'3 c' 1F622'87
+' 1F626'==
1=' Calc&late the len$th "! the cental c&.e' a' 1'4 b' ='7
+' 3'8
c' <'<
86' The -e#*ete "! a t#an$le #s <3 * wh#le #ts aea #s 261 * 1 ' Dete*#ne the +#a*ete "! the c#cle that *a)be #nsc#be+ #n th#s t#an$le' a' ='22 b' 3'<7 c' 4'18 +' 7'33 82' 0hat #s the len$th "! the ch"+ c"**"n t" c#cles 1 F )1 ? 27 an+ 1 F )1 F21 ? 65 a' 7'81 b' 4'26 c' 3'77 +' '43 81' Pant#n" $"&- "! c"*-an#es #s -lann#n$ t" &n+eta(e a -"ect e%&##n$ #n#t#al #n.est*ent #! P 264 *#ll#"n' The -"ect #s e-ecte+ t" $eneate ann&al -"!#t "! P 14 *#ll#"n !" )eas' C"*-&te the -a)"&t -e#"+ #n )eas' a' 8'4 b' 2'1 c' 4'3 +' 7'1 88' A -aab"l#c ach #s 7 *' h#$h an+ 11 * w#+e at the b"tt"*' H"w h#$h ab".e the b"tt"* can a 24 * bea* be #nstalle+ h"#"ntall) ac"ss the ach5 a' 1'= b' 8'12 c' 22 +' '4 83' E-an+ J8 c#s 26#n+ the *"*ent "! #net#a "! the aea b"&n+e+ b) the c&.e ? <) , the l#ne ? 3 an+ the @ a#s' a' 14'7 b' 12'< c' 82'7 +' 87'3 87' San+ #s -"&#n$ !"* a s-"&t at the ate "! 14ccs' It !"*s a c"ne wh"se he#$ht #s alwa)s 28 the a+#&s "! #ts base' At what ate #n c*s #s the he#$ht #nceas#n$ when the c"ne #s 46 c* h#$h5 a' 6'666<4 b' 6'666123 c' 6'66633 +' 6'666843 SITUATION 8: Ch#na, Un#te+ States, an+ R&ss#a ae the c"&nt#es w#th the *"st cell&la s&bsc#bes #n the w"l+' T"$ethe , thee c"&nt#es ha.e 83'<; "! the w"l+ cell&la s&bsc#bes' I! the -ecent "! the w"l+ s&bsc#bes #n Ch#na #s 8'2; less than 3 t#*es the -ecent "! the w"l+ s&bsc#bes #n R&ss#a an+ the -ecent "! the w"l+ s&bsc#bes #n Un#te+ States #s 3'8; *"e than the -ecent "! the w"l+s s&bsc#bes #n R&ss#a, 8' Calc&late the -ecenta$e !" R&ss#a' a' 4'7; b' 2='8
c' ='=
+' 74'1
8<' Calc&late the -ecenta$e !" Un#te+ States' a' 4'7; b' 2='8
c' ='=
+' 74'1
8=' Calc&late the -ecenta$e !" Ch#na' a' 4'7; b' 2='8
c' ='=
+' 74'1
36' I! the s&* "! the !#st 28 te*s "! tw" AP ae #n the at#" "! :8 , what #s the at#" "! the# c"es-"n+#n$ se.enth te*s5 a' 3 b' 3 c' 8 +' 3 32' These ae chea-e .a#et) "! -a#nts #n wh#ch chal( #s &se+ as base an+ wate #s &se+ as a ca#e' a' 9#t&*en b' Plastes c' D#ste*-es +' P#$*ents 31' Tw" -"#nts l#e #n a h"#"ntal l#ne +#ectl) n"th "! a b&#l+#n$ 46 * h#$h' The an$les "! +e-ess#"n t" the -"#nts ae 13 an+ 41 ' The +#stance between the -"#nts #s cl"sest t": a' <4'4 * b' <'< c' =7'8 +' 8'1 38' It #s a elat#.el) th#n +e-"s#t "! *#neal between +e!#n#te b"&n+a#es' a' Ve#n b' St#(e c' D#+' D#!t 33' Dete*#ne the $a+#ent "! the !&nct#"n 1 F )1 F 1 at -"#nts PJ2,1,8K a' '3< b' <'14 c' 7'66 +' '12 34' In a b"wl#n$ $a*e, h"w *an) wa)s can the 26 -#ns !all #n the !#st th"w5 a' 2618 b' 871<<66 c' 2 +' 1613 37' Ele.en w"(es "! the sa*e s(#ll wee "n a "b that c"&l+ be !#n#sh #n 24 +a)s' A!te 26 +a)s, 4 w"(es %&#t' H"w *an) +a)s *"e w"&l+ #t ta(e the e*a#n#n$ w"(es t" !#n#sh the "b5 a' 7'= b' ='88 c' ='2 +' 4'4
3' I!
tan B =
x
√ 1
x
2
−
, !#n+ c"s 9'
b' s%t 2@ 1
a'
c' s%t 1 2
+' 2
3<' >#n+ the +#!!eent#al e%&at#"ns "! a !a*#l) "! c#cles th"&$h the "#$#n' a' + F ) +) ? 6 b' +) F ) + ? 6 c' + ) +) ? 6 +' +) ) + ? 6 2 2 y
3=' >#n+ the #nte$al "!
∫∫ x 0
a' 11'4
2
dxdy
0
b' 27'8
c' 26'
+' ='4
46' A t&n"&t has a !"$ n&*be "! = w#th a len$th "! heel s-ea+ e%&al t" 887'22 **' C"*-&te the len$th "! heel' H#nt: >N ? / "! heel / "! s-ea+ a' 8164 b' 8864 c' 8614 +' 82=4 42' M' C&) can !#n#sh a "b #n 7 +a)s an+ M' Th"*-s"n can !#n#sh the sa*e "b #n 21 +a)s' H"w *an) +a)s the "b w#ll be +"ne #! b"th "! the* w#ll w"( t"$ethe5 a' 8 b' 3 c' 4 +' 1 41' A c&.e has an e%&at#"n 1 s#n 1 ? 7' Tans!"* t" ectan$&la c""+#nates' a' ) ? 8 b' ) ? 7 c' ) ? 8 +' ) ? 7 48' The #nte#" an$les "! a %&a+#lateal ae as !"ll"ws: An$les A 9 C D
Val&e =1 << 2 226
C"*-&te the c"ecte+ .al&e "! an$le C' a' =287 b' <3<
N"' "! Meas&e*ents 1 3 8 7 c' 633
+' 26<1
43' It #s a +e.#ce wh#ch +ete*#nes "nes +#ect#"n w#th e!eence t" *a$net#c n"th' a' c"*-ass b' *ac' le.el +' "+"*ete 44' A b" ha.#n$ a s%&ae base an+ "-en t"- #s t" be *a+e w#th the least *ate#al t" be &se+' 0hat sh"&l+ be the base e+$e #n *etes #! the ."l&*e #s t" be 81 * 8' a' < b' 3 c' 7 +' = 47' The +at&* l#ne !" +es#$n "! -"t !ac#l#t#es #n acc"+ance w#th chats wh#ch #s be#n$ &se+ b) the Ph#l' P"ts A&th"#t)
a' b'
/"w 0ate /e.el Res#+&al 0ate /e.el
c' H#$h 0ate /e.el +' Mean /"w 0ate /e.el
4' A *ach#ne c"st#n$ P 34666 #s est#*ate+ t" ha.e a sal.a$e .al&e "! P3846 when et#e+ at the en+ "! 7 )eas' De-ec#at#"n c"st #s c"*-&te+ &s#n$ a c"nstant -ecenta$e "! the +ecl#n#n$ b""( .al&e' 0hat #s the ann&al +e-ec#at#"n #n -ecent5 a' 18'77; b' 81'14; c' 36'1; +' 27'<4; 4<' I! a s&* "! *"ne) t#-les #n a ceta#n -e#"+ "! t#*e at a $#.en ate "! #nteest, c"*-&te the .al&e "! the s#n$le -a)*ent -esent w"th !act"' a' 6'14 b' 6'4 c' 6'888 +' 6'4 4=' S"l.e !" that sat#s!) 1 F 87 ? = 1 1 a' 7# b' 8#
c' =#
76' A se#es "! n&*bes wh#ch ae -e!ect s%&ae n&*bes #s (n"wn as a' >#b"nacc# b' >"&#e c' >e*at 72' Inte$ate tan Jln K a' ln c"sJln K F C b' ln secJln K F C
+' @=#
Se#es +' E&le
+ c' tan1 Jln K F C +' tan Jln K1 FC
71' Data "n ta!!#c acc#+ent ec"+e+ "n a ceta#n #ntesect#"n !" the -ast 4 )eas has an acc#+ent ate "! 3276 -e *#ll#"n ente#n$ .eh#cles' I! the a.ea$e +a#l) ta!!#c ente#n$ the #ntesect#"n #s 463 , !#n+ the t"tal n&*be "! acc#+ents +&#n$ the 4 )ea -e#"+' a' 331 b' 8<17 c' 4682 +' 761 78' The +e!lect#"n an$les "! tw" #nte*e+#ate -"#nts A an+ 9 "n a c#c&la c&.e ae 864 an+ <24 es-ect#.el)' I! the ch"+ +#stance A9 #s 36 * , c"*-&te the len$th "! the c&.e !"* PC t" A' a' 17'4 b' 18'=6 c' 11'4 Q+' 13'67 73' 9")les law states that !" a $as "! c"nstant te*-eat&e, the ."l&*e "! a !#e+ *ass #s #n.esel) -"-"t#"nal t" #ts abs"l&te -ess&e' I! a $as "cc&-#es a ."l&*e "! 2'4 * 8 at a -ess&e "! 166666 Pascals' Dete*#ne the ."l&*e when the -ess&e #s <66 26 8 Pascals' a' 6'84 *8 b' 6'44< c' 6'184 +' 6'<< 74' An a#-lane !lew !"* Man#la J2387 N, 21264 EK at a c"&se "! S 86 E *a#nta#n#n$ a ceta#n alt#t&+e an+ !"ll"w#n$ a $eat c#cle -ath' I! #ts $"&n+s-ee+ #s 846 (n"ts, a!te h"w *an) h"&s w#ll #t c"ss the e%&at"5 a' 1'< hs b' 1'1 c' 8'2 +' 8'= SITUATION 3: A l#ne 266 * l"n$ was -ace+ b) a s&.e)" !" !"& t#*es w#th the !"ll"w#n$ +ata: 231 , 234 , 234'4 , an+ 237' Then, an"the l#ne was -ace+ !" !"& t#*es a$a#n w#th the !"ll"w#n$ es<s: <=8 , <=8'4 , <=2 , an+ <=4'4' 77' Dete*#ne the -ace !act"' a' 6'62 b' 6'7<4
c' 6'7=2
+' 6'11
7' Dete*#ne the n"' "! -aces !" the new l#ne' a' <=8'14 b' <=8'72 c' <=8'47
+' <=8'
7<' C"*-&te the +#stance "! the new l#ne' a' 72'43 b' 72'13
+' 72'83
c' 72'33
7=' >#n+ the /a-lace tans!"* "! !JtK ? 8t c"s J1tK' a' 8Js1 F 3KJs1F3K1 c' 8Js1 3K Js1 F3K b' 8Js1 3K1 Js1 @3K1 +' 8Js1 3KJs1F3K1 6' C"n.et t" Aab#c N&*be s)ste*: MCMXCIII' a' 2==8 b' 2<=8 c' 1228
+' 16=8
2' >#n+ the -e#*ete "! the ell#-se $#.en e ?6'4 an+ +#stance between the !"c# #s 7' a' 13'<< b' 1<'34 c' 81'74 +' 14'32 1' 0#th #nteest ate "! =; c"*-"&n+e+ c"nt#n&"&sl), what #s the -esent w"th "! a -e-et&#t) "! P <666 -a)able *"nthl)5 a' P 2143148 b' P2248147 c' P 26717== +' P 2437816
8' A -aab"l#c c&.e has a +escen+#n$ $a+e "! @6'<; wh#ch *eets at an ascen+#n$ $a+e "! 6'3 ; at Sta' 26F616' The *a' all"wable chan$e "! $a+e -e 16 * stat#"n #s 6'24' Ele.at#"n "! 26F616 #s 136'76 *' 0hat #s the len$th "! the c&.e' a' 216 b' 276 c' 246 +' 236 SITUATION 4: Answe the !"ll"w#n$ te*s #n Anal)t#c Ge"*et)' 3' It #s a l"c&s "! -"#nts #n wh#ch e.e) -"#nt thee#n #s e%&#+#stant t" a !#e+ -"#nt' a' c#cle b' -aab"la c' ell#-se +' h)-eb"la 4' It #s a ch"+ -ass#n$ th"&$h the !"c&s an+ -aallel t" the +#ect# "! a c"n#c sect#"n' a' *a" a#s b' eccent#c#t) c' lat&s ect&* +' .ete
