ELEVENTH YEAR MATHEMATICS

Transcription

1 The University f the State f New Yrk REGENTS HIGH SHOOL EXAMINATION ELEVENTH YEAR MATHEMATIS Mnday, June 8, 973- :5 t 4 :5 p.m., nly The last page f the bklet is the answer sheet. Fld the last page alng the perfratins and, slwly and arefully, tear ff the answer sheet. Then fill in the heading f yur answer sheet. The "Referene Tables fr Mathematis" whih yu may need t answer sme questins in this examinatin are stapled in the enter f this bklet. When direted t d s, pen the bklet and arefully remve the referene tables. Nte that the remainder f Referene Table is lated n a separate page. Fld this page alng the perfratins and, slwly and arefully, tear ff this page. Then lse the' bklet and leave it fae up. DO NOT OPEN THIS EXAMINATION BOOKLET UNTIL THE SIGNAL IS GIVEN

2 Part I Answer all questins in this part. Eah rret answer will reeive 2 redits. N partial red'it will be allwed. Unless therwise speified, answers may be left in terms f?t r in radial frm. Write yur answers in the spaes prvided n the separate answer sheet. Slve fr.r : \}2.: + 2 := 6 2 If (-6 + 3i) - (a - Zi) value f Q. 3 Fatr mpletely: 3 tan" A + tan A Find the value f.'r if lg.; + ) 27 =: 3. (0 + 5i), find the 5 A vehile travels at 40 m.p.h, fr the first 5 hurs and at 48 m.p.h. fr the next 3 hurs. What is its average rate f travel during the 8 hurs? 6 The measures f the sides f a right triangle are expressed as :r.," +, and % -. Find the num-erial value f x. 7 If s..'" == %3 and s y == %, find the value f s (x + y). 8 Tw sides f a parallelgram are 0 and 20 and ne angle has a measure f 30 0 Find the area f the parallelgram. 9 If tan f) =.2099 and f) is an aute angle, find the measure f f) t the nearest minute. Diretins ( 0-30): W-rite in the spae prvided n the separate answer sheet the numeral preeding the expressin that best mpletes eah statement r answers eah questin. 0 If 2 2 :D - 2 == 6, then the value f x is () 8 (3) 3 (2) 9 (4) 4 The ratinal number is equivalent t () (.006)2 (3) 3.6 X 0-6 (2) 3.6 X (4) 0,000 2 The graph f the equatin xy == 8 des nt pass thrugh the pint whse rdinates are () (-2,-4) (3) (~,6) (2) (0,8) (4) (4,2) 3 Whih is the slutin set f the inequality Iwt~-2 < 5? ( ) {..~ -5 < x < 5} (2) {..t" -3 <..~ < 7} (3) {..t"..'r < -5 r x > 5} (4) {..t".r < -3 r x > 5} 4 If ==. then whih an nt be a value f b? 2b +. () - (2) -h (3) 0 (4) % 5 Expressed in radians, an angle f 08 0 is equal t () ~ (3) ~ (2) 6?r 5 (4) 37' 0 6 The equatin ;.~ + (2 + 3x) ==.'t" + (2 + 3x) IS an illustratin f the () distributive prperty (2) assiative prperty fr additin ( 3) mmutative prperty fr additin ( 4) reflexive prperty fr equality 7 Fr whih value f 8 is the expressin tan" 8s 2 f) == sin" (j nt defined? () (2) 2 3 (3) (4) 8 The value f tan (Ar s -0) is 2 () (2) - 9 Whih slutin set gives all values fk whih will plae the turning pint f the graph f y:=x 2-6.t" + K belw the..r-axis P () {K > 36} (2) {K > 9} 4 6 (3)\}2 (4) -\}2 (3) {K<9} (4) {K< 36} Math. II-June '73 [2]

3 --, 20 If 'In == 0 and 'In =F, the mplex fratin m is equivalent t ()n- (2) t + (3) (4) 'Jt - 2 If a == 8, b == 8, and m L A == 30, then the triangle AB ( ) must be an aute triangle (2) must bea right triangle ( 3) must be an btuse triangle (4) may be either an btuse r an aute triangle 22 The perid f the graph f y == 3 s 2.:r is () (3) 3 (2) ~ (4) ~ - Jt 27 In triangle AB) a == 9, == 0, and nlla ==. Whih statement an be used t find the value f angle? () sin = 0 9 (2) sin == (3) sin == (4 ) sin == 9 sin sin sin The sum f the rts f a quadrati equatin is -5 and the prdut f the rts is -6. What is the quadrati equatin? ( ) :r 2-5z - 6 = 0 (2) :r 2 + 5z - 6 = 0 ( 3)..'r 2-5"t" + 6 = 0 (4),,-r 2 + 5",; + 6 = 0 23 A value f..t.. fr whih s,,'t: == sin (..: - 20 ) is () SS (3) 90 (2) 70 (4) The expressin sin (-60 ) written as a funtin f a psitive aute angle is () -sin 70 (2) -sin The expressin () (3) s 20 (4) sin In the diagram belw, a unit irle is drawn with radius DA ==. Angle () is in quadrant I as shwn. AD. D and Be. D. Whih line segment has tan fj + se () is equivalent t a length that is equivalent t tan ()? + sin f) (3) + sin () s () (2) + sin () (4) s f) 29 The graphs f the equatins 2% - Y 2y + x == 5 are tw lines whih ( ) interset at right angles (2) interset, but nt at right angles ( 3) inide (4) are parallel () AD (2) B 5 and 26 The lengths f tw sides f.a triangle are 3 and 5 and the inluded angle is 20. What is the length f the third side f the triangle? () 5 (3) 7 (2) 6 (4) 8 (3) O (4) OD Math. II-June '73 [3] [OVER]

4 '' --, 20 If 'J't =F 0 and t #, the mplex fratin --- is equivalent t () n - (2) n, + (3) (4), - 22 The perid f the graph f y == 3 s 2:r is () (3) 3 (2) ~ (4) ~ -- 'JIJ, 2 If a = 8, b == 8, and m LA == 30, then the triangle AB ( ) must be an aute trian.gle (2) must be a right triangle ( 3) must be an btuse triangle ( 4) may be either an btuse r an aute triangle - nt 23 A value f.:r fr whih s.;\,- == sin (..t ) is () 55 (3) 90 (2) 70 (4) The expressin sin (-60 ) written as a funtin f a psitive aute angle is () -sin 70 (3) s 20 (2) -sin 20 (4) sin The expressin () tan6+s() is equivalent t l+s~nb (3) + sin 6 s 6 (2) + sin () (4) s B 26 The lengths f tw sides fa triangle are 3 and 5 and the inluded angle is 20 0 What is the length f the third side f the triangle? () 5 (3) 7 (2) 6 (4) 8 27 In triangle AB} a == 9, == 0, and nlla ==. Whih statement an be used t find the value f angle? (l)sin= 0 9 (2) sin == (3) sin == (4) sin == 9 sin sin sin The sum f the rts f a quadrati equatin is -5 and the prdut f the rts is -6. What is the quadrati equatin? () x 2 - Sx - 6 = 0 (2) x 2 + 5x - 6 = 0 (3) ~lt2-5.;t~ + 6 = 0 (4)..'" t" + 6 = 0 29 The graphs f the equatins 2% - Y 2y +..t~ == 5 are tw lines whih ( ) interset at right angles (2) interset, but nt at right angles ( 3) inide (4) are parallel 30 In the diagram belw, a unit irle is.drawn with radius OA ==. Angle () is in quadrant r as shwn. AD.L O and B..L. Whih line segment has a length that is equivalent t tan (J? () AD (2) B (3) O (4) OD 5 and Math. II-June '73 [3] [OVER]

5 Answers t the fllwing questins are t be written n paper prvided by the shl. Part II Answer fur questins frm this part. Shw all wrk unless therwise direted. 3 a The sum f an irratinal number..: and its reipral (multipliative inverse) is 4. Find the value(s) f x. [Answers shuld be left in radial frnt.] [7] b hek ne f the answers fund in part a. [3] 32 a On the same set f axes, sketh the graphs f y == 2 sin x and y == s ~x in the interval L.,'t L 2'7r. [8] b Fr hw many values f ~t" in the interval 'T' L.. L. 3'7l' ---,; des 2 sin x == s ~x? [2] 33 a If sin A = ys and A is in the send quadrant, 3 find the value f s 2A and f sin 2A. [4] b In what quadrant des the lie? [2] 34 Slve and hek: [8,2] 2z2 - y2 _ 8 = 0 y==x+2 terminal side f 2A Fr all values f X" fr whih the expressin is defined, shw that the fllwing is an identity: [4] - tan 2.,t. 2 s" X' - == ~ + tan" x 35 The frmula t = O~ 7f'~ relates t the mtin f a 36 Answer either a r b but nt bth: [0] a In trapezid ABeD) AB is parallel t D;t AB == 22 in., mla == 90, rle = 00, and nldb == A 22 in. Find D t the nearest inh. OR b A metal brae in the frm f a triangle, AB, has rn L B == 06, AB == 2 m., and Be = 24 em, Find the length f A t the nearest entimeter. *37 a On the mplex rdinate axes, using vetrs, represent the fllwing mplex numbers, r, s, t where r = 3 + 4i) s == 2-2i) and t = i. [4] b Shw graphially the vetr sum r + s. [3] Shw graphially the vetr differene t - r, [3] * This questin is based n an ptinal tpi in the syllabus. B pendulum. When r == 3.4, L == 0.74, and g == 980, find, by using lgarithms, the value f t t the nearest thusandth. [0] Math. II-June '73 [4]

6 en u ee D f!.t= Q) 'I-.. en en E s: D.i:.a e 0' t--.j D : U 0 D. E E 0 D 0 E ~~ mmn Lgarithms f Numbers N i Ji I S SO N mmn Lgarithms f Numbers N S S _ _ M IS SHi N This table gives the mantissas f numbers with the deimal pint mitted in eah ase. harateristis are determined frm the numbers by inspetin.

7 en.-..u :J.L J -- (l) E 0' - a- I-. fi) D ::J - Values f Trignmetri Funtins Angle Sin s Tan t I 0 00' ' SO ' ' ' SO DO' ' ' ' S 00' ' SO ' ' SO ' ' SO I 8 00' ' SO g 00' ' s Sin t Tan I Values f Trignmetri Funtins Angle Sin s Tan t g 00' ' ' ' ' ' ' SO ' ' SO' ' ' ' ' ' ' ' ' ' ' ' ' s Sin t Tan Angle

8 us..u :J La.!:! I) E ' ~ 't., :J Values f Trignmetri Funtins OB I Tan t Angle I Sin I 8 00' ' ' ' SO ' ' ' ' ' ' ' ' ' SO ' ' SO ' ' ' I.5095 I.9626 I 63 00' I s Sin I t Tan I Values f Trignmetri Funtins Angle I Sin Ol Tan t 27 00' ' SO ' ' SO ' ' SO ' ' SO SO ' ' SO 20 ~ ' ' ' ' ' ' ' S 00' ' ' s Sin I t I Tan Angle

9 @ Values f Trignmetri Funtins Lgarithms f Trignmetri Funtins's Angle Sin s Tan I t Angle L Sin L s L Tan L t en...u ::J La.. u.. 4D e.- J' '- t- en D :I ' ' SO ' ' ' ' ' SO ' ' ' ' SO ' ' SO ' ' SO ' ' ' ~ ' SO ' ' Ol Sin t Tan AnJlle U) e.;: U ::J LL ei: D E at.t: I-. en E s:.t: ' ' ' ' ' ' SO ' ~ ' 0 8~ SO ' ' SO ' ' SO ' ' ' ' SO SO ' ' SO g 00' ' L e L Sin L t L Tan AnKle *These tables give the lgarithms inreased by 0. Hene in eah ase 0 shuld be subtrated.

10 0 s he Lgarithms f Trignmetri Funelns'" Angle L Sin L Ol L Tan L t g 00' ' ' ' SO ' ' ' ' ' ' ' ' ' ' ' ' ] ' ' ' ' L s L Sin L t L Tan An&:le Lgarithms f Trignmetri Funtins. Tan L t Angle ILSin I LOl IL 8 00' ' ' e 00' ' ' " SO ' ' ' ' ' ' ' ' ' ' i i ' ' SO 20 9.~ ~ ' ? ' I L Ol LSin L t L Tan Angle * These tables give the lgarithms inreased by 0. Hene in eah ase hrmlrl snbtrar-ted.

11 Lgarithms f Trignmetri Funtins. Lgarithms f Trignmetri Funtins" *. :.- U ::J La u '- D E 0' ~... en E s:.-. a 0' Angle L Sin L Ol L Tan L t 27 00' ' ' ' SO ' ' ' ' , ' ' ' ' ' ' ' ' SO ' ' SO ' ' L s L Sin L t L Tan Anal. * These tables give the lgarithms inreased by 0. Hene in eah ase 0 shuld be subtrated. Angle L Sin L s L Tan L t 36 00' ' SO ' ' ' ' ' ' ' ' ~ ' ' SO 20' SO ' ' ' ' ' ' ' i ' L s L Sin I L t I L Tan Angle

12 E Mnday, June 8, :5 t 4 :5 p.m., nly Just befre the start f the examinatin perid, distribute ne examinatin bklet, fae up, t eah pupil. Instrut the pupils t read the diretins n the ver f the examinatin bklet and t detah the answer sheet and fill in its heading. When eah pupil has reeived a bklet and finished filling in the heading f the answer sheet, instrut the pupils t pen their examinatin bklets and arefully remve the Referene Tables whih are stapled in the enter f the bklet. Instrut the pupils t als arefully remve the remainder f Referene Table whih is lated n a separate perfrated page. After these tables have been remved frm the bklets, diret the pupils t pen their examinatin bklets and begin wrk. Use nly red ink r penil in rating Regents papers. D nt attempt t rret the pupil's wrk by making insertins r hanges f any kind. Use hekmarks t indiate pupil errrs. Unless therwise speified, mathematially rret variatins in the answers will be allwed. In prblems invlving lgarithms, answers shuld be left rret t fur signifiant digits unless diretins say therwise. Units need nt be given when the wrding f the questins allws suh missins. SORING KEY Part I Allw 2 redits fr eah rret answer; allw n partial redit. Fr questins 0-30, allw redit if the pupil has written the rret answer instead f the number, 2, 3, r 4. () 8 () 3 (2) 3 (2) -6 (2) 2 (22) 2 (3) (3 tan A - 2) (tan A + ) (3) 2 (23) (4) 2 (4) 2 (24) 2 (5) 43 m.p.h. (5) 3 (25) 3 (6) 4 (6) 4 (26) 3 (7) 3%5 (7) (27) 4 (8) 00 (8) (28) 2 (9) 50 26' (9) 3 (29) (0) 3 (20) 2 (30) 2 [OVER]

13 ELEVENTH YEAR MATHEMATIS - nluded Part II Please refer t the Department's pamphlet Suggestins n the Rating f Regents E..xaminatin Papers in Mathematis, are shuld be exerised in making dedutins as t whether the errr is purely a mehanial ne r due t a vilatin f sme priniple. A mehanial errr generally shuld reeive a dedutin f 0 perent, while an errr due t a vilatin f sme ardinal priniple shuld reeive a dedutin ranging frm 30 perent t 50 perent, depending n the relative imprtane f the priniple in the slutin f the prblem. (3) a 2 + v'~ 2 - V3 b hek [3] [7] (37) I I t Y J \ (32) b [2] (33) a s 2A sin 2A - bill [2] (34)..t" = 6, y = 8 x = -2, y = y is 9 [8] [2] [2] ~ \ IOli ~ t-r \, ~ "- ~ r-, \ 5- r-, ~ I r ~ \ il -,, I r s -, \ ~, ", ~- ~-"" ~ L, ~. 5 0 " 5- -x--, ~ s (35) [0] (36) a 73 OR b 36 [0] [0] a b ~5i I [4] [3] [3] DO YOU KNOW... that pratially all bjetive questins used n the Regents examinatins have been "pretested" n a representative sample f students in New Yrk State shls? Over 6,000 questins in 6 subjet areas were tried ut in May 972. These questins were assembled int 267 pretest frms that uld be administered in a singlelassrm perid. Sme 53,000 students in 355 shls thrughut New Yrk.State partiipated in this pretesting f questins fr pssible use.. in future Regents examinatins.