SHW5-01 Total: 21 marks

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Geoffrey Moody
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1 SHW5-0 Total: arks.. 3. x ( ) (5 ) (Pyth. theore) x 5 3 x cos60 8 x 4 9 sin x 5 3 sin x 5 x Consider the area of C, we have 5 x 63 x Consider the area of C, we have x 5 x 6 6. BC cos60 8 BC 4 AC sin60 8 AC 4 3 Area of C BC AC 3 A A A x (8 ) (5) (Pyth. theore) x Consider the area of D, we have x y 85 7y 0 y 7.06(cor.to3sig.fig.) 9. In D, AD co5 45) AD cos60 6 In ACD, CD tan 45 6 CD 6tan 45 6 In D, BD sin(5 45) ( x 6) sin 60 x (sin 60 6) 0. In C, x sin x 5 y BC CD 4.39 (cor.to 3 sig.fig.) x x tan 30 tan A M+ +A 7. In BCD, x ( ) ( 5 ) (Pyth. theore) x ( 5) 3 In D, x cos y 4 3 cos y 4 y 4.4 +A 8. In D,

2 SHW5-A Total: 8 arks. Area of C 66sin Area of R 88sin (cor.to3sig.fig.) 3. Area of C 7 xsin x 7 x 9 9. Area of the shaded region area of sector AOB area of AOB sin Area of C sin 33 sin or (rejected) 54.6 (cor. to 3sig.fig.) BC (property 7 Area of CD 3.5 of rhobus) area of 7 7 sin 40 C BC AD (opp. sides of // gra) 9 Area of ACD areaof C 69sin Area of PRS area of S area of R 4 ( 3) sin 68 4 sin PRQ 60 (property of equil. ) SR PR 8 Area of quadrilateral RS area of R area of PRS 88sin 60 88sin

3 SHW5-B Total: arks. Let a b c s, where a 5, b 4 and c 5. a b c s 7 s a s c (75) s b (7 4) 3. Let Area of C s a)( s b)( s c) a b c s, where a, b 5 and c s 8.5 s a (8.5) 6.5 s b (8.55) 3.5 s c (8.50) 8.5 Area of C s a)( s b)( s c) Let a b c s, where a 7, b and c s.5 s a (.57) 5.5 s b (.5) 0.5 s c (.56) 6.5 Area of C s a )( s b )( s c) BC AP AP AP Let a b c s, where a 6, b and c. 6 s 5 s a (56) 9 s b (5) 3 s c (5) 3 Area of C s a )( s b )( s c) BC AP AP 8775 AP.7 5. (a) : QR : RP 4x : 5x : 6x 4 : 5 : 6 (b) (i) A The perieter of R is 45. 4x 5x 6x 45 5x 45 x 3 (ii) Fro (b)(i), we have 4 3, QR 5 3 5, PR Let p q r s, where p 5, q 8 and r. 5 8 s.5 s p (.55) 7.5 s q (.58) 4.5 s r (.5) 0.5 Area of R s p)( s q)( s r) QR RS 8 PR PS RS (Pyth. theore) PR Consider R. Let RQ PR s s 3 s s QR (38) 5 s PR (30) 3 Area of R s )( s QR)( s PR) Area of quadrilateral RS area of PSR + area of R A 7. OB OA 6 (radii) 4

4 Let OA OB s s s OA s OB ( 6) 5 s (0) Area of the shaded region s OA)( s OB)( s ) Consider R. Let QR PR s. s 7 s s QR (7) 6 s PR (7) 5 Area of R s )( s QR)( s PR) 3060 Area of R area of PRS 3060 PS sin 3 PS 7.4 +A 5

5 SHW5-C Total: 8 arks. By the sine forula, AC sin C sin B x 6 sin 55 sin 48 6 sin 55 x sin A B C 80 ( su of ) 80 B 5 80 B 48 By the sine forula, AC sinc sin B x 0 sin5 sin 48 0sin 5 x sin (cor.to3sig.fig.) 3. By the sine forula, AC sin B sin C 9 6 sin sin 38 9 sin 38 sin or.5579 (rejected) By the sine forula, PR QR sin Q sin P sin sin 45 8 sin 45 sin (rejected) or By the sine forula, a b sin A sin B a 8 sin 50 sin 75 8 sin 50 a sin By the sine forula, b a sin B sin A 6 5 sin B sin 46 6 sin 46 sin B 5 B or or 0 7 By the sine forula, b a sin B sin A 5 sin B sin 85 sin 85 sin B 5 B or (rejected) A B C 80 ( su of ) 6 40 C 80 C 78 By the sine forula, a b sin A sin B a 7 sin 6 sin 40 7 sin 6 a sin By the sine forula, c b sin C sin B c 7 sin 78 sin 40 7 sin 78 c sin (cor.to3sig.fig.) +A 6

6 9 By the sine forula, b c sin B sin C 4 9 sin B sin 30 4 sin 30 sin B 9 B or (rejected) 9.9 A B C 80 A A By the sine forula, a c sin A sin C a 9 sin sin30 9 sin a sin In ACD, by the sine forula, AC CD sin ADC sin CAD x 8 sin 40 sin 5 8 sin 40 x sin ACB CAD ADC (ext. of ) Consider C. sin ACB AC y sin y.0 + RS PS 9 PRS RPS 36 (base s, isos. ) PRS RPS RSP 80 ( su of ) RSP 80 RSP 08 In S, by the sine forula, PS sin PSQ sin S x 9 sin08 sin 5 9sin 08 x sin A 7

7 SHW5-D Total: 8 arks. By the cosine forula, x 4 5 (4)(5) cos 50 x 4 5 (4)(5) cos (cor. to 3sig. fig.). By the cosine forula, cos (7)(6) By the cosine forula, a b c bc cos A a.3 9 (9)() cos By the cosine forula, a b c cos C ab (0)(3) C (4)(3) cos DC (opp. sides of // gra) 3 Consider BCD. By the cosine forula, BD BC DC ( BC)( DC) cos BCD BD 6. By the cosine forula, b a b c 4 0 (4)(0) cos ac cos B 9.07 b c a cos A bc (9.0659)(0) A A B C 80 ( suof ) C A b c a cos A bc (7)(8) A a c b cos B ac (5)(8) B 60 A B C 80 ( su of ) 8.8 C Consider BCD. By the cosine forula, CD BC x BD 9 6 (9)(6) cos Consider ACE. By the cosine forula, CE AC CE y Consider MNC. MN MN ( BC)( BD) coscbd AE (8 9) (8 9)() cos80 y 0. MC 5 9 NC ( AC)( AE) coscae (Pyth.theore) NC cos MCN MC 5 MCN Consider C. By the cosine forula, AC BC ( AC)( BC) cos ACB.7 ++A + (6 5) (6 ) (6 5)(6 ) cos By the cosine forula, 0

8 SHW5-E Total: 0 arks. tan APB PB tan tan (cor.to3sig.fig.) The height of the building is QPM QM tan QPM QM tan QM 70 tan The distance between Q and M is Consider APB. tan APB PB 40 tan 35 PB 40 PB tan QB 30 tan 35 Consider AQB. tan AQB QB tan 35 AQB 55.9 The angle of elevation of A fro Q is M+ 5. sinpmq PM sin36 9sin5 sin 9sin5sin36 sin.7 The height of the tower is.7. M+ XPR PRY (alt. s, PX // YR) 3 QPR RQS YRQ (alt. s, QS // YR) 7 Consider R. By the sine forula, QR sinqpr sinprq QR 60 sin77 sin(3 7) 60sin77 QR sin40 Consider RQS. RS sinrqs QR RS sin 7 60sin 77 sin 40 60sin 77sin 7 RS sin The height of RS is 4.3. M+ PNQ NPR (alt. s, QN // PR ) 5 MPN PNM PMQ (ext. of ) MPN 5 36 MPN Consider PMN. By the sine forula, PM MN sin PNM sin MPN PM 9 sin 5 sin 9sin 5 PM sin Consider M.

9 6. 7. APB QAP (alt. s, BP // AQ ) 55 Consider P. sin APB AP 50 sin 55 AP 50 AP sin 55 AQP QPD (alt. s, AQ // PD ) 65 In AQC, ACQ AQP CAQ (ext. of ) In APC, by the sine forula, CP AP sin CAP sin ACP 50 CP sin 55 sin(0 55) sin sin 75 CP sin 55sin 45 Consider CPD. CD sin CPD CP CD sin 65 50sin 75 sin 55sin 45 50sin 75sin 65 CD sin 55sin The height of CD is M+ (a) PRQ SPR (alt. s, RQ // SP ) 8 Consider PRQ. tan PRQ RQ tan tan The height of is 3.7. (b) PHRHRQRH 360 ( suof polygon) PHR 5 90 (90 40) 360 Consider PRQ. RQ cosprq PR 7 cos 8 PR 7 PR cos 8 In HRP, by the sine forula, HR PR sinhpr sinphr 7 HR cos8 sin(40 8) sin88 PHR PHR 88 7sin68 HR cos8sin The distance between H and R is

10 SHW5-F Total: 5 arks cosqpr PR cos 48 5k 5cos 48 k 6.7 k The distance between P and Q is 6.7 k. OP tan O OQ 0 9 O (a) BC AC cos BCA ( BC)( AC) 3 5 ()(3) BCA 73.6 (cor.to 3 sig.fig.) EAC BCA (alt. s, AE // BC ) 73.6 The copass bearing of C fro A is S73.6W. R O (alt. s, RP// QO) 48.0 (cor.to3sig.fig.) The copass bearing of Q fro P is S48.0ºE. 5. BAC By the cosine forula, BC AC ( )( AC) cosbac 8. BC 5 6 (5)(6) cos (cor.to3sig.fig.) The distance between B and C is (b) OQB AOQ (alt. s, QB// AO) 40 OQP By the sine forula, OP sin OQP sin POQ OP 30k sin 50 sin(5 40) 30sin 50 OP k sin k (cor.to3sig.fig.) The distance between O and P is 5.4 k. OOQPPOQ80 ( suof ) O50 (5 40) 80 O 65 O POQ 65 OQ (sides opp. equal s) 30 k The distance between O and Q is 30 k. (a) By the cosine forula, BC AC cos C ( )( BC) 5 3 (5)() C 56.3 (cor. to 3 sig.fig.) D C (alt. s, DA // BC ) 56.3 The copass bearing of B fro A is N56.3W. (b) By the cosine forula, 4

11 3. By the cosine forula, LS PL SP cos SLP ( LS)( PL) 30 3 (30)() SLP By the sine forula, PR QR sin R sin QPR PR 8 k sin sin 30 8 sin PR k sin k (cor. to 3sig.fig.) The distance between P and R is 5.8 k BLS ASL (alt. s, LB // AS ) 35 BLP CPL BLP (alt. s, CP // LB ) Reflex CPL ( s at a pt.) 3 (cor.to3sig.fig.) The true bearing of L fro P is 3º. M+ (a) QPR By the sine forula, QP QR sin QRP sin QPR 5 k 8 k sin QRP sin 30 5 sin 30 sin QRP 8 QRP APR PRB80 50 ( QRB) 80 (int. s, AP// BR) QRB 60.4 The copass bearing of Q fro R is N60.4W. (a) (b) OA 35 3 k 05 k OB 45 3 k 35 k DOA OAC (alt. s, DO // AC ) 35 DOA AOB EOB 80 (adj. s on st. line) 35 AOB AOB 95 By the cosine forula, OA OB ( OA)( OB)cos AOB (05)(35) cos 95 k k 78 k The distance between A and B is 78 k. By the cosine forula, OA OB coso ( OA)( ) (05)(78.035) O The true bearing of B fro A 80 BAC (adj. s on st.line) 80 ( ) 66 (cor.to 3 sig.fig.) (b) R QPR QRP80 ( suof ) R R

Trigonometric ratios 9B 1 a d b 2 a c b

Trigonometric ratios 9B 1 a d b 2 a c b Trigonometric ratios 9B 1 a a Using sin A sin B 8 sin 72 sin 30 8sin 72 sin 30 As 72 > 30, > 8 cm 15.2 cm ( ) ABC 180 68.4 + 83.7 27.9 Using a 9.8 sin 27.9 sin 83.7 9.8sin 27.9 a sin 83.7 4.61 cm ( ) 2