SHW6-R1 1M+1A 1M+1A 1M+1A. 11. (a) 14. (a) With the notations in the figure, With the notations in the figure, AG BH 800 m Consider ACG.

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1 SHW6-R (a) 4. (a) With the notations in the figure, With the notations in the figure, AG BH Consider G. ΑG tan G tan 50 tan 50 Consider CHG. GH tan H GH tan 70 tan 50 tan 70 GH tan 50 The speed of the aeroplane tan 70 tan 50 0 s 84 /s (cor. to sig. fig.) (b) Consider CHG. cosh CH cos 70 tan 50 CH CH tan 50 cos 70 Consider BCH. BH tan BCH CH tan 50 cos 70 BCH. (cor. to sig.fig.) The angle of elevation of B fro C is.. (b) 5 ) Consider AOX..5 cos OA.5 OA cos (cor. to sig.fig.) With the notations in the figure, VA VO OA (Pyth. theore) VO.0 Area of base ABCDE 5 area of AOB.5 cos sin 54 cos Volue of the right pyraid fored area of base ABCDE VO (cor. to sig.fig.) M

2 7. (a) (i) Consider ADF. DF sin DAF AD DF sin 0 4 DF HK DF Consider AHK. HK sin HAK AH sin 6 AH AH sin Consider BHK. HK sin HBK BH sin 0 BH BH sin (cor. to sig. fig.) 5.85 (cor. to sig. fig.) (ii) Consider ABH. By the cosine forula, we have AH BH AB cos AHB ( AH)( BH) (4.56)(5.8476) AHB (cor.to sig.fig.) (b) Tie required for the blue toy car to reach H /s 5. s (cor. to sig. fig.) Tie required for the green toy car to reach H /s 4.6 s (cor. to sig. fig.) The two toy cars will not reach H at the sae tie. 8. (a) (i) BD BD AD OB BD AB In VOB, VB OB VB (Pyth. theore) (property of square) VO 8 (Pyth. theore) (cor. to sig. fig.) (ii) Consider VAB. VA VB 4 By the cosine forula, we have VB AB VA cos VBA ( VB)( AB) (b) ( 4) 0 ( 4) ( 4)(0) VBA AN sin VBA AB AN 0 sin (cor. to sig. fig.) (cor. to sig. fig.) A (c) VAB VCB CN VB and CN = AN The angle between the planes VAB and VBC is ANC. BD 00 Consider N. By the cosine forula, we have AN CN cos ANC ( AN)( CN) ( 00) (8.857)(8.857) ANC 06 (cor. to sig.fig.) The angle between the planes VAB and VBC is 06.

3 6. (a) Consider Figure (a). AE CE (Pyth. theore) AE (5 5) 6 8 ABD ~ E (AAA) AD AB AE (corr. sides, ~ s) AD AD 4 (b) (i) The angle between the line and the horizontal table is CAE. Consider DAE in Figure (b). By the cosine forula, we have EA AD DE ( AD)( DE) cos ADE EA 4 4 (4)(4) cos Consider CAE in Figure (b). CE tan CAE EA 6.76 CAE (cor.to sig.fig.) The angle between the line and the horizontal table is (ii) Consider CAE in Figure (b). CE sin CAE 6 sin sin Consider B in Figure (b). By the cosine forula, we have AB BC cos CBA ( AB)( BC) sin (5)(5) CBA 8.55 Area of ABC AB BC sin CBA 5 5 sin (cor. to sig.fig.) M+ 9. (a) Let M be the projection of Q on CD. AFP CHQ (RHS) FP HQ (b) (i) PE QE QM ED 6 Consider PEQ. PE QE QE PQ ( ) 9 QE CM CD MD 9 6 Consider CMQ. CQ QM CQ CM (Pyth. theore) 9 (Pyth. theore) 7.4 (cor. to sig.fig.) Let X and Y be the id-points of PQ and respectively. Let Z be the projection of Y on the plane EFGH. The angle between the planes QP and D is XYD, i.e. XYD. Consider BCD. BD BC BD CD YD BD 7 7 Consider EXQ. (Pyth. theore)

4 EQ EX EX XQ 9 XZ EZ EX YD EX 7 (Pyth. theore) Consider XYZ. YZ tan YXZ XZ 6 7 YXZ 65.4 (cor. to sig.fig.) XYD YXZ (alt. s, YD// ZE) 65.4 (cor.to sig. fig.) M+ (ii) XY is a line of greatest slope of the inclined plane QP. The angle between the planes QP and D, i.e. is greater than the angle between the line QC and the plane D. To s clai is agreed.

5 SHW6-P (a) (i) Consider ABC. (b) (i) By the cosine forula, we have BC AB cos B ( )( BC) (5)(5) B (cor.to sig.fig.) BCX B 5. (ii) In BCX, BCX BXC CBX 80 ( su of ) CBX By the sine forula, we have CX BC sin CBX sin BXC 5 sin CX sin (cor. to sig.fig.) Let A be the projection of A on the horizontal plane. AX CX (5.676).64 BX BC sin BCX sin BXC 5 sin BX sin A AX sin X AX cos B X XB Consider A BC. C B BC ( B)( BC) cos CBX C (cor. to sig. fig.) Consider AA C. C A (Pyth. theore) (cor. to sig. fig.) M+ (ii) The tetrahedron ABXC has a axiu height when AXB is perpendicular to the horizontal plane. i.e. The tetrahedron ABXC has a axiu volue when AXB is perpendicular to the horizontal plane. When increases fro 5 to 8.7, the volue increases; when increases fro 8.7 to 5, the volue decreases. +A. (a) (i) Consider CPQ. By the cosine forula, we have PQ CP CQ ( CP)( CQ)cos PCQ PQ 6 5 (6)(5) cos (cor. to sig. fig.) (ii) Consider P. cos P CP cos 5 6 6cos AP sin P CP AP sin 5 6 AP 6sin Consider BCQ. BC cos BCQ CQ BC cos55 5 BC 5cos BQ sin BCQ CQ BQ sin 55 5 BQ 5sin Let N be a point on BQ such that PN BQ. NQ BQ AP ( ) Consider NPQ. NP NQ NP PQ (Pyth. theore) AB NP 9.60 Consider ABC. By the cosine forula, we have BC AB cos B ( )( BC) B (4.5009)(8.606) 09 (cor. to sig. fig.) M+

6 (b) D B 80 (adj. s on st. line) D Consider D. By the cosine forula, we have AD DC ( )( DC)cos D AD (4.5009)(8.606) cos Consider ADP. DP AP DP AD (Pyth. theore) (cor. to sig. fig.) DP is shorter than CP. Sion s clai is disagreed. +A. (a) Let M be the id-point of FD. AF = AD and EF = ED with DM = FM AM DF and EM DF (prop. of isos. ) DF is the line of intersection of the planes AFD and EFD with AM DF and EM DF. The angle between the planes AFD and EFD is AME. Consider EFM. EM FM EF (Pyth. theore) (b) (i) Consider AEM. AE tan AME EM AE tan AE 64 tan (cor. to sig. fig.) Distance between XY and BC EM tan (cor. to sig.fig.) M+ M+ (ii) XF is not a line of greatest slope on the plane XFDY. The angle between the planes XFDY and EFD ust be greater than that between the line XF and the plane EFD. +A EM 0 64

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