PLMGS(S) 4049 AM Prelim P1 Solutions 2021

Preliminary Examination 2021 Secondary 4 Express Additional Mathematics Qn No Working Total 1(i) 1 1 3Given tan 4 3 tan 4 Since principal values are 90 tan 90 , hence angle in 4th quad. 4cos 5 x x x x           ALTERNATIVE 22 2 sec 1 tan 3sec 1 4 25 16 5 4 4cos 5 xx x x        1 1(ii) 2cos 1 2sin 2 1 cossin 22 4Sub cos , 5 41 5sin 22 1sin 2 10 11sin or sin22 10 10 10 (reject) = 10 xx xx x x x xx          2 ̶ 3 O C x 4 α y

2            22 22 22 2 22 2 21 2 2 2 3 28 3 243 0 3 28 3 3 243 0 3 84 3 243 0 Let 3 , 84 243 0 ( 81)( 3) 0 ( 81) 0 or ( 3) 0 81 3 Hence, 3 81 or 3 3 3 xx xx xx x xx x y yy yy yy yy                     4 22 3 By comparison, 4 or 1 2 or 1 xx xx     4 3             2 2 2 22 2 22 2 1 ............ (1) 21 ............ (2)2 (1) (2), 21 1 = 2 1 2 2 1 2 2 2 1 1 3 2 3 0 4 3 2 4 1 3 9 12 4 12 y mx xxy x xxmx x mx x x x mx mx x x x m x m x b ac m m mm                                        2 2 2 22 12 4 3 Since curve doesn't meet line, there's no real roots. 4 0 Hence, 4 3 0 2 3 0 2 3 2 3 0 m m b ac m m mm           33 22 m   5 - m

4 (Alt seg theoreom) (Alt seg theoreom) (Alt angles, / / ) ( s in the same seg) i.e. , hence, bis ABF FEB y CBD BED x CBD FDB x FD AC FEB BED CBD yx FEB CBD BE                  ects . DEF 4 5(i) 2 2 dGiven 24 cm /s,d 4 d 8d d d d d d d d24 8 d d 24 d8 d3 cm/sd A t Ar A rr A A r t r t rr t r tr r tr               3 5(ii) 3 2 2 3 3 4 3 d 4d d d d d d d d3 4d d 12d When = 10 cm, d 120 cm /sd Rate of decrease in volume is 120 cm /s. Vr V rr V V r t r t V rtr V rt r V t             3 6(i) When t = 0, 2

0.6(0) 95 000 2 498 95 000 2 498(1) 190 N e N N      Initially, there are 190 people who were infected by the virus. 6(ii) When N = 0.08(95 000) = 7600,   0.6 0.6 0.6 0.6 0.6 0.6 95 000 76002 498 95 000 7600 2 498 95 000 2 4987600 10.5 498 7 332 7ln ln 332 70.6 ln 332 7ln 332 0.6 6.432041366 t t t t t t e e e e e e t t t                 The authority will issue home quarantine orders after 7 days. 3 6(ii) When t is very large, 0.6te  0, 95 000 4750020N   Hence, all 95 000 people will never be infected after a long period of time. 2 7(a)(i) (2 − 3𝑥)5 = 32 + 5(16)(−3𝑥) + 10(8)(9𝑥2) + 10(4)(−27𝑥3) + ⋯ 2

= 32 – 240x + 720x2 – 1080x3 + …