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Transcription

1 Maths In Focus Mathematics Etension Preliminar Course Answers Chapter : Basic arithmetic Problem Eercises.. (a) Rational Rational (c) Rational (d) Irrational (e) Rational (f) Irrational (g) Irrational (h) Rational (i) Rational (j) Irrational. (a) 8 (c) (d) (e) -. (f) (g) 9 (h) (i) (j) 0 8. (a) 08. o 00. oo (c) 0. oo (d) 0. o (e) 0. o (f) 0. oo (g) 0. 8 o o or 0.8 (h) 8. oo 8 9. (a) (c) (d) (e) (f) (h) (i) (j) (a) 0. o. (c) 0. o (d) 08. oo (e). oo. (a) 8 8 (c) 8 (d) 8 (e). %..%..%..% (g). (a)..8 (c) 8.80 (d). (e) -. 0 (f) 0. (g) 0. (h).0 (i) -. (j) $ $ Eercises.. 0 (d) (a) 0 0 (c) 0 (e). $... g.. $. 8 km cents m. $ g. $.0. (a) (c) 0. (d). (e) Eercises cm 9. (a).9. (c) 9 (d). (e). (f) 0. (g) (h) 0 (i) 0.08 (j) $.0..9 ml.. g. $0...9%. 0. g.. m. $ m $898 Problem minutes after o clock. Eercises (a) 00 (c) (d) (e) Eercises.. (a) 000 (c) 0 (d). (a) 0..8 (c) 0. o (d) 0. oo. (a) 0 (c) (d) 000. (a) (c) 0.00 (d) 0.0. (a) % % (c) % (d) 0.%. (a) % 0% (c) 0.% (d).9%. (a) 0. ; 00. ; 00 9 (d) 09. ; 00 (c) 08. ; 9 (e) 0. ; (f) 0. ; (a).. (c) 0.8 (d). (e) -. (f) 0.. (a) a 0 (c) a - (d) w (e) (f) p 0 (g) (h) (i) 0 (j) 8-8 (k) a (l) (m) w 0 (n) p (o) - - (p) a b (q) - or b or a. (a) a - (c) m (d) k 0 (e) a - 8 (f) (g) mn (h) p - (i) 9 (j). (a) p q b a (f) a (c) b 0 (d) 9 a 0 b (e) 8 m k (g) (h) (i) a (j) - 8

2 ANSWERS. 0. (a) pq r. 8 Eercises.. (a) (g) (n) (s) (c) (h) 8 (o) (t). 9. (i) (j) 8 (p) (a) a b. 8. (d) (k) 8 (q) 8 (e) (f) (l) 9 (m) (r). (a) (c) (d) (e) (f) (g) 9 (h) 9 (i) (j) (k) (l) (m) (n) 8 8 (o) (p) (q) - (r) - (s) (t) 8. (a) m - - (c) p - (d) d 9 (e) d (f) - z - (g) - (h) (i) z - or (j) t - 8 (k) m - - (l) (m) (n) ] + g - (o) ] a + bg -8 (p) ] - g - (q) ^p + h - (r) ] t - 9g - ] g (s) + - ] a + bg - (t) 9. (a) t (c) (d) n 8 (e) w 0 (f) - (g) m (h) (i) (j) 8 n (k) (l) ] + g 8 + z p (m) ( n) (o) (p) 0 (q) ] k - g ^ + h (r) ] a + bg (s) (t) e o + w - z Eercises.. (a) 9 (c) (d) (e) (f) 0 (g) (h) 8 (i) (j) (k) (l) (m) 0 (n) (o) (p) (q) (r) (s) (t). (a).9.0 (c). (d) 0.0 (e) 0.90 (f) 0.9. (a) or _ i (c) (d) + (e) (f) q + r (g) or - ] + g ^ + h. (a) t (c) (d) ] 9 - g (e) ] s + g (f) t ] g (g) ^ - h (h) ] + g (i) ] - g - (j) ^ + h - (k) ] g (l) - - a k (m) + - _ i - +. (a) - (c) (d) (e). (a) + + a - b (c) p + p - + p. (a) - (d) + + (d) a - b ^ h + Eercises (e) - + ^ - h (c) (e) 9 ] + 8g ] a + g. (a) 8. # 0. # 0 (c) 9. # 0 (d).# 0 (e) 8. # 0 9 (f). # 0 (g) 9# 0 (h). # 0 (i) # 0 (j) 8# 0. (a). # 0 -. # 0 - (c) # 0 - (d).# 0 - (e) # 0 - (f) 8# 0-8 (g).# 0 - (h).# 0 - (i) 8.# 0 - (j) # 0 -. (a) (c) 9 0 (d) (e) (f) 0.0 (g) (h) (i) (j) (a) (c) 000 (d) 0. (e). (f) 9.0 (g) (h) 0 (i) 900 (j) 9.. (a). 0.8 (c) 8. (d).0..0 # # 0-0 Eercises.9. (a) (c) (d) 0 (e) (f) (g) (h) (i) (j). (a) (c) (d) (e) (f) (g) (h) (i) (j) 0. (a) (c) (d) (e). (a) a - a (c) 0 (d) a (e) a (f) 0 (g) a + (h) -a - (i) - (j) -. (a) a + b a + b ` a + b # a + b a + b a + b ` a + b # a + b (c) a + b a + b ` a + b # a + b (d) a + b a + b 9 ` a + b # a + b (e) a + b 0 a + b 0 ` a + b # a + b. (a) (c) (d) (e) 9. (a) + for -and - - for - b - for b and - b for (c) a + for a -and -a - for a - (d) - for and - for (e) + 9for -and - - 9for - (f) - for and - for (g) k + for k - and -k - for k - (h) - for and - + for (i) a + b for a -b and -a - b for a - b (j) p - q for p q and q - p for p q 8.! 9.! 0.!,!

3 8 Maths In Focus Mathematics Etension Preliminar Course Test ourself 9. (a) 0. (c) 0. (d) (e).% 0 00 (f).%. (a) (c) 9. (a) 8.8. (c). (d) (e) 0.0. (a) (c) 9 (d) (e) - 0 (f) - (g). (a) 9 (c) a b (d) (e). (a) 0 (c) (d) (e). (a) (c) 9 (d) (e) (f) (g) (h) (i) (j) 8. (a) a 0 8 (c) p 9 (d) b (e) 8 9. (a) n - (c) ^ + h - (d) ] + g (e) ] a + bg (f) - (g) - 0. (a) n a (f) a + b (g) (h) (i) ] + g (j) 9 m - (c) + (d) (e) - t ] - g (h) b (i) ] + g (j). a + b a + b 8 ` a + b # a + b ml. (a) h (d).%. $8 0. 0% 8.. # 0 9. (a) - (c) ] + g (d) ] - g - (e) (c) 8 0. (a). # 0 -. # 0. (a) 9 0. (a) a + (c) a b c m. 00. LHS - + -, RHS So a + b # a + b since #. Challenge eercise % h , %,, 0. o # 0. 8% k k. LHS ^ - h + + k+ k+ - + k+ : - k ^ + -h RHS k k k ` ^ - h ^ -h ,,. o, 0,. %. when -, when % # a + b a + b when a 0, b 0or a 0, b 0; a + b a + b when a 0, b 0or a 0, b 0; ` a + b # a + b for all a, b Chapter : Algebra and surds Eercises... a. z. a. b. r k. 9 t. 0 w. - m b. b a - b ab. m - m +. p - p ab + 0b. bc - ac 8. a Eercises.. 0 b. 8. 0p. - wz. ab. z. 8 abc 8. d 9. a ab. - 0ab. pq. ab. - 8n 0. kp 8. t 9. - m 0. Eercises a. 8 a. a. 8. ab 9... cd. p a.. ab z z a b Eercises.. p q. ab c b 8. a h a qs. -. a - ab. ab + ab 8. n - 0n k +. t b m +. 8h - 9. d - 8. a - a ab - a b + b b. t -. a + Eercises.. a + a m - m h - 0 h a - a n a - b a -. z - 9

4 ANSWERS ab + b - b - a a -. a + 8a k - 8k t - t +. 9a + ab + b a + ab + b. a - b. a + ab + b 8. a - ab + b 9. a + b 0. a - b Eercises.. t + 8t +. z - z q + q + 9. k - k + 9. n + n + 8. b + 0 b a - ab + b. d + 0de + e. t p -. r a a ab- c a ^ - h a 9. ] a + bg + ] a + bg c + c a + ab + b + ac + bc + c 0. ] + g - ] + g a. - z n a + 0a + 0a Problem a, b, c 9, d, e, f 8, g 0, h, i Eercises.. ^ + h. ] - g. ] m - g. ] + g. ^ - h. ] + g. mm ] - g 8. ^ + h 9. a] - ag 0. ab ] b + g. ] - g. mn ^n h. ] - zg. a] b + - ag +. ^ - + h. q q _ - i. b ] b + g 8. a b ] b - ag 9. (m + )( + ) 0. ^ - h^ - h. ( + )( - ). ] a - g] + g. ] t + g^ - h. ] - g] a + b - cg. ] + g. q _ pq i. ab ^a b h ] - g 9. mn^mn h 0. ab ^ab h - +. r rr ] + hg. ] - g] + g. ( + )( + ). -] a + g. ( a + )( ab - ) Eercises.8. ] + g] + bg. ^ - h] a + bg. ] + g] + g. ] m - g] m + g. ] d - cg] a + bg. ] + g^ + h. ] a - g] b + g 8. ^ - h^ + h 9. ^ + h] a + g 0. ] + g] - g. ( + )( + a). (m - )( - ). ^ + h^ - h. ^ a + b h] ab - g. ] - g] + g. ( + )( - ). ] - g^ - h 8. ] d + g] - eg 9. ] - g^ + h 0. ] a + g] - bg. ( - )( + ). ^q - h^p + qh. ] - g^ - h. ] a - bg] + cg. ^ + h] - g. ( - )( - ). ( - )( + ) ( - )( + ) 8. ( a + b)( a + ) 9. ( - )( + ) 0. ] r + g] rr - g Eercises.9. ] + g] + g. ^ + h^ + h. ] m + g. ] t + g. ] z + g] z - g. ] + g] - g. ] v - g] v - g 8. ] t - g 9. ] + 0g] - g 0. ^ - h^ - h. ] m - g] m - g. ^ + h^ - h. ] - 8g] + g. ] a - g. ] - g] + g. ^ + h^ - 9h. ] n - g] n - g 8. ] - g 9. ^p + 9h^p - h 0. ] k - g] k - g. ] + g] - g. ] m - g] m + g. ^q + 0h^q + h. ] d - g] d + g. ] l - 9g] l - g Eercises.0. ( a + )( a + ). ^ + h^ + h. ( + )( + ). ( + )( + ). ( b - )( b - ). ( - )( - ). ^ - h^ + h 8. ] + g] + g 9. ^p - h^p + h 0. ] + g] + g. ( + )( - ). ] - g] + g. ( t - )(t - ). ( + )( - ). ^ - h^ + 8h. ] n - g] n - g. ] t - g] t + g 8. ^q + h^q + h 9. ] r - g] r + g ] r - g] r + g 0. ] - g] + g. ^ - h^ - h. ^p - h^p + h. ( 8 + )( + ). ] b - g] b - 9g. ( + )( - 9). ] + g. ^ + h 8. ] k - g 9. ] a - g 0. ] m + g

5 0 Maths In Focus Mathematics Etension Preliminar Course Eercises.. ^ - h. ( + ). ( m + ). ( t - ). ( - ). ] + g. ] b - g 8. ] a + g 9. ] - g 0. ^ + h. ^ - h. ] k - g. ] + g. ] 9a - g. ] m + g. dt + n. d - n 8. d + n 9. c + m 0. dk - n k Eercises.. (a + )( a - ). ( + )( - ). ( + )( - ). ] + g] - g. ( + )( - ). ( + )( - ). ( + z)( - z) 8. ] t + g] t - g 9. ] t + g] t - g 0. ] + g] - g. ( + )( - ). ^ + h^ - h. ] a + bg] a - bg. ^ + 0h^ - 0h. ] a + 9bg] a - 9bg. ^ + + h^ + - h. (a + b - )( a - b + ) 8. ] z + w + g] z - w - g 9. d + nd - n 0. e + oe - o. ^ + + h^ - + h. ( + )( - ) ( + )( + )( - ). _ + i_ - i. _ + i^ + h^ - h. (a + )( a + )( a + )( a - ) Eercises.. (b - )( b + b + ). ] + g^ - + 9h. ] t + g^t - t + h. (a - )( a + a + ). ( - )( + + ). ^ + h_ i. ( + z)( - z + z ) 8. ( - )( + + ) 9. ^ + h_ i 0. ] ab - g^a b + ab + h. ( 0 + t)(00-0t + t ) d + ne - + o a b a ab b. ^ + - h_ i d - ne + + 9o. ^ + zh _ - 0z + z i. -9^a - a + h. d - ne + + o 9 8. ^ + + h_ i 9. ^ + - h_ i 0. ( a + - b)(a + a + ab + b + b + ) Eercises.. ] + g] - g. ^p + h^p - h. ^ - h _ + + i. ab ^a + b)( a - h. ] a - g. -] - g] - g. zz ] + g] z+ g 8. ab ] + abg] - abg 9. ] + g] - g 0. ] - g] + g. ] m - g] + ng. - ] + g. ^ + h^ + h^ - h. ] - g] + g^ - + h. ] + g^ - + h] - g^ + + h. ] + g] - g. ] + g ( - ) 8. ( + )( - ) 9. ] - bg^ + b + b h 0. ] - g] + g. ] - g. ( + )( + )( - ). zz ] + g. ] + g] - g] + g] - g. ] + g] - g^ + h _ - + i. aa ( + )( a- ). ] - g^ + + h 8. (a + )( a - )( a + )( a - ) 9. kk ( + ) 0. ( + ) ( - ) ( + ) Eercises ] + g. b - b + 9 ] b - g ] - g ^ + h. m - m + 9 ] m - g. q ^q + 9h. + + ] + g 8. t - t + ] t - 8g ] - 0g 0. w + w + 8 ] w + g ] - g. + + d + n d - n. a + a + da + n d + n.. k k - + dk - n - + d - n ^ + h 9. a - ab + b ] a - bg 0. p - 8pq + q ^p - qh Eercises.. a +. t p +.. p -. p - p + ] b - ag a - a + a a + b. a - b s - s + +. d b + b + b

6 ANSWERS Eercises.. (a) b. (a) a - (d). (a) ab (c) a + 8 (d) ^ p - h_ q - q + i q ] - g ^p + qh^p - qh+ (e) p + q (g) ] + g] - g ] - g] - g (e) ] - g] - g a + b + (c) a + b p - q + p + q a + (h) ] a + g _ + + i (i) ^ + h^ + h^ - h ] + g. (a) 8_ - + 9i p + (e) b ^ + h (c) 0] b - g (d) + - ] - g (f) ] + g] - g -] + g (j) ] + g] - g] + g ^ + h^ + h b - b - 0 (c) (d) (e) ] - g] - g b ] b + g - -. (a) ] - g] - g] + g ] + g] - g p + pq - q (c) pq ^p + qh^p - qh ^ + + h (e) ^ + h^ - h Eercises.8 a - ab - b + (d) ] a + bg] a - bg. (a) (c) 8. (d) -. (e) 0. (f). (g) Eercises.9. (a) (c) (d) (e) (f) 0 (g) (h) (i) (j) (k) (l) 0 (m) 8 (n) 9 (o) (p) (q) (r). (a) 0 (c) 8 (d) (e) (f) 8 (g) (h) 0 (i) 0 (j). (a) 8 0 (c) (d) 8 (e) (f) 0 (g) (h) 98 (i) (j) 008. (a) (c) (d) 0 (e) (f) (g) 0 (h) 88 (i) (j) 90 Eercises Eercise Eercises (a) (c) + 8 (d) - (e) (f) + (g) - - (h) - (i) + 0 (j) + + (k) (l) 0 - (m) 0-0 (n) (o) -. (a) (c) (d) (e) - 0 (f) (g) (h) - (i) - (j) (k) (l) - (m) - (n) + 0 (o) - (p) + (q) + (r) - (s) (t) + 0. (a) 8 08 (c) (d) 9 + (e) 9. (a) a, b 80 a 9, b -. (a) a - p - - p ^p - h. k a, b 0 9. a 0, b units 8

7 Maths In Focus Mathematics Etension Preliminar Course Eercises.. (a) (e) (h) + (f) - (c) - (i) (d) 0 (g) (j) - 0 -^ + h. (a) - ^ - h -^ - 8h -^ - h (c) 9 9 -^9-8 h 8-9 (d) (e) (f) (a) - ^ h (c) 9 (d) -^ h (e) - (f) (g) (h) (j) (i) + 9 (k) (l). (a) a, b 0 a, b 8 (c) a -, b 8 (d) a -, b - (e) a, b # - + # + - ^ - h^ - h + ^ h So rational. (a) (c) # + # + - ^ - h ^ h # So rational 9. - ^ + h 0. Test ourself b + b + b -. (a) - a + (c) - k (d) (f) (g) + (e) a - 8b. (a) ] + g] - g ] a + g] a - g (c) ab ] b - g (d) ( - )( + ) (e) ^n - p + h (f) ( - )( + + ). (a) b (c) m + (d) (e) p - (f) - - a (g) - (h) (a) b ^ a + a + 9h ] m - g - 9. V.. (a). + ] + g] - g 9. (a). (a) 8. (a) - (c) (d) (e) 8 0. d. +. (a) (a) ( - )( + ) ] - g] + g (c) ^ + h_ - + i. (a) -. (a) 99. (a) a - b a + ab + b (c) a - ab + b. (a) ] a - bg ] a - bg^a + ab + b (a) b + a ab h - 0

8 ANSWERS. (a) - 8 (c) (d) (f) n m (g) - (e) 0a b. (a) (a) (c) (d) (e) - (d). (a) (d) k - (e) (e) (c) a - - (c) ( + )( - ) (a) n 8 n (c) n 9 (d) n (e) n.., (c) 8. (d) 9. (a), (d) 0. (c). (c).. (a). (d). Challenge eercise. (a) ab- 8ab+ a -.. (c) b b b + + d + n a a a. (a) ] + g] + 9g _ - i_ + i ( + )( - ) _ + i (c) ] + g^ - + 9h (d) ] b - g] a + g] a - g ] + g. 9.. ] a + g a - a + or a a 0. d + nd - n b b ] - g] + g] - g + ] - g +. (a) ] - g d + n i (a) r r r r 8. a, b - Chapter : Equations Eercises.. t -. z -... w b 9. n - 0. r. 9. k. d m a b -.. a -. t -... a.. 9. p 0. Z. Eercises. b 8. t b.... k t w. t p. t q b 0.8. a t - Eercises.. t 8.. l. b 8. a.. r n h.. (a) BMI.9 w 9. (c) h.9. r t..!. r.. r r. Eercises.. (a) # (a) t $ (c) p - (d) $ - (e) - 9 (f) a $ - (g) $ - (h) - (i) a # - (j) (k) b - 8 (l) 0 (m) # (n) m (o) b $ (p) r # - 9 (q) z 8. s +

9 Maths In Focus Mathematics Etension Preliminar Course (r) w (s) $ (t) t $ - 9 (u) q - (v) - (w) b # -. (a) # p (c) (d) - # # (e) Eercises.. (a)!! 8 (c) - a (d) k $, k # - (e), - (f) - 0 # p # 0 (g) 0 (h) a, a - (i) - (j) b $ 0, b # - 0. (a), - 9 n, - (c) a, a - (d) # # (e), - (f), - (g) - (h) $ 9, # - (i)! (j) # a # 0. (a) a, - (c) b (d) No solutions (e) - (f) (g) m, (h) d, - (i), - (j) No solutions. (a), -, (c) a - 0, (d), - (e) d, -. (a) t, - - t Eercises.. (a)! 8 (c) n! (d)! (e) p 0 (f)! (g)! (h) w (i) n! (j) q -. (a) p!.. (c) n.99 (d)!.9 (e).89 (f) d!. (g) k!. (h). (i)! 8. (j).0. (a) n t (c) (d) t 8 (e) p (f) m (g) b (h) (i) a 8 (j) t 8. (a) a (c) (d)! (e) n (f) a (g)! (h) b 9 (i)! (j) b!. (a) (e) (c) a (d) 8 k 9 (f) (g) 8 (h) n 8 (i) b 8 (j) m. (a) n (c) m 9 (d) (e) m 0 (f) (g) (h) (i) (j) k. (a) (c) - (d) n (e) 0 (f) (g) (h) (i) (j) a 0 8. (a) m (c) (d) k - (e) k - (f) n (g) (h) n (i) k - (j) 9. (a) - - (c) k - (d) n (e) - (f) - (g) - (h) - (i) (j) 8 0. (a) m k - (c) (d) k 8 (e) n (f) n - (g) (h) b - 8 Puzzle (i) - (j) m. All months have 8 das. Some months have more das as well.. 0. Bottle $.0; cork cents. each time. Frida Eercises.. 0, -. b, -. p, -. t 0,. -, -. q!.! 8. a 0, -

10 ANSWERS 9. 0, - 0.!. -, -., -. b,., -. 0,.,. 0, 8. -, 9. n, 0.,. m -,. 0, -,-., -,-., -. m 8, - Eercises.8. (a)! - a! + (c)! + (d)! - (e) p! -! - (f)! 8 +! + (g)! 88-0! - 0 ^! - h (h)! + (i) n! -! (j) +. (a).,-. -9.,-. (c) q 0. 0, - 8. (d).,- 09. (e) b -.,-. (f).,. (g) r., - 0. (h) -08.,-. (i) a 0.,-. (j) 0., Eercises.9. (a) -0.,-.,. (c) b.,-. (d), - 0. (e) - 0., 08. (f) n 0.,- 8. (g) m -, - (h) 0, (i), - (j)., 08.!. (a) -!! 8 (c) q!! 8! (d) h ! 0! 0 (e) s! (f) -! (g) d -!! (h)! (i) t (j)! Eercises.0., ,. 0 m #. -, > 0. - # b z - 9. # 0.,. - # a -, a -.. -, - 9. # -,. p 8 8. # -, - 9. t #, t m 0. -, n #, n $., - 0. m # -,- m #. # -, #. # -, #. $, - # 8. n -, n 9. -, #, # Eercises n # 0, n $. # -, $. n -, n. - # n #. c -, c 8. - # # b # -, b $ -. a -, a. -,. #, $. b -, b. - # # -. - # #. -, 8. - # a # # -, $. 0. # a #. -, # $. m -, m. # #. 0. 0, $ 8. -, - 9. n # 0. # -8, -.,. #,. # -, -. -,. - # -

11 Maths In Focus Mathematics Etension Preliminar Course Eercises.. a, b.,. p, q -.,. - 0,. t, v. -, 8. -, , - 0. m, n. w -, w. a 0, b. p -, q., -. -, -. s, t -. a -, b 0 8. k -, h 9. v -, v 0., Z. Problem adults and children. Eercises.. 0, 0 and,. 0, 0 and -,. 0, and, 0., - and, -. -, -., 9. t -, and t, 8. m -, n 0 and m 0, n - 9., and -, , 0 and,., and -, -. 0,., and,., and -, -. t -, h., 0. 0, 0 and -, - 8 and, 8. 0, 0 and, and -, 9., 0. -, - Eercises.. -, - 8, z -. a -, b -, c. a -, b, c. a, b, c -., 0, z -. 0, -, z. p -, q, r 8., -, z 9. h -, j, k - 0. a, b -, c - Test ourself. (a) b 0 a - (c) - (d) # -, - (e) p #. (a) A.8 P (a) ] - g k + k + ] k + g. (a) -,, and -, - 8. (a). (a) b, - g, (c) $, #. (a) A b 8., 9. - # 0. (a) , -. 0.,-. (c) n 0. 89, -.. (a) V. r.9.., 9..,.. (a) V 00 r.9. (a) ii i (c) ii (d) iii (e) iii. a, b, c - 8. n 0, n (a) - # n # 0 (c) (d) (e), - (f) t $, t # - (g) - # # (h) - (i), - (j) # -, $ (k) (l) - # b # (m) No solutions (n) t, (o) - (p) m # -, m $ (q) t -, t 0 (r) (s) n # (t) - Challenge eercise #.. - a, a. a, b!..,-.. # -, 0 #. ] + g] - g] - g^ + + h ;!,., and -, 0 8. b ;! + Z 8., ! 0. - t. - # # 8.. r.. No solutions.! b + a + a. # -, #. P.! 0 8. ^ h 9. -, ,

12 ANSWERS Chapter : Geometr Eercises.. (a) c 9c (c) m c (d) 0c (e) b 0c (f) c (g) a 0c (h) c (i) 0c (j) 80c. (a) c c9l (c) c8l. (a) c cl (c) cl. (a) (i) c (ii) c (i) 9c (ii) 99c (c) (i) c (ii) c (d) (i) c (ii) c (e) (i) c (ii) c (f) (i) cl (ii) 0cl (g) (i) cl (ii) cl (h) (i) cl (ii) cl (i) (i) cl (ii) cl (j) (i) 8cl (ii) 8cl. (a) 9c c (c) c. (a) c, z c c, 8c, z 8c (c) a c, b c, c 0c (d) a 9c, b d c, c c (e) a 8c, b c, c 8c (f) a 0c, b 0c (angleof revolution) ABE ( )- 0 0c + EBC ( )- 0 0c + ABE + + EBC 0c+ 0c 80c ` + ABC is a straight angle + DBC ( ) + 0 0c + DBC + + EBC 0c+ 0c 80c ` + DBE is a straight angle ` AC and DE are straight lines 8. + DFB 80c-( 80 - ) c ( + AFB is a straight angle) ` + AFC (verticall opposite angles) + CFE 80c- ( + 80c- ) ( + AFB is a straight angle) ` + AFC + CFE ` CD bisects+afe 9. + ABD + + DBC c So + ABC is a straight angle. AC is a straight line AEB + + BEC + + CED c So + AED is a right angle. Eercises.. (a) a b e f 8c, c d g c z 0c, 0c (c) c, c, z 89c (d) c, z c (e) n e g a c z 98c, o m h f b d w 8c (f) a 9c, b 8c, c c (g) a c, b c, c 8c (h) c, c, z a c, b c (i) c (j) c. (a) + CGF 80c- c ( FGH is a straight angle) 9c ` + BFG + CGF 9c These are equal alternate angles. ` AB < CD + BAC 0c- 9c 8c (angle of revolution) ` + BAC + + DCA 8c+ c 80c These are supplementar cointerior angles. ` AB < CD (c) + BCD 80 - ( + BCE is a straight angle) 0c + ABC + BCD 0c These are equal alternate angles. ` AB; CD (d) + CEF 80-8 ( + CED is a straight angle) c + CEF + ABE c These are equal corresponding angles. ` AB; CD (e) + CFH 80 -] + g ( + EFG is a straight angle) c `+ BFD c (verticall opposite angles) + ABF + + BFD 8c+ c 80c These are supplementar cointerior angles. ` AB; CD

13 8 Maths In Focus Mathematics Etension Preliminar Course Eercises.. (a) 0c c (c) m c (d) c (e) 0c (f) 0c (g) c (h) a c (i) a c, b c, c c (j) a c, b c, c c (k) c, z 9c, w c. All angles are equal. Let them be. Then (angle sum of D) 80 0 So all angles in an equilateral triangle are 0 c.. ] 90 - g c. + ACB 0c + ABC 80 c- (0c+ c) (verticall opposite angles) (angle sum of D) 8c ` + DEC + ABC 8c These are equal alternate angles. ` AB < DE. + ACB 80c- c c + CBA + 8c c + CBA c- 8c c ` + CBA + ACB c ` D ABC is isosceles. 8c ( DCB is a straight angle) (eterior angle of D). (a) c c, c (c) c (d) a 9c, b 0c 8. + HJI 80 c- (c+ c) 0c + IJL 80c- 0c 0c + JIL 80 c- (90c+ 0 c) 0c + ILJ 80 -(0c+ 0 c) 0c (angle sum of D HJI) ( HJL is a straight angle) (angle sum of D IKL) (angle sum of D JIL) Since + IJL + JIL + ILJ 0 c, D IJL is equilateral + KJL 80c- 0c ( KJI is a straight angle) 0c + JLK 80c- ( 0c+ 0c) ( angle sum of D JKL) 0c ` + JLK + JKL 0 ` D JKL is isosceles 9. BC BD `+ BDC c (base angles of isosceles triangle) + CBD 80 - # 88c `+ CBD + BDE 88c These are equal alternate angles. ` AB; ED 0. + OQP 80 -] + g (angle sum of triangle) c ` + MNO + OQP c These are equal alternate angles. ` MN ; QP Eercises.. (a) Yes AB EF cm (given) BC DF cm (given) AC DE 8cm (given) ` D ABC / D DEF ( SSS ) Yes XY BC.m (given) + XYZ + BCA 0c (given) YZ AC.m (given) ` D XYZ / D ABC ( SAS ) (c) No (d) Yes + PQR + SUT 9c (given) + PRQ + STU c (given) QR TU 8 cm (given) ` DPQR / D STU ( AAS ) (e) No. (a) AB KL + B + L 8c BC JL (given) (given) (given) ` b SAS, DABC / DJKL + Z + B 90c (given) XY AC (given) YZ BC (given) ` b RHS, DXYZ / DABC (c) MN QR 8 (given) NO PR 8 (given) MO PQ (given) ` b SSS, DMNO / DPQR (d) + Y + T 90c (given) + Z + S c (given) XY TR. (given) ` b AAS, DXYZ / DSTR (e) BC DE (given) + C + E 90c (given) AC EF (given) ` b SAS, DABC / DDEF. (a) + B + C (base angles of isosceles D) + BDA + CDA 90c AD is common (given) ` b AAS, DABD / DACD

14 ANSWERS 9 ` BD DC (corresponding sides in congruent Ds) ` AD bisects BC. + ABD + BDC (alternate angles, AB < CD) + ADB + DBC ( alternate angles, AD < BC) BD is common ` b AAS, D ABD / DCDB ` AD BC (corresponding sides in congruent Ds). (a) OA OC (equal radii) OB OD (similarl) + AOB + COD (verticall opposite angles) ` DAOB / D COD ( SAS ) AB CD (corresponding sides in congruent triangles). (a) AB AD (given) BC DC (given) AC is common ` DABC / D ADC ( SSS ) + ABC + ADC (corresponding angles in congruent triangles). (a) OA OC (equal radii) OB is common + AOB + COB 90c (given) ` DOAB / D OBC ( SAS ) + OCB + OBC (base angles of OBC, an isosceles right angled triangle) But + OCB + + OBC 90c (angle sum of triangle) So + OCB + OBC c Similarl + OBA c ` + OBA + + OBC c+ c 90c So + ABC is right angled 8. (a) + AEF + BDC 90c (given) AF BC (given) FE CD (given) ` DAFE / D BCD ( RHS ) + AFE + BCD (corresponding angles in congruent triangles) 9. (a) OA OC (equal radii) OB is common AB BC (given) ` DOAB / D OBC ( SSS ) + OBA + OBC (corresponding angles in congruent triangles) But + OBA + + OBC 80c ( ABC is a straight angle) So + OBA + OBC 90c OB is perpendicular to AC. 0. (a) AD BC (given) + ADC + BCD 90c (given) DC is common ` DADC / D BCD ( SAS ) AC BD (corresponding sides in congruent triangles) Eercises.. (a).. (c) m. (d) a c, i c, b 8c (e) b. (f) a c, 9c,. (g) p 9.. a 8., b BAC + EDC ( alternate angles, AB < ED) + ABC + DEC (similarl) + ACB + ECD ( verticall opposite angles).. ` since pairs of angles are equal, D ABC DCDE + GFE + EFD (given) GF. 0. o EF. EF. 0. o DF 8. GF EF ` EF DF Since two pairs of sides are in proportion and their included angles are equal, then D DEF DFGE AB. 0. DE 8. AC. 0. DF 88. BC EF 8. AB AC BC ` DE DF EF Since three pairs of sides are in proportion, D ABC DDEF c. (a) OA OB OC OD OA OB ` OD OC + AOB + COD (equal radii) (similarl) (verticall opposite angles) Since two pairs of sides are in proportion and their included angles are equal, OAB OCD AB. cm. (a) + A is common + ABC + ADE + ACB + AED ( corresponding angles, BC < DE) (similarl)

15 0 Maths In Focus Mathematics Etension Preliminar Course ` since pairs of angles are equal, D ABC DADE., ABF + BEC ( alternate angles, AB z CD) + CBE + BFA ( similarl, BC z AD) `+ C + A ( angle sum of Ds) ` since pairs of angles are equal, D ABF DCEB 9. + A is common AD. 0. AB AE AC AD AE ` AB AC Since two pairs of sides are in proportion and their included angles are equal, D AED D ABC, m. AB CD. BC 09. AC 9. AC AD 0. AB BC AC ` CD AC AD Since three pairs of sides are in proportion, D ABC D ACD, 09c, c. (a).8 m 0., p. (c).. (a) (d).,. (e)., 9. (c) Also ` Also ` AB BC AD DE AB BC BD CE AD AE BD CE AD DE AF FG AF FG AD AE DF EG DF EG Also ` AB AC AD AE AB AC AD AE AF AG AF AG. a 8., b ,. Eercises.. (a).. (c) b. (d) m.. (a) p t 8 (c) (d). s m.. CE. cm. AB 8, CB, CA AB + CB 8 + CA ` D ABC is right angled. XY YZ ` D XYZ is isosceles YZ XY, XZ YZ + XY + XZ ` D XYZ is right angled. AC AB + BC ^ h + BC + BC BC ` BC AC # BC 8. (a) AC AC, CD, AD 9 AC + CD + 9 AD ` D ACD has a right angle at + ACD ` AC is perpendicular to DC 9. AB b d ] 0 - tg + ] - tg 00-0t + 9t + - 0t + t t - 80t +. mm. 8 m.. m.. cm.. m.. cm and... +.!. so the triangle is not right angled ` the propert is not a rectangle 9. No. The diagonal of the boot is the longest available space and it is onl. m. 0. (a) BC - 0 BC 0 AO cm (equal radii) So AC - 0 AC 0 Since BC AC, OC bisects AB + OCA + OCB 90c (given) OA OB (equal radii) OC is common ` DOAC / D OBC ( RHS ) So AC BC (corresponding sides in congruent triangles) ` OC bisects AB Eercises.. (a) 9c 0c (c) c (d) 0c (e) c (f) 0, (g),

16 ANSWERS. D ABE is isosceles. ` + B + E c ( base + s equal) + CBE + DEB 80c- c 0c ( straight + s) + D + c+ 0c+ 0c 0c(angle sum of quadrilateral) + D + 0c 0c + D 90c ` CD is perpendicular to AD `. (a) + D 80c - ( + Aand + Dcointerior angles, AB < DC) + C 80c-( 80c- ) ( + C and + Dcointerior angles, AD < BC) 80c- 80c+ `+ A + C + B 80c - ( + Band + C cointerior angles, AB < DC) `+ B + D 80c - Angle sum c- + 80c- 0c. a 0c, b c. (a) a m, b m, z 08 c, c c, c, z c (c) cm, a b 8c (d) a c, b c, i c (e) 0c (f),. + ADB + CDB + CDB + ABD + ADB + DBC ( BD bisects + ADC) (alternate angles, AB < DC) (alternate angles, AD < BC) ` + ABD + DBC ` BD bisects+ ABC. (a) AD BC 8. cm AB DC. cm (given) (given) Since two pairs of opposite sides are equal, ABCD is a parallelogram. AB DC cm (given) AB < DC (given) Since one pair of opposite sides is both equal and parallel, ABCD is a parallelogram. (c) + X + + M c+ c 80c These are supplementar cointerior angles. ` XY < MN Also, XM < YN ` XMNY is a parallelogram (d) AE EC cm DE EB cm (given) (given) (given) Since the diagonals bisect each other, ABCD is a parallelogram. 8. (a) cm, i c a 90c, b c, c c (c) cm, cm (d) 8c, 9c (e) cm 9.. cm 0. + ECB 9 c, + EDC c, + ADE 9c. cm. c Eercises.8. (a) 0c 0c (c) 080c (d) 0c (e) 800c (f) 880c. (a) 08c c (c) 0c (d) c (e) c. (a) 0c c (c) c (d) c. 8cl. (a) c8l.. 0c 8. 0c 9. 8cl 0. Sum n ( n - ) # 80c n 80n - 0 n 0. n But n must be a positive integer. ` no polgon has interior angles of c.. (a) 9 (c) 8 (d) 0 (e) 0. (a) ABCDEF is a regular heagon. AF BC (equal sides) FE CD (equal sides) + AFE + BCD (equal interior angles) ` DAFE / D BCD ( SAS ) S ] n - g # 80c ( - ) # 80c 0c 0c + AFE 0c Since AF FE, triangle AFE is isosceles. So + FEA + FAE (base angles in isosceles triangle) 80-0c ` + FEA (angle sum of triangle) 0c + AED 0-0c 90c Similarl, + BDE 90c So + AED + + BDE 80c These are supplementar cointerior angles ` AE < BD. A regular octagon has equal sides and angles. AH AB (equal sides) GH BC (equal sides) + AHG + ABC (equal interior angles) ` DAHG / D ABC ( SAS ) So AG AC (corresponding sides in congruent triangles) S ] n - g# 80c ( 8 - ) # 80c 080c 080c ` + AHG 8 c + HGA + HAG (base angles in isosceles triangle)

17 Maths In Focus Mathematics Etension Preliminar Course 80 - c `+ HAG (angle sum of triangle) c0l + GAC - # c0l 90c We can similarl prove all interior angles are 90c and adjacent sides equal. So ACEG is a square. ] # 80c. + EDC - g 08c ED CD (equal sides in regular pentagon) So EDC is an isosceles triangle. `+ DEC + ECD (base angles in isosceles triangle) 80-08c + DEC (angle sum of triangle) c + AEC 08 - c c Similarl, using triangle ABC, we can prove that + EAC c So EAC is an isosceles triangle. (Alternativel ou could prove EDC and ABC congruent triangles and then AC EC are corresponding sides in congruent triangles.) 0. (a) p Each interior angle: p 80p 0 - p p 80p - 0 p 80^p - h p Eercises.9. (a). m. 8 cm (c) 8. mm (d) m (e) cm (f) 8 m (g) 8. cm. 8m.. (a). 88 cm 9. m (c). cm (d). m (e) 00. cm. (a) m 0. 8 cm (c) 9. m (d) 0. 9 cm (e) cm ^ + h cm.. 9 cm. $ (a). m 89 m (c) 0. m 9. (a) 8 cm cm 0. w units Test ourself. + AGF i (verticall opposite + HGB) So + AGF + CFE i These are equal corresponding + s. ` AB < CD. 8.8 cm. (a) + DAE + BAC (common) + ADE + ABC (corresponding angles, DE < BC) + AED + ACB (similarl) `DABC and DADE are similar ( AAA). cm,. cm. c. 00. cm. m 8. (a) AB AD BC DC (adjacent sides in kite) (similarl) AC is common ` Δ ABC and Δ ADC are congruent (SSS) AO CO BO DO + AOB + COD (equal radii) (similarl) (verticall opposite angles) ` Δ AOB and Δ COD are congruent (SAS) 9.. cm 0. + ^ h ` Δ ABC is right angled (Pthagoras) AF AD. AG AE AD AB AE AC AF AB ` AG AC (equal ratios on intercepts) (similarl). (a) AB AC (given) + B + C (base + s of isosceles D) BD DC ( AD bisects BC, given) ` DABD / D ACD ] SASg + ADB + ADC (corresponding + s in congruent Ds) But+ ADB + + ADC 80c (straight + ) So + ADB + ADC 90c So AD and BC are perpendicular.. + ACB 8c ( base + s of isosceles D) + CAD 8c- c ( eterior + of D) c ` + CAD + ADC c So Δ ACD is isosceles ^base + s equalh.. (a) c, c, z c c (c) a 9c, b 0c, c 8c (d) 0c (e) r. cm (f). cm, 8. cm (g) i c

18 ANSWERS + DAC + ACB + BAC + ACD ( alternate + s, AD < BC) (alternate + s, AB < DC) AC is common `DABC / DADC (AAS) ` AB DC (corresponding sides in congruent Ds) Similarl, AD BC ` opposite sides are equal. (a) cm cm BFG + + FGD 09c- + + c 80c These are supplementar cointerior + s. ` AB < CD 8. cm 9. + ACB 80c -] + A + + Bg ( + sum of D) 80c ACD 80c - + ACB ( straight + ) z 80c-( 80c- - ) 80c- 80c (a) + A + E ^givenh AC 9.. EF. AB 9.. DE. AC AB ` EF DE So Δ ABC and Δ DEF are similar (two sides in proportion, included + s equal).. cm Challenge eercise. 9c. c, c, z 9c. 0c, cl. + BAD + DBC + ABD + BDC ` + ADB + DCB ` since pairs of angles are equal, D ABD < ; DBCD d.cm (given) (alternate angles, AB < DC) ( angle sum of D). 8. Let ABCD be a square with diagonals AC and BD and + D 90c AD DC (adjacent sides of square) ` DADC is isosceles `+ DAC + DCA (base angles of isosceles D) + DAC + + DCA 90 (angle sum of D) ` + DAC + DCA Similarl, + BAC + BCA (other angles can be proved similarl) Let ABCD be a kite AD AB (given) DC BC (given) AC is common `b SSS, DADC / DABC ` + DAC + BAC (corresponding angles in congruent Ds) AD AB (given) + DAE + BAE (found) AE is common `b SAS, DADE / DABE ` + DEA + BEA ( corresponding angles in congruent Ds) But+ DEA + + BEA 80c ( DEB is a straight angle) ` + DEA + BEA 90c ` the diagonals are perpendicular. AB DC (given) + A + + D c+ 9c 80c + A and + D are supplementar cointerior angles ` AB < DC Since one pair of opposite sides are both parallel and equal, ABCD is a parallelogram... m 9. + MNY + 8 c (c+ c) (eterior angle of DMNZ) ` + MNY c + XYZ + 9c c ( eterior angle of D XYZ) ` ` + XYZ c + MNY + XYZ c These are equal corresponding angles. ` MN < XY 0. m.. (a) m 0 + ^ + h m. 8. cm,. 8 cm..0 m,.9 m

19 Maths In Focus Mathematics Etension Preliminar Course. (a) AB BC + ABE + CBE (adjacent sides in square) (diagonals in square make c with sides) EB is common. `b SAS, DABE / DCBE ` AE CE (corresponding sides in congruent Ds) Since AB BC and AE CE, ABCE is a kite. BD + DE BD units Practice assessment task set. p 9. ^ + h^ - h. (a) ABC + EDC 90 + ACB + ECD AB ED `b AAS DABC / DEDC (given) (verticall opposite angles) (given) ` AC EC (corresponding sides in congruent triangles) ` D ACE is isosceles c.. # , r !.!. 9., or -, -., cm. ] - g^ + + h $ $8.. c, 9c, w z 90c.. cm -. a b 0 b a , , # - $.. Given diagonal AC in rhombus ABCD : AB BC (adjacent sides in rhombus) + DAC + ACB (alternate + s, AD < BC) + BAC + ACB (base + s of isosceles D ABC) ` + DAC + BAC ` diagonal AC bisects the angle it meets. Similarl, diagonal BD bisects the angle it meets.. ] + g c, c (a) - 8 (c) ] + g (e) (f) (g) z ] + g] - g (h) (i) 8 (j) aa ] + bg] + bg - -. c +..,... r cm r.. cm (d) + z. Let + DEA Then + EAD (base + s of isosceles D) + CDA + (eterior + of DEAD) ` + ABC (opposite + s of < gram are equal) ` + ABC + DEA. 8. % 9.. # 0 kmh cl.. m 0. k 0. (a) ] a + b - g ^a - a - ab + b + b + h ] a + bg] a - b + cg. - # 8. BC < AD ( ABCD is a < gram) BC < FE ` AD < FE Also BC AD (BCEF is a < gram) ^ opposite sides of < gramh BC FE ^ similarlh ` AD FE Since AD and FE are both parallel and equal, AFED is a parallelogram.. b.9 m 8. (a) cm 0 cm

20 ANSWERS m. BD bisects AC So AD DC + BDC + BDA 90c (given) BD is common ` DBAD / D BCD ( SAS ) ` AB CB (corresponding sides in congruent. triangles) So triangle ABC is isosceles + 9. (d) 80. (d).. (c). (a). 8. Chapter : Functions and graphs Eercises.. Yes. No. No. Yes. Yes. Yes. No 8. Yes 9. Yes 0. No. Yes. No. Yes. No. Yes Eercises.. f] g, f] - g 0. h] 0g -, h] g, h] - g. f] g -, f] - g -, f] g -9, f] - g ! z, - 0. f^ ph p - 9, f] + hg + h - 9. g ] - g +. f] kg ] k - g^k + k + h. t - ; t, f] g, f] g, f] - g -. f] g- f] - g+ f] - g (a) Denominator cannot be 0 so the function doesn t eist for. (c). f] + hg- f] g h + h - h. + h +. ] - cg. k +. (a) 0 (c) n + n + Eercises.. (a) -intercept, -intercept - -intercept -0, -intercept (c) -intercept, -intercept (d) -intercepts 0, -, -intercept 0 (e) -intercepts!, -intercept - (f) -intercepts -, -, -intercept (g) -intercepts,, -intercept (h) -intercept -, -intercept (i) -intercept -, no -intercept (j) -intercept!, -intercept 9. f] - g ]-g - - f ( ) ` even function. (a) f ^ h + f] ga + +. (c) f] - g - + (d) Neither odd nor even g] - g ]- g + ] -g - ] -g g ( ) ` even function. f] - g - -f] g ` odd function. f] - g ]-g - - f ( ) ` even function. f] - g ] -g-]-g - + -^ - h -f] g ` odd function 8. f] - g ]- g + ]-g + f] g ` even function f] g- f] - g (a) Odd Neither (c) Even (d) Neither (e) Neither 0. (a) Even values i.e. n f,,, Odd values i.e. n,,, f. (a) No value of n Yes, when n is odd (,,, ). (a) (i) 0 (ii) 0 (iii) Even (i) (ii) (iii) Neither (c) (i) - (ii) -, (iii) Neither (d) (i) All real! 0 (ii) None (iii) Odd (e) (i) None (ii) All real (iii) Neither Eercises.. (a) -intercept, -intercept - -intercept -, -intercept (c) -intercept, -intercept (d) -intercept -, -intercept (e) -intercept, -intercept -

21 Maths In Focus Mathematics Etension Preliminar Course. (a) (e) (f) (g) (c) (d) (h)

22 ANSWERS (i) (j) (a) " all real,, " all real, " all real,, " :, (c)! : -+," allreal, (d)! : +," allreal, (e)! all real +," :,. (a) Neither Even (c) Neither (d) Odd (e) Odd. Eercises.. (a) -intercepts 0, -, -intercept 0 -intercepts 0,, -intercept 0 (c) -intercepts!, -intercept - (d) -intercepts -,, -intercept - (e) -intercepts, 8, -intercept 8. (a) (c) (, -)

23 8 Maths In Focus Mathematics Etension Preliminar Course (d) (h) (e) (i) (f) (j) (g) (a) (i) -intercepts,, -intercept (ii) {all real }, ( : $ - (i) -intercepts 0, -, -intercept 0 (ii) {all real }, " : $ -, (c) (i) -intercepts -,, -intercept -8 (ii) {all real }, " : $ - 9, (d) (i) -intercept, -intercept 9 (ii) {all real }, " : $ 0, (e) (i) -intercepts!, -intercept (ii) {all real }, " : #,. (a) {all real }, " : $ -, {all real }, " : $ - 9,

24 ANSWERS 9 (c) {all real }, ( : $ - (d) {all real }, " : # 0, (e) {all real }, " : $ 0,. (a) 0 # # 9 0 # # (c) - # # (d) - # # (e) - 8 # #. (a) (i) 0 (ii) 0 (i) 0 (ii) 0. (c) (i) 0 (ii) 0 (d) (i) (ii) (e) (i) - (ii) - f] - g -]-g - f ( ) ` even (c) (a) Even Even (c) Even (d) Neither (e) Neither (f) Even (g) Neither (h) Neither (i) Neither (j) Neither Eercises.. (a) -intercept 0, -intercept 0 No -intercepts, -intercept (c) -intercepts!, -intercept - (d) -intercept 0, -intercept 0 (e) -intercepts!, -intercept (f) -intercept -, -intercept (g) -intercept, -intercept (h) -intercept -, -intercept (i) -intercept, -intercept (d) (e) (j) No -intercepts, -intercept 9. (a) (f)

25 80 Maths In Focus Mathematics Etension Preliminar Course (g) (h) (i) (j) (a) {all real }, " : $ 0, {all real }, " : $ -8, (c) {all real }, " : $ 0, (d) {all real }, " : $ -, (e) {all real }, " : # 0,. (a) (i) (ii) (i) 0 (ii) 0 (c) (i) (ii) (d) (i) 0 (ii) 0 (e) (i) 0 (ii) 0. (a) 0 # # -8 # # - (c) 0 # # (d) 0 # # (e) - # # 0. (a) - 0 (c) 9 (d) (e) -. (a)!, - (c) - # # (d) -, - (e) (f), (g) - (h) - # # (i), 0 (j) #, $ (k) - # # (l) # 0, $ (m), - (n) No solutions (o) 0 (p) (q) 0, - (r) No solutions (s) ( t) 0, Eercises.. (a) (i) {all real :! 0}, {all real :! 0} (ii) no -intercept (iii) (i) {all real :! 0}, {all real :! 0} (ii) no -intercept (iii)

26 ANSWERS 8 (c) (i) {all real :! - }, {all real :! 0} (ii) (iii) (iii) (d) (i) {all real :! }, {all real :! 0} (ii) - (iii) (e) (i) {all real :! - }, {all real :! 0} (ii) (iii) (f) (i) {all real :! }, {all real :! 0} (ii) (g) (i) {all real :! }, {all real :! 0} (ii) - (iii) (h) (i) {all real :! - }, {all real :! 0} (ii) - (iii)

27 8 Maths In Focus Mathematics Etension Preliminar Course (i) (i) ' all real :!, {all real :! 0} (ii) - (iii) Eercises.8. (a) (i) (j) (i) {all real :! - }, {all real :! 0} (ii) - (iii) (ii)! : - # # +," : - # #, (i) (ii)! : - # # +," : - # #,. f] - g - - -f ( ) ` odd function (c) (i). (a) # # # # (c) - # # - 9 (d) # # (e) - # # (, ) (a) # # # # (c) - # # 0 (d) # # (e) # #

28 ANSWERS 8 (ii)! :0# # +," : - # #, (iii)! : - # # +," : - # # 0, (d) (i) (i) Above -ais (ii) (iii)! : - # # +," : 0 # #, (c) (i) Above -ais (ii)! : - # # +," : - # #, (ii) (e) (i) (-, ) (iii)! : - # # +," : 0 # #, (d) (i) Below -ais (ii) (ii)! : - # # -+," : 0 # #,. (a) (i) Below -ais (ii) (iii)! : -8 # # 8+," : -8 # # 0, -

29 8 Maths In Focus Mathematics Etension Preliminar Course (e) (i) Below -ais (ii). (a) { : - 9 # # } { : 0 # # 9} (c) { : - 8 # # } (d) ' : # # (e) { : 0 # # } (f) { : - # # } (g) { : - # # 0} (h) " : - # # 8, (i) { : - # # } (j) ' : - # # - -. (a) {all real :! - } -intercept: This is impossible so there is no -intercept (c) {all real :! 0} (iii) " : - # #,,# : - # # 0-8. (a) {all real :! 0} {all real :!! } 9. (a). (a) Radius 0, centre (0, 0) Radius, centre (0, 0) (c) Radius, centre (, ) (d) Radius, centre (, -) (e) Radius 9, centre (0, ) 0. (a) (c) (d) (e) (f) (g) (h) (i) (j) Eercises.9. (a) {all real }, {all real } {all real }, {: -} (c) {: }, {all real } (d) {all real }, { : $ -} (e) {all real }, {all real } (f) {all real }, ' : # (g) { : -8 # # 8},{ : - 8 # # 8} (h) {all real tt :! }, {all real ft (): ft ()! 0} (i) {all real z: z! 0}, {all real g^zh: g^zh! } (j) {all real }, { : $ 0} (a) { : $ 0}, { : $ 0} { : $ }, { : $ 0} (c) {all real }, { : $ 0} (d) {all real }, { : $ - } (c) (e) ' : $ -,{ : # 0} (f) {all real }, { : # } (g) {all real }, { : 0} (h) {all real }, { : 0} (i) {all real :! 0}, {all real :! } (j) {all real :! 0}, {all real :! }. (a) 0, -,, (c) 0,, (d) 0,! (e)!. (a) - # # { : - # # } (a) { : # -, $ } { t: t # -, t $ 0} -

30 ANSWERS 8 (d) (g) (a) " : $, " : $ 0, (e) (f) (a) (i) {all real }, {all real } (ii) All (iii) None (i) {all real }, " : -, (ii) 0 (iii) 0 (c) (i) {all real :! 0}, {all real :! 0} (ii) None (iii) All! 0 (d) (i) {all real }, {all real } (ii) All (iii) None (e) (i) {all real }, " : 0, (ii) All (iii) None. (a) - # # (i) { : - # # }, { : 0 # # } (ii) { : - # # }, { : - # # 0} Eercises.0. (a) - 0 (c) 8 (d) (e) (f) (g) 0 (h) - (i) (j) (k) - (l) - -0 (m) + - (n) c. (a) Continuous Discontinuous at - (c) Continuous (d) Continuous (e) Discontinuous at!

31 8 Maths In Focus Mathematics Etension Preliminar Course. (a). (a) (c) (c) (d) Eercises.. (a) 0 0 (c) 0 (d) (e) (f) (g) (h) 0 (i) (j). (a) RHS LHS from above (c) from below (e). (a) from below from above. (a)

32 ANSWERS 8 (f). # 8. -, - 9. #, 0. - # - Eercises.. (a) (g) (h) (i) (j) (c) Eercises #. - # 0.. $ -,

33 88 Maths In Focus Mathematics Etension Preliminar Course (d) (g) (e) (h) (f) (i)

34 ANSWERS 89 (j) (c) (d). (a) - $ - (c) $ + (d) - (e) $. (a) (e) (a) - + (c) + 9 (d) + 8 (e),

35 90 Maths In Focus Mathematics Etension Preliminar Course. (a) (c) (c) (a) (d)

36 ANSWERS 9 (e) (h) (f) (i) (g) (j)

37 9 Maths In Focus Mathematics Etension Preliminar Course. (a) (d) (e) (a) (c)

38 ANSWERS 9 (e) Test ourself. (a) f ]- g f ] ag a - a - (c), - (c). (a) (c) (d) (d) - (e) - -

39 9 Maths In Focus Mathematics Etension Preliminar Course (f) 8. (g) 9. (h) 0.. (a) Domain: all real ; range: $ - Domain: all real ; range: all real (c) Domain: - # # ; range: - # # (d) Domain: - # # ; range: 0 # # (e) Domain: - # # ; range: - # # 0 (f) Domain: all real! 0; range: all real! 0 (g) Domain: all real ; range: all real (h) Domain: all real ; range: $ 0. (a) # + (c) $ -, # 0. (a) Domain: all real!, range: all real! 0.. (a) (c) 9 (d) (e).. (a). (i), - (ii) - (iii), -

40 ANSWERS 9. (a) (c). (a) -intercept -0, -intercept -intercepts -,, -intercept -. (a) i iii (c) ii (d) i (e) iii. (a) (c) - (d) 8. - (c) 9. (a) Domain: $, range: $ 0 - (d) 0. (a) f ( ) f ( ) ]-g ] g f ( ) So f ] g is even. f ( ) f ( ) ]- g ( ) - + -( ) -f ( ) So f ] g is odd. (e). (a)

41 9 Maths In Focus Mathematics Etension Preliminar Course Challenge eercise. f ] g 9, f ]- g, f ] 0g. b -,... Domain: all real!! ; range: # -, Domain: $ 0; range: $ ,, -... h] g+ h] -g- h] 0g ]- g -.

42 ANSWERS f^( - a) h ( -a )- a - fa ( ) 0. Domain: $ ; range: $ 0. Domain: - # #..!. (a) RHS + + ] + g LHS + ` Domain: all real! - ; range: all real! (c). (a) 0.

43 98 Maths In Focus Mathematics Etension Preliminar Course Chapter : Trigonometr Eercises.. cos i, sin i, tan i. sin b, cot b, sec b. sin b, tan b, cos b. cos, tan, cosec 9. cos i, sin i. tan i, sec i, sin i. cos i, tan i 8. tan i, sin i 9. (a) c 0 9 (c) sin c, cos c, tan c 0. (a) sin 0c, cos 0c, tan 0c (c) sin 0c, cos 0c, tan 0c. sin c cos c sec 8c cosec 8c. 9. tan 8c cot c.. (a) cos cor sin 9c 0 (c) 0 (d) (e). 80c. c. p c 8. b c 9. t 0c 0. k c Eercises.. (a) (c) 0.9 (d) 0.98 (e).9. (a) c 0l c 0l (c) cl (d) c l (e) 9c0l Eercises.. (a).. (c) b.9 (d). (e) m.9 (f). (g) 0.0 (h) p. (i). (j) t 8. (k). cm (l).9 cm (m) 0. cm (n) 0. mm (o). m (p) k 0. cm (q) h. m (r) d. m (s). cm (t) b. m.. m. 0. cm..9 m. (a) 8. cm.8 cm. 0 cm and 0. cm.. mm m 9. (a). cm. cm (c) 9.0 cm 0. (a).8 cm. cm. 8 cm Eercises.. (a) 9c 8l a c 0l (c) i c 9l (d) a 0c l (e) a 8c l (f) b 0c l (g) c 0l (h) i 0c l (i) a 9c l (j) i c l (k) a c l (l) i c l (m) i c 8l (n) i c l (o) b cl (p) a 9c l (q) i c l (r) a c 8l (s) i 9 l (t) c c 9l. c l. cl. cl. 0c. (a). cm c l. a c8l, b cl 8. (a) m cl 9. (a) c9l cm 0..9 cm and. cm. (a).9 m c l Eercises.. (a) North. (a) c 8c (c) 9c (d) c (e) c. (a) cl 8c l (c) 9c l (d) c l (e) c 0l. (a).c.c (c).8c (d) 8.c (e) 8.c. (a) 9c l cl (c) 8c l (d) c l (e) cl Beach house 00c Boat

44 ANSWERS 99 Jamie North (f) North Farmhouse c 0c Campsite (g) Dam North (c) North House 0c Jett 00c (h) North Mohammed (d) Seagull North Alistair Town 80c Mine shaft 0c Bus stop (i) Yvonne North (e) North Plane 9c School B Hill 8c

45 800 Maths In Focus Mathematics Etension Preliminar Course (j) North. (a) nd -. (a) nd. (a) st. (a) (c) - (d) (e) - (f) - (g) (h) - (i) - (j) - Boat ramp 8. (a) - - (c) (d) - (e) - 80c Island (f) - (g) (h) - (i) (j) -. (a) 8c c (c) 080c (d) c (e) 80c. 080c. 0c. 0c. 0. m. m 8..9 m 9. c8l 0. (a) 0. km.8 km (c) c. 8. m..8 km. m. c. 0c. 9. m. m km 9. c l 0.. m. 9cl. 9.9 km.. m. 9c. 98 m..8 km. 9. m 8. c 9. (a). km. km 0. (a). m 0cl Eercises.. (a) + (c) + (g) (h) (d) (e) ^ + h (i) (f) (j) - ^ + h (k) 0 (l) (m) ^ - h (n) - (o) (p) - (q) (r) - (s) (t) 9. (a) (c) p. 0c. m. m.. (a) m m m m Eercises. 0 m ^ + h m. (a) st, th st, rd (c) st, nd (d) nd, th (e) rd, th (f) nd, rd (g) rd (h) rd (i) nd (j) th 9. (a) - (c) (g) (h) (i) (j) 0. sin i -, cos i -. cos i -, tan i -. cos 8, cosec - 89 (d) (e) - (f) 89. cosec -, cot -, tan -. cos -, sin -. tan i -, sec i 9 8. tan, sec -, cosec - 8. (a) sin 9 cos -, tan cot a -, sec a, cosec a - 9. sin i, cot i (a) sin i cos (c) tan b (d) - sin a (e) - tan i (f) - sin i (g) cos a (h) - tan Eercises.8. (a) i 0c9l, 9cl i 0c, 0c (c) i c, c (d) i 0c, 0c (e) i 0c, 0c (f) i 0c, 0c (g) i 0c, 0c, 0c, 00c] 0c# i # 0cg (h) i 0 c, 0 c, 90 c, 0 c, 0 c, 0c ] 0c # i # 080cg. (a) rd -. (a) th -

46 ANSWERS 80 (i) i 0c, 0c, 0c, 0c (j) i c, c, c, 0c, c, c, 9c, c, c, 8c, c, c.. (a) i! 9cl i 0c, 0c (c) i c, - c (d) i -0c, - 0c (e) i 0c, - 0c (f) i! 0c,! 0c (g) i c0l, c0l, -c0l,-c0l (h) i! c,! c,! c,! 0c,! c,! c (i) i c, - c (j) i! 0c,! 0c,! 0c,! 0c Eercises.9. (a) cos i - tan i (c) cos i (d) tan i (e) - sec a. (a) sin i sec i (c) cosec (d) cos (e) sin a (f) cosec (g) sec (h) tan i (i) cosec (j) sin (k) (l) sin icos i i. 0c, 80c, 0c c, 0c c. 0c, 80c, 0c. 0c, 80c, 0c. - 0c, 90c. (a) LHS cos - - sin - -sin RHS So cos - - sin LHS sec i + tan i sin i + cos i cos i sin i + cos i RHS sin i So sec i + tan i + cos i (c) LHS + tan a ( + tan a) sec a cos a - sin a RHS So + tan a - sin a

47 80 Maths In Focus Mathematics Etension Preliminar Course (d) LHS sec - tan tan + - tan cosec - cot RHS So sec - tan cosec - cot (e) LHS ] sin - cos g ] sin - cos g] sin - cos g ] sin - cos g^sin - sin cos + cos h ] sin - cos g] - sin cos g sin - sin cos - cos + sin cos RHS So ] sin - cos g sin - sin cos - cos + sin cos sin i sin i (f) RHS - + sin icos i cos i + sin i sin icos i cos i sin i + sin icos i sin icos i cos i + sin i cos i cot i + sec i LHS sin i sin i So cot i + sec i - + sin icos i (g) LHS cos ] 90c - ig cot i sin icot i cos i sin i # sin i sin icos i RHS So cos ] 90c - ig cot i sin icos i (h) LHS ] cosec + cot g] cosec - cot g cosec - cot + cot - cot RHS So ] cosec + cot g] cosec - cot g sin icos i (i) LHS - cos i sin icos i - cos i cos i sec i - sin i tan i + -( - cos i) tan i cos i tan i + cos i RHS - sin icos So cos i i tan i + cos i.. cot b (j) LHS + - cos b cosec b cot b cos bcosec b + - cosec b + cot b - cos b # sin b cosec b cot b cot b + - cosec b cosec b sin b LHS sec b RHS tan b + cot b sec b sin b cos b + cos b sin b sec b sin b + cos b sin bcos b sec b sin bcos b cos bsin b sec b # cos bsin b # cos b sin b RHS + cot b So - cos b sin b cosec b LHS + ] cos ig + ] sin ig cos i + sin i ( cos i + sin i) ] g RHS So + LHS + ] 9cos ig + ] 9sin ig 8 cos i + 8 sin i 8( cos i + sin i) 8] g 8 RHS So + 8

48 ANSWERS 80 Eercises.0. (a) cm (c) a 0.0 (d) b 0. m (e) d 8.0. (a) i c l a c l (c) c 0l (d) a 8c 0l (e) i 0c l. c l. (a). mm mm. (a).8 m. m.. cm. (a) 0. m 9. m 8. (a) 0c l c 9l 9. (a). cm. cm 0. (a). mm. mm Eercises.. (a) m.8 b 0. m (c) h. cm (d) n. (e) 9.. (a) i c 0l i 0c l (c) c l (d) b c l (e) i c 9l..9 mm.. cm and.9 cm. (a).9 cm cl (c) 8cl. + XYZ + XZY c0l, + YXZ c0l. (a) 8. mm 80c9l 8. (a). cm. cm 9..9 cm 0. (a) cm 0c Eercises... cm and. cm. (a) 00c 0c.. m. 0c..9 m. c. (a). km minute 8. m 9. 0 m. 8 sin c 9 0. (a) AC l i c 0 l sin 0c l. h 8... km.. km and. km. 8 km..8 m. 89cl. 9.9 km 8.. km 9.. m 0. 9 km. (a). cm c 0l. c. (a). cm cl. (a). km cl. (a) c l (i). m (ii) 0. m Eercises.. (a). cm. units (c) 9.9 mm (d) 0. units (e). cm. m.. cm.. cm..8 cm.. m. cm 8.. mm 9. (a).8 cm 80.8 cm 0. (a). cm 8. cm (c) 9. cm Eercises.. (a) m. m (c) c l. (a).9 m 9c l. (a) 09 cm c 0l. c 9l. (a) 9 m cl. (a) m 89. m. (a) 8 m 8. m (c) 9. m 8. 8 m 9. c 0l 0. c0l Eercises.. (a) sin acos b - cos asin b cos pcos q - sin psin q tan a + tan b (c) - tan a tan b (d) sin cos 0c+ cos sin 0c tan 8 + tan (e) (f) cos icos a + sin isin a - tan 8c tan tan - tan (g) cos cos c- sin sin c (h) + tan tan tan a - tan b (i) sin a cos b - cos a sin b (j) + tan a tan b. (a) sin ] a + bg tan c (c) cos c (d) sin^ + h (e) tan i (f) sin c (g) sin acos b (h) cos sin (i) sin sin (j) cos mcos n +. (a) (c) (d) - ^ + h - ^ + h - - (e) + (g) - + (f) + - -

49 80 Maths In Focus Mathematics Etension Preliminar Course + (h) - ^ + h - ^ + h - (i) sin e - + o+ cos e o (j) cos cos +. tan. (a) + + (c) - + tan i. (a) sin i cos i cos i - sin i (c) - tan. (a) sin icos i - sin i tan i - tan i cos i - sin icos i (c) - tan i 8. (a) tan i sin i cos i - cos isin i 9. cos cos - sin sin 0. (a) (c) - (d) - (e). (a) (c) - (d) (e) -. (a) cos cos [ cos c+ cos ]-cg]. (a) sin cos - sin sin (c) cos sin (d) cos cos - sin sin + sin cos - cos sin tan _ + tan i (e) - tan tan tan i. (a) sin b cos b - tan tan ^ tan + h (f) - tan tan i i (c) cos i - sin i (d) sin cos + cos sin sin _ cos - sin i+ cos sin cos (e) cos a cos b - sin a sin b ^cos a - sin ahcos b - sin a cos a sin b tan + tan (f) - tan tan tan - tan tan + tan - tan - tan tan (g) sin icos d - cos isin d sin icos icos d - cos isin d + sin isin d (h) cos icos c + sin isin c cos i_ cos c - sin ci+ sin isin c cos c tan - tan z tan - tan tan z - tan z (i) + tan tan z - tan z + tan tan z (j) sin cos - cos sin sin cos _ cos - sin i - sin cos ^cos - sin h. (a) sin cos (c) tan 0i (d) cos (e) sin i (f) + sin (g) cos a (h) cos 80c (i) tan b (j) - sin. (a) (g) (c) (h) (i) 9. cos -, sin 8. (a) (j) - (c) 0 (d) sinicosi^cos i - sin ih sin icos i - sin icos i 0. (a) tan (a) RHS sin itan i ( sin icos i) tan i sin i sin icos i cos i sin i LHS ` sin i sin itan i cos i RHS - sin i i i -dcos - sin n i i sin cos i i - cos + sin i i sin cos i i sin + sin i i sin cos i sin i i sin cos i sin i cos i tan LHS i cos i ` tan - sin i (d) (e) (f)

50 ANSWERS 80. RHS sin isin i sin ( i + i) sin ( i - i) ( sin icos i + cos isin i) ( sin icos i - cos isin i) sin icos i - cos isin i sin i( - sin i) -( - sin i) sin i sin i - sin isin i - sin i + sin isin i sin i - sin i LHS ` sin i - sin i sin isin i. LHS cos i cos ( i + i) cos icos i - sin isin i ( cos i - sin i) cos i - sin icos i cos i - sin icos i - sin icos i cos i - sin icos i cos i - ( - cos i) cos i cos i - cos i + cos i cos i - cos i RHS ` cos i cos i - cos i. sin - sin Eercises.. (a) tan i cos i (c) tan 0c (d) cos 0c. (a) (e) sin i (f) cos i + t. (a) t - t + t (e) - t + t (i) - t + t - t (c) (d) 0 + t (f) - t t^ - t h (j) ^ + t h - t (c) t t + - t (d) + t - t + 8t (g) + t (h) t. t - t - + t - t ^ + t h. (a) sin ] i + c l g sin ] i + 0cg (c) (e) (g) (i) sin ] i + cg (d) 9 sin ] i + c 8l g sin ] i + c l g (f) 0 sin ] i + 8c l g sin ] i + c9l g (h) sin ] i + 0cl g sin ] i + 8c 0l g (j) sin ] i + 9c l g. (a) sin ] i - cg sin ] i - c l g (c) sin ] i - 0cg (d) sin ] i - 0cg (e) 9 sin ] i - c 8l g 8. 0 cos ] i - 8c l g 9. cos ] i + 0cg 0. (a) 8 sin ] i + c l g 8 cos ] i - c 8l g Eercises.. (a) c, c 0c, 0c (c) 0c, 0c, 80c, 00c, 0c (d) 0c, c, 80c, c, 0c (e) 90c, 0c, 0c (f) 0 c, 0 c, 00 c, 0c ( g) 0 c, c, 80 c, c, 0c (h) 0 c, 80 c, 0c (i) 0 c, c, 0 c, c (j) 0 c, 0c. (a) i cl, 0cl i c8l, 89cl (c) i 0 c, 0c (d) i 80 c, 0c (e) i 0cl, cl (f) i 90 c, 80c (g) i 90c, 0cl (h) i cl, cl (i) i cl, 90cl (j) i 0cl, 0c. (a) i 80n + ]-g n # 0c a 80n + 0c (c) i 0n! 0c (d) 80n -]- g n # 0c (e) i 80n - c (f) b 0 n! c (g) c 80n! 0c (h) i 80n + 0c (i) i 0n! c 9l (j) a 80n + ]- g n # c l. + sin i - cos i LHS + sin i + cos i t - t t + t t - t t + t + t + t - + t + t + t + t + - t + t t + t + t tt ] + g ] + t g t RHS + sin i - cos i ` t + sin i + cos i. c0l, 8c0l, -9c0l,-c0l. 80n + ]-g n # 0c, 0n! 90c. - 80c, 0c, 90c, 80c. (a) i 80n 0n (c) 80n n (d) i 80n + (- ) 0c (e) 0n! 90c 8. (a) (i) 0c, 0c (ii) 80n + ]-g n! 0c (i) cl, 8cl (ii) 0n! c l (c) (i) c, cl (ii) 80n + c l (d) (i) cl, cl (ii) 80n - 8c l (e) (i) c (ii) 80 n + (-) n 90c- c

51 80 Maths In Focus Mathematics Etension Preliminar Course n n 9. 80n + ]- g # 0 c, 80 n + (-) 0c 0. (a) 0c, 0c, 0c, 0c 0n! 0c, 0n Test ourself. cos i, sin i. (a) cos (c) cosec A (d) cos i (e) cos 0i. (a) 0..8 (c) 0.9. (a) i c l i c l (c) i c l. cos i LHS - sin i ^ - sin ih - sin i ] + sin ig] - sin ig - sin i ( + sin i) + sin i RHS cos i So + sin i - sin i. b 0c. (a) 0 (d) - (e) c, 0c 9. 90c, 0c - (c) - 0. km.. (a). cm 8. m. (a) i c l i 8cl (c) i 9c l.. cm. (a)! 0c,! 0c c, 0c, -c,- c (c) 0c,! 80c, 0c, - 0c. sin i -, cot i. (a) 09c 09c n 8. i 80n + ]- g 0c+ c8l 0 sin 9c 9. (a) AD 8. m sin 99c + ^ + h 0. 9 km. (a) - ^ - h (c). (a) 0n! 0c 80n + c (c) 80n + ]-g n # 0c. i 0c, 0c, 0c. a c 0l. (a) cos ^ + h cos ] + g cos cos - sin sin cos - sin ^ - sin h - sin - sin Challenge eercise. 9c 8l. 0. km.. cm. sin 9c l. (a) AC h. cm.. km sin c l. - cos. cm c0l, c0l, 0c0l, 9c0l 0. i c l.. m. i 0c, 0c.. km. -. m. 9 cos i ] sin i + cos ig LHS - sin i ] sin i + cos ig cos i cos i sin i + cos i cos i tan i + RHS (a) m c l 9. (a) c l 9 m 0. 0c 8l. LHS cos icos i - sin isin i cos ( i + i) cos 0i cos i - sin i cos i -( - cos i) cos i - RHS ` cos icos i - sin isin i cos i m, 0. ms -. i 0, 0, 0 n. i 80 n + (- ) 0c. - t

52 ANSWERS 80 Chapter : Linear functions Eercises.. (a) 0 (c). (a) (c) 8 (d). (a) 9.8. (c).. units. Two sides,side 8. Show AB BC 8. Show points are units from ^, -h 8. Radius units, equation Distance of all points from ^0, 0h is, equation + 0. a. a! -. All sides are units.. a 0, -. MQ NP, QP MN 0, so parallelogram. BD AC 98. (a) AB AC 0, BC OC OB units 9. AB 9, BC, AC AB + BC 9 + AC So triangle ABC is right angled (Pthagoras theorem) 0. XY, YZ 0, XZ Since XY YZ, triangle XYZ is isosceles. XY + XZ + 0 YZ So triangle XYZ is right angled. (Pthagoras theorem) Problem 0. Eercises.. (a) ^, h ^, -h (c) ^-, h (d) ^-, h (e) ^-, h (f) ^-, h (g) d, n (h) d, n (i) d, n (j) d0, n. (a) a 9, b - a -, b (c) a -, b - (d) a -, b - (e) a, b. + ] - g 0, P Q ^, -h. ^, h. is the vertical line through midpoint ^, h.. Midpoint of AC midpoint of BD d, n. Diagonals bisect each other 8. AC BD, midpoint AC midpoint BD d, - n ; rectangle 9. ^-8, h 0. (a) X d-, n, Y d, n, Z ^, h XY 0, BC 0 0; XZ, AC ; YZ, AB Eercises.. (a) (c) - (d) - (e) (f) - 8 (g) - (h) - (i) (j) (a) Show Lines are parallel. (-, ) (, ) m m (, ) (, -) -. Gradient of AB gradient of CD Gradient of BC gradient of AD 0. Gradient of AB gradient of CD - Gradient of BC gradient of AD Gradient of AC -, gradient of BD - 8. Gradient of AC, gradient of BD - 9. (a) Show AB + BC AC Gradient of gradient of AB, BC -

53 808 Maths In Focus Mathematics Etension Preliminar Course 0. (a) F ^, - h, G d, n Gradient of FG gradient of BC Gradient of ^, -h and ^, - h gradient of ^, -h and ^, h cl. 08c l 8. (a) (c) - - -]-g 9. m m tan i - tan i ` i 80c- c ^ nd quadranth c ^ + h 0. Eercises.. (a) (i) (ii) (i) (ii) (c) (i) (ii) - (d) (i) - (ii) 0 (e) (i) - (ii) (f) (i) (ii) - (g) (i) - (ii) (h) (i) - (ii) (i) (i) 9 (ii) 0 (j) (i) (ii) -. (a) (i) - (ii) (i) - (ii) - (c) (i) (ii) - (d) (i) (ii) (e) (i) - (ii) (f) (i) (ii) (g) (i) - (ii) - (h) (i) - (ii) (i) (i) (ii) - (j) (i) (ii). (a) - (c) 0 (d) - (e) - (f) - (g) (h) (j) (k) (l) (m) (n) (o) - (p) - (q) (r) - (s) (t) - 8 Eercises.. (a) (c) - (i) (d) + 0 (e) (f) (g) - (h) (a) (c) (d) (e) (a) Eercises.. (a) - (c) (d) (e) (f) - (g) (h) (i) (j). (a) (c) (d) (e) (f) (g) m m so parallel. mm - # - so perpendicular. m m. m # m - # -. k - 8. m m 9. AB < CD _ m m i and BC < AD dm m - n 8 0. Gradient of AC: m, gradient of BD :, m - m # m #- -. (a) (c) (d) Eercises.. (a) ^, -h ^-, -h (c) ^, h (d) ^0, -h (e) ^, -h (f) ^-, h (g) ^, h (h) ^, 0h (i) ^, h (j) d, - n. Substitute ^, -h into both lines 9 9. ^, h, ^, h and ^-, -h. All lines intersect at ^, -h. All lines meet at ^-, 0h

54 ANSWERS 809 Eercises.8. (a). (c). (d). (e) 8. (a).8.0 (c) 0.8 (d).09 (e).. (a). d d d (c). A: d, B: d (d) (e) Opposite signs so points lie on opposite sides of the line. ^, - h: d, ^9, h : d 0 Same signs so points lie on the same side of the line. ^-, h: d -, ^, h : d Opposite signs so points lie on opposite sides of the line 8. d d so the point is equidistant from both lines 9. ^8, - h: d, ^, h : d 0 9 Same signs so points lie on same side of the line 0., : d ^- h -, ^, h : d Opposite signs so points lie on opposite sides of the line. d d so same distance. 8 units or -. b or -. m - or Show distance between ^0, 0h and the line is 9. Show distance between ^0, 0h and the line is greater than 0 0. (a) ^, -h, d, n, ^-, h,, 9 Eercises.9. (a) 8c l 9c l (c) 8c l (d) c l (e) 0c 9l (f) c 9l (g) cl (h) 8c l (i) c l (j) c l. (a) 9c l 9c l (c) c 8l (d) cl (e) c. c 0l. c 8l. cl, 0c8l, 8c. m, -. m -.,. 8. k Z -., (a) + A + C cl, + B + D cl c l 0. + A cl, + B + C 9cl Eercises.0 8. (a) d-, n d, n (c) d-, n (d) d, - n (e) d, - n (f) d-, n 0 (g) d, n (h) d-, - n (i) d-, n (j) d, - n. (a) d-, n d, n (c) ^9, h (d) d, n (e) ^0, h (f) d9, - n (g) d-, - n (h) d9, n (i) ^-8, 0h (j) ^0, h. (a) E d, n F d, n (c) EF, AC ` AC EF. A B (, ) (, ) (, ) (-, ). P d, n, Q ^, - 9h, PQ units. B d9, - n. p, q 0 8. (a) d, n Each ratio gives d, n. This means that the intersection of the medians divides each median in the ratio : a 8, b 8 0. P d,9 n 9 Test ourself.. units. d, -n. (a) - (c) (d). (a) (c) + 0. (d) (e) units. m -, m so m m - ` lines are perpendicular.. -intercept, -intercept - 8. (a) (c) units 9. m m, so lines are parallel

55 80 Maths In Focus Mathematics Etension Preliminar Course. ^-, h. a, b. c 8l. Solving simultaneousl, and have point of intersection ^, -h. Substitute ^, -h in - - 0: LHS # - #- - 0 RHS ` point lies on - - 0: Substitute ^, -h in : LHS # - # RHS ` point lies on : ` lines are concurrent. d, - n c -, ^, h 0.. 9cl 8. ^-, h: d -, ^, h : d Opposite signs so points lie on opposite sides of the line. (a) AB : ^-, h lies on the line (show b substitution) - : or : -., -. m , (a) P d, n Q d, n (c) PQ has gradient m 0 AC has gradient m 0 Since m m, PQ AC < (d) R d, 0n (e) PR has gradient m - BC has gradient m - Since m m, PR BC <. c l Challenge eercise. k Show AC and BD have the same midpoint ^, h and m # m - AC BD Chapter 8: Introduction to calculus Eercises 8... Show distance of all points from ^0, 0h is ; radius ; equation OBA c; a b ( sides of isosceles D) 9. cl BC AC 8, AB, so D is isosceles; m # m -, so D is right angled. BC AC. ^, -h.. a, b c b, - 9. d,- n, d,- n 0. m - m + mm m - m ` + mm mm+ m- m mm m- m- m - m or - + mm m - m - - m m mm m- m- -p - p -. P f, p p - p -.

56 ANSWERS Eercises 8.. Yes, 0. Yes,. No. Yes, 0. Yes,,. Yes, 0. Yes, - 8. Yes, 9. Yes, -, 0. Yes, - # 0. Yes, 90c, 0c. Yes, 0. No. No. Yes,!. Eercises 8.. (a) - (c) (d) 8 (e) (f) - (g) (h) - (i) 0 (j) -. (a) - - (d) - (e) - + (f) + (g) - (h) (i) - (j) Eercises 8... (a).0.99 (c). (a)..00 (c).90 (d) (a) f ] + hg + h+ h f( + h) - f( ) + h + h - h + h f] + hg- f] g h + h (c) h h h] + hg h + h f ] + hg- f ] g (d) fl( ) lim h " 0 h lim ( + h) h " 0. (a) f ( + h) ] + hg - ( + h) + ( + h + h )- - h + + h + h - - h + f ( + h) - f ( ) ( + h+ h- - h+ ) -( - + ) + h + h - - h h + h - h

57 8 Maths In Focus Mathematics Etension Preliminar Course (c) f ] + hg- f ] g h + h - h h h h] + h - g h + h - (d) fl] g -. (a) f ] g f] + hg h + h + (c) f] + hg - f] g h + h (d) f] + hg - f] g h h + h h hh ] + g h h + (e) fl] g. (a) f ]- g - f] - + hg - f] - g h - h + h (c) 8. (a) f ] g 8 f] + hg- f] g h + h (c) fl] g 9. (a) fl] g - 0. (a) + Substitute _ + d, + di : + d ] + dg + ( + d) + d + d + + d Since + d d + d + d d d + d + d d d d] + d + g d + d + d (c) + d. (a) (c) - (d) (e) - 9 d. (a) fl] g + d d (c) fl] g 8 - (d) 0 - d d (e) (f) fl] g + d d (g) - + (h) fl] g - d. (a) (c) (a) (c) Eercises 8.. (a) (c) + (d) 0 - (e) + - (f) - + (g) - + (h) (i) (j) 0 -. (a) (c) (d) (e) + -. (a) (c) (f) - +. fl] g (d) (e) d 9 ds t - 0 d dt - dv dh 8. gl] g t t dt dt dv. rr.. (a) - (c) dr. (a)!. 8 Eercises 8.. (a) - (c) (d) - 8 (e) 8 (f) (g) (h) (i) - (j) 9. (a) - (c) 0 (f) (g) - (h) 0 (d) - (e) 0 (i) - (j) - 8. (a) (i) (ii) - (i) 8 (ii) - 8 (c) (i) (ii) - (d) (i) - 8 (ii) 8 (e) (i) (ii) -. (a) (c) (d) (e) t - v (a) (c) (d) (e) (a) (i) (ii) (i) (ii) (c) (i) (ii) (d) (i) (ii) (e) (i) (ii) ! 8. (, ) and (-, 0) 9. (-, - ) 0. (0, ). (, ). d-, - n. (a) (, - ) t - h

58 ANSWERS 8 Eercises (a) (c) (d) (e) (a) (f) - (g) - (f) - (g) - (h) (j) (h) - 0 (c) (d) - (e) (i) (a) (9, ).. d, n, d-, - n Eercises 8.8. (a) ] + g ] - g (c) 0 ^ - h (d) 8] 8 + g (e) - ] - g 8 (f) ] + 9g (g) ] - g (h) ^ + h^ + h (i) 8] + g^ + - h (j) ^ - h^ - + h (k) - - ] g (l) ] - g - (m) - ^ - 9h - (n) ] g (o) ^ - + h^ - + h (p) + 8 (q) - (r) - (s) - ] - g ^ + h - (t) - ] + g ^ h (w) - ^ - + h (u) - ] - g + () (v) - ] + g 0 () ] - g (, )., Eercises 8.9. (a) (c) 0 + (d) - (e) 0 - (f) ] + g] + g (g) 8] 9 - g] - g (h) ] - g] - g (i) ] 0 + g] + g (j) 0^ + - h^ + h + ^ + 0h^ + h ^ h^ + h (k) ] + g (l) + ] - g - - ] - g ! Eercises (a) ] - g (d) ] + g - (g) ^ - h - ^ - h (c) ] + g ^ - h ^ - h - + (e) - + (f) ] + g - (h) ] - g - (i) ] - g ] - g (k) (l) ^ - h ] - g ] - g -8 + ] + g (m) (n) ^ - h ] + g ] + g (o) ] + g (q) ^ - - h (s) (p) ] + g ] + g - 8] - g] + g (t) ] + g ] + g - ] + g (r) + - (j) ] + g ] + g 8 ] - g ] + g - ] + g ] - g (u) ] - g ] + g ] - g ] - g (v) ] + g (w) ] - g - ] - g () ] + g] - 9g - ] - 9g ] - 9g + ] + g ] - 9g ] 0 + g ] + g - ] - 9g ] - 9g + ] - 9g

59 8 Maths In Focus Mathematics Etension Preliminar Course ,. - 9, Eercises 8.. (a) Substitute Q into both equations. (c) - has m has m - (d) 8c l d d (a) d d d d ] + g d 8 (c) 9 ( + )( + - ) d d (d) 0] - g + ] - g ] - g (0 - ) d d d 0 (e) (f) - d d. (a). dv t -. (a) dt 8. (a) - (c) P ^, 9h (c) m (d) 0c. 8c 8l. c l. c l. (a) X ^, h, Y ^-, h At Xm :, m At Y: m - 8, m - (c) At X: clat Y: c9l d 9. (a) fl] g ] + 9g - d ] - g d d (c) ] 9 - g] - g (d) - d d (e) fl] g 0.. cl, 8c8l 8. (a) (0, 0), (, 8), (-, - ) c l at (0, 0), c l at (, 8), c l at (-, - ) 9. At (0, 0), m 0and m At (, ), m and m 0 Angle at both is c 8l 0. c l at (0, 0), 8c l at (-, - ), c9l at (, ) Test ourself 8. (a) ds (, ). 8rr dr. (-, ), (, - ) ds u + at, t dt cl at (, 9), c8l at (-, ). c l at (, ), c 0l at (, )

60 ANSWERS 8 Challenge eercise 8. f] g -, fl] g d. 8t + 00t ; t 0, -. dt. + 0, - - 0, ^, h, ^-, - h, + - 0, c 80c 0c 0c. ] + g ] - 9g + ] - 9g ] + g 0] + g ] - 9 g ( - ) 8. ] - 9g - ] + g] - 9g ] - 9g - ] + g ] - 9g 8! 0! !. a -. P d-, n., , Q ^0, h, PQ 0. (a) Substitute (, ) into both curves: ] - g : LHS RHS ] # - g LHS So (, ) lies on the curve ] - g - : + LHS # - RHS + LHS - So (, ) lies on the curve + ` (, ) is a point of intersection c l. n 8 8. e, o, ,-, 0. (a) 90c, 0c. ^-, -h (a) + + 0, m $ m - # - So perpendicular. 0,,. a -, b. dv 8. p 9. dr ] - g - 8r 0. k a -, b, c 8. S 8rr - 8r + rrh -] + g. (a) - ] - g] - g ] - g +.! 8. (a) Q d-, n 9 Practice assessment task set ^, -h (a) cm AC cm, BD cm 8. mm # - - ; A d-, n 8. c 9.

61 8 Maths In Focus Mathematics Etension Preliminar Course i 0c, 0c a c l Show perpendicular distance from ^0, 0h to the line is units, or solving simultaneous equations gives onl one solution. 0. (a) g] g, g] - g m. -, -. (a) AB.0 m.8 m. c 9. Domain: all real! ; range: all. real! 0. cos i. (a) P^-, 0h, Q^0, h (c) units 8. m 9. units 0. f( - ) ]-g -]-g f ( ). ^ + h + ^ + h ^8 + h^ + h. - # #9. -. (a) (c) (d) R ^-0, 0h 8. units. Domain: all! - ; range: all! ] + g km ] + g f] - g -, fl] - g 8. a, b ^, -h. sin i. units c l. cl, cl

62 ANSWERS 8.. (a) cos i cos ^i + bh (c) tan a..,.. c l at both points 8. (a) domain: $ range: $ 0 domain: all real! - range: all real! 0 (c) domain: - # # range: - # # 0 9. a -, b - 0. cos i. (a) (0, 0), (, ), (-, - ), (, 0) c l at (0, 0), c 0l at (, ), 0c l at (-, - ), 0c l at (, 0). (a) 0n! c 80n + 0c (c) 80n + ]-g n # 0c. (a) (, ) units (c) - (d) (a)., (d). (a). (c) 8. (c) 9., (d) 80. (c) Chapter 9: Properties of the circle The proofs given as answers to this chapter are informal. Also, the ma not be the onl wa to answer the question. Eercises 9.. (a) i c 8 cm (c) i a 8c0l (d) i c (e) 9 mm (f) i c0l. (a) c, c, z c 9c (c) c, c (d) c, c (e) c, b c (f) c, z 8c, v c, w c (g) c (h) 0c (i) c0 l, c0l (j) c, c, z c. (a) + DCE + ACB (verticall opposite +s) + EDC + BAC ( + s in the same segment) + DEC + ABC (similarl) ` Since all pairs of + s are equal, D ABC. cm ;< DDEC. 0 (angle at centre is double the+ at the circumference) ( 80-0 ) ' ( + sum of isosceles D) c. 0 - # 0 ( + at the centre is double the + at the circumference) ` 0 0 (similarl) 8. + ABC 90c ( + in semicircle) ` + BAC 90c- 9c ( + sumof D) c ` c ( + in same segment) 9. + STV + WUV ( + in same segment) + TSV + UWV (similarl) + TVS + UVW (verticall opposite + s) ` Since all pairs of angles are equal, DSTV DWUV cm B 90c ( + in semicircle) AC AB + BC AC Radius AC cm. r cm 9. (a) i 9c 8c (c) a 8c, b c (d) 8c (e) 0 cm (f) 9c (g) c, 0c, z c (h) c, c, z 9c (i) b 0c (j) 9c. + OAC 0c(Base+ s of isosceles D) + BAO c(similarl) ` + CAB 0c+ c c + CAB ( + at the centre is double the + at the circumference) # c 0c

63 88 Maths In Focus Mathematics Etension Preliminar Course. (a) c, c AC BD (equal diameters) Diagonals are equal so ABCD is a rectangle. ` AD BC (opposite sides of a rectangle). + ECB c (angles in same segment) + EBC 80 -] + g (angle sum of triangle) `+ ECB + ADE These are equal alternate angles. ` AD < BC. (a) + AOB 90c (given) + ABC 90c (angle in semi-circle) `+ AOB + ABC `+ A is common `D AOB DABC ( AAA ) (Note pairs of angles equal means pairs will be equal b angle sum of triangle.) AO BO (equal radii) AB r + r r # r r B similar triangles AO BO AB BC But AO BO so AB BC So BC r. Obtuse + BOD i (angle at centre double angle at circumference) Refle + BOD 0 - i (angle of revolution) + BCD + BOD (angle at centre double angle at circumference) (0 - i) 80 - i So + BCD and + DAB are supplementar (add to80c) Eercises 9.. (a) cm cm (c). m (d) c (e) z 90c (f) Z 0. m (g) m, m (h) m Z. cm (i) Z cm (j) mm. cm. mm.. cm. CE CD # 9. (perpendicular from O bisects chord) 8. AB. OB 8. cm.. m, 8. m 8. Z. m, a 8 c, b 8 c, i c 9. OA r AC (perpendicular from the centre bisects a chord) OC r -d n (Pthagoras theorem) r - r - r - r - CD r + r - r + r - 0. (a) + ECD + ACB (verticall opposite angles) + A + E (angles in same segment) `D ABC DCDE ( AAA ) B similar triangles AC BC CE CD AC. CD BC. CE Eercises 9.. (a) 0c, 9c i c, c 90c (c) c, c, z 8c (d) 9c, c (e) b c, a 0c, c 0c (f) c, c (g) c, c (h) w 89c, 8c, c, z c (i) w 9c, c, 8c, z 98c (j) 8c. (a) c, c c, 0c (c) 88c, c (d) c, 8c, z c (e) 90c, c (f) 8c, c (g) 8c, 9c (h) 8c, 8c (i) 0c, c (j) a 8c, b c, c 8c, d c, e 8c. (a) + A 80c- 8c ( + A and + B cointerior angles, AD ; BC) + D 80c- 8c ( + C and + D cointerior angles, AD ; BC) So + A 80c - + C and + D 80c - + B Since opposite angles are supplementar, ABCD is a cclic quadrilateral. + B + D 90c (given) ` + B 80c - + D Let + A + C 0 -] g (angle sum of quadrilateral)

64 ANSWERS A Since opposite angles are supplementar, ABCD is a cclic quadrilateral. (c) + CDA 80 - i (straight angle) ` + B 80c - + CDA Let + A + C 0 -] g (angle sum of quadrilateral) A Since opposite angles are supplementar, ABCD is a cclic quadrilateral. Eercises 9.. (a) i c m (c). cm (d) c (e) a c, b c (f) i c (g) p Z cm (h) 0 mm (i) Z.9 cm (j) c, c. (a) 0 cm c, c (c) cm (d) c, c (e) cm (f) c, c (g) c, c (h) c, 90c (i) m c, n c, p c, q c (j) c, c. + OAB 90c (tangent to radius) ` z 90c- 8c c ( + sum of D AOB) OA OC ( equal radii) ` + OAC + OCA (base + s of isosceles D) ( 80c- 8c) ' ( + sum of D OAC) c + ACD 80c -+ AED (opposite + s of cclic quad.) ` + u 80c- c c+ u 8c u c + BAC + OAB -+ OAC ` 90c- c c v + AOC ( + at centre twice + at circumference) # 8c c. cm. AC + BC AB.. ` AB AC + BC ` + ACB 90c ( b Pthagoras theorem) ` A lies on a diameter of the circle (tangent radius). (a) c Z. cm (c) c, c (d) c, c (e) 8.9 m, Z. m (f) c, c (g) 98c, c, z c (h) c, c (i) c, c (j) c, 0c, z c. (a) c, c, z 8c 8 c, c, z 8c (c) z c (d) 0c, c (e) 0c, c, z 0c (f) Z. cm (g) Z cm. (h) c, c (i) Z. cm, Z. cm (j) c, c, z c 8. AB Z m Test ourself 9. i c.. mm.. m. cm. z 9c ( + sinsamesegment) 80c -( c + 9c) ( + sum of D) 0c 0c ( + sinsamesegment). 0 cm. a c, b c, c c 8. a # 00c ( + at centre twice + at circumference) 0c + OCA 90c (tangent perpendicular to radius) ` b 90c- 8c c OC OE (equal radii) ` D OCE is isosceles ` + OCE + OEC c c + 00c 80c ( + sum of D) c 80c c 0c Refle+ COE 0c- 00c ( + of revolution) 0c d 0c- ( 0c+ 0c+ c) ( + sum of quadrilateral) c 9. cm 0.. m. a 0c, b 98c. a c, b 9c..9 cm..9 m. 8 cm. a c, b c. + D 80c- ( 80c+ c) ( + sum of T) c ` c ( + sinsamesegment) c ( + s in alternate segment) 8. c, c, z c

65 80 Maths In Focus Mathematics Etension Preliminar Course 9. + C is common + A + CBD ( + s in alternate segment) ` D BCD DABC ] AAAg. Let + ODC and + OAB. Then ou can find all these angles (giving reasons). 0. (a) + OCB + OCA 90 c (given) OA OB ( equal radii) OC is common ` DOAC / D OBC ] RHSg AC BC (corresponding sides in / D s) OC bisects AB Challenge eercise 9. cm. Let+ DOB + DCB (base + s of isosceles D ODC) Then+ EDO EO DO ( et. + of D) ( equal radii) ` + OED + EDO ( base of+ s of isosceles DEOD) + EOD 80 c -( + OED + + EDO) ( + sum of D EOD) 80c - + AOE 80c -( + EOD + + DOB) ( + AOC straight + ) 80c-( 80c- + ) ` + AOE + DCB + AOC + + COB + + BOD + + AOD 0c ( + of revolution) 90c COB c AOD 0c + COB + + AOD + 80c 0c ` + COB + + AOD 80c. B. Let+ DAB and+ CAB Then+ DAC + + ACB + DAB ( + s in alternate segment) + ADB + CAB (similarl) + DBA 80 c -( + ) ( + sum of D ADB) + CBA 80 c -( + ) ( + sum of D ACB) + DBA + + CBA 80c ( DBC is straight + ) ` 80c- ( + ) + 80c- ( + ) 80c ` 80c ( + ) ` 90c + ` + DAC 90c. (a) AD DB BE EC CF FA (equal radii) ` AB BC CA `D ABC is equilateral r r units (c) r - rr r - r e o units. + BDE + ABD + + BAD ( et. + of D BAD) ` a + ABD + a a + ABD ` D BAD is isosceles with AD BD + CDE + ACD + + CAD ( et. + of D CAD) ` b + ACD + b b + ACD ` D CAD is isosceles with AD CD ` AD BD CD So a circle can be drawn through A, B and C with centre D. 8. A D Let ABCD be a kite with AB AD and BC DC, and + ADC + ABC 90. AC is common. b SSS (or RHS) DABC / D ADC `+ BAC + DAC and+ BCA + DCA (corresponding s in congruent D s) Let+ BAC + DAC a Then+ BAD a + BCA + DCA 90 c - a ( + sum of D ) ` + BCD 80c - a Opposite angles are supplementar. ABCD is a cclic quadrilateral, and A, B, C and D are concclic points Since + ABC 90c, AC is a diameter. ( + in semicircle) rr 8 units C

66 ANSWERS 8 9. Let interval AB subtend angles of at + ADB and + ACB. Now ABCE is a cclic quadrilateral, so + AEC + + B 80c (opposite + s supplementar) Also, + D + + B 80c (given) + D + AEC These are equal corresponding angles, so DA < EA (this is impossible!) A, B, C and D must be concclic ABCD is a cclic quadrilateral. Chapter 0: The quadratic function Assume A, B, C and D are not concclic. Draw a circle through A, B and C that cuts AD at E. Eercises 0.. Ais of smmetr -, minimum value -. Ais of smmetr -., minimum value -. Then + AEB + BCA ( + s in same segment) But + AEB and + EDB are equal corresponding angles. ` EB DB (this is impossible!) A, B, C, D must be concclic 0. Let ABCD be a quadrilateral with opposite angles supplementar. i.e. + A + + C 80c and + B + + D 80c Assume the points are not concclic. Draw a circle through A, B and C, cutting CD at E.. Ais of smmetr -., minimum value Ais of smmetr 0, minimum value -. Ais of smmetr, minimum point d, 8 8. Ais of smmetr, maimum value -. Ais of smmetr -, maimum point ^-, h 8. Minimum value -, solutions 9. Minimum value., no solutions n 0. Minimum value 0, solution. (a) - ; (-, -) - ; (-, ) (c) ; d, n (d) - ; d-,- n 8 (e) -; ^-, -h. (a) (i) - (ii) - (iii) (-, -) (i) (ii) (iii) (, )

67 8 Maths In Focus Mathematics Etension Preliminar Course. (a) Minimum (-, 0) Minimum (, -) (c) Minimum (-, -) (d) Minimum (, -) (e) Minimum (, -) (f) Minimum d-, - n 8 (g) Maimum (-, ) (h) Maimum (, ) (i) Maimum d, n (j) Maimum (, -). (a) (i) - (ii) Minimum 0 (iii) (c) (i).8, 0. (ii) Minimum -8 (iii) (d) (i) -, 0 (ii) Minimum - (iii) (i) -, (ii) Minimum - (iii) (e) (i)! (ii) Minimum (iii) (f) (i) -, (ii) Minimum -

68 ANSWERS 8 (iii) (i) (i) 0., -. (ii) Minimum (iii) (j) (i).8, -0.8 (ii) Maimum (g) (i)., -. (ii) Maimum (iii) (iii) (a) None (h) (i)., -. (ii) Maimum (c) (iii)

69 8 Maths In Focus Mathematics Etension Preliminar Course. (a) None (c) Graph is alwas above the -ais so 0 for all ` for all 0.. (a) 8 - None (c) 8. (a) , (c) # # Graph is alwas above the -ais so 0 for all ` for all Graph is alwas below the -ais so 0 for all ` for all

70 ANSWERS 8. Graph is alwas below the -ais so 0 for all ` for all Eercises 0.. -,. - # n # 0. a # 0, a $. -,. 0 # #. 0 t. -, 8. p # -, p $ - 9. m, m 0. # -, $. h. - # #. - # k #. q, q. All real. n # -, n $ # t # 9. -, 0. # -, $ #. - # 0.. $ -, -. # 8. -, - 9. #, 0. - # - Eercises (a) 0 - (c) - (d) 9 (e) 9 (f) - (g) 0 (h) (i) (j) 0. (a) unequal real irrational roots -9 no real roots (c) unequal real rational roots (d) 0 equal real rational roots (e) unequal real irrational roots (f) - no real roots (g) 9 unequal real rational roots (h) - no real roots (i) unequal real rational roots (j) 8 unequal real irrational roots. p. k!. b # -. p. k a 0 b - ac ]-g - ] g] g -8 0 So for all 9. k # -, k $ 0. 0 k. m -, m. k # -, k $.. p # -, p $ p -. 0 # b #. Solving simultaneousl: + () + () Substitute () in (): b - ac ]-g - ] g] -g 0 So there are points of intersection () + + () From (): - + () Substitute () in (): b - ac 8 - ] g] -g 8 0 So there are points of intersection () () Substitute () in (): b - ac - ] g] g - 0 So there are no points of intersection 9. - () + - () Substitute () in (): b - ac ]-g - ] g] g 0 So there is point of intersection ` the line is a tangent to the parabola 0. p. (c) and (d)

71 8 Maths In Focus Mathematics Etension Preliminar Course Eercises 0.. (a) a, b, c - a, b -, c (c) a, b, c - (d) a, b, c 8 (e) a, b -, c - (f) a, b, c (g) a, b -, c - 9 (h) a, b - 8, c (i) a -, b 0, c - (j) a -, b 0, c -. m, p -, q. - + ] - g- ] + g + +. RHS a] - g] + g+ b] - g+ c ] - g] + g+ ] - g RHS ` true. A, B, C -. a, b, c -. K, L, M. 8. ] + g+ ] - g a 0, b -, c - 0. (a) (c) - + (d) (e) Eercises 0.. (a) a + b -, ab a + b., ab - (c) a + b 0., ab - 8. (d) a + b -, ab (e) a + b, ab. (a) - (c) - 0. (d). (a) (c) (d) (e) m 0.. k -. b. k 8. p 9. k - 0. m!. k -. n -,. p, r -. b -, c 8. a 0, b -. ab ` b a. (a) k - k -, 0 (c) k -.8 (d) k (e) k # -, k $ 0 8. (a) p! p # -, p $ (c) p! 9. (a) k k - (c) k (a) m m, m (c) m - Eercises 0.. (a) -, -, (c) -, (d) n -, (e) a -, (f) p, (g), - (h) k, (i) t, - (j) b -, -. (a) -,, (c), (d), (e),. (a)!!,! (c)! (d).,-., 09.,- 9. (e) a -, -!. (a) 0, p (c) (d) (e),.!,!. -.! 9.,! 0.,! 9.,! (a) 0c, 90c, 80c, 0c 90c, 80c, 0c (c) 90c, 0c, 0c (d) 0c, 90c, 0c, 00c (e) 0c, 80c, 0c, 0c 9. (a) 0c, c, 80c, c, 0c 0c, 80c, 0c (c) 0c, 0c, 0c, 80c, 0c (d) c, 0 c, c, 0 c, c, 0 c, c, 00c (e) 0 c, 0 c, 0 c, 0 c, 0 c, 0 c, 00 c, 0c ( + ) #( + ) + #( + ) #( + ) ] + g ] + g + ] + g ] + g - ] + g+ 0 u Let u + - u + 0 b - ac ]-g - ] g] g 0 So u has real irrational roots. ` + and so has real irrational roots Test ourself 0. (a) 0 # # n -, n (c) - # #. a, b - 9, c. (a) -. a 0 D b - ac ]-g - # # - 0 ` positive definite

72 ANSWERS 8. (a) (c) (d) 8 (e) 0.,. (a) iv ii (c) iii (d) ii (e) i 8. a - 0 D b - ac - #(-)#(-) - 0 ` for all 9. (a) ] - g + ] + g -. 0c, 0c, 0c. (a) k k (c) k (d) k (e) k 9. -,. m -. 0,. (a) i i (c) iii (d) i (e) ii. (a) iii i (c) i (d) ii 8. For reciprocal roots b a c ab a k a a k LHS RHS roots are reciprocals for all. 9. (a) ,. (a) -, - n # -, n (c) (d) # -0, - (e) # Challenge eercise 0. D ] k - g $ 0 and a perfect square real, rational roots a, b -, c.!.. n -.. p Show D 0 9.! 0. A, B - 9, C or A -, B, C k #, k $!. 0c, 90c, 0c.,. 0c, 90c, 0c, 00c. - Chapter : Locus and the parabola Eercises.. A circle. A straight line parallel to the ladder.. An arc. A (parabolic) arc. A spiral. The straight line - or. A circle, centre the origin, radius (equation + i 8. lines! 9. lines! 0. line. Circle + (centre origin, radius ). Circle, centre ^, -h, radius. -. Circle, centre (, ), radius. -..! 8 8.! 9. Circle, centre ^-, h, radius 0. Circle, centre ^-, h, radius Eercises ! , , ! , ,

73 88 Maths In Focus Mathematics Etension Preliminar Course Problem , Eercises.. (a) Radius 0, centre (0, 0) Radius, centre (0, 0) (c) Radius, centre (, ) (d) Radius, centre (, ) (e) Radius 9, centre (0, ). (a) (c) (d) (e) (f) (g) (h) (i) (j) (a) Radius, centre (, ) Radius, centre (, ) (c) Radius, centre (0, ) (d) Radius, centre (, ) (e) Radius, centre (, ) (f) Radius, centre (, 0) (g) Radius, centre (, ) (h) Radius 8, centre ( 0, ) (i) Radius, centre (, ) (j) Radius 0, centre (, ). Centre ^, -h, radius. Centre ^, h, radius. Centre ^-, -h, radius. Centre (, ), radius 8 8. Centre d-, n, radius Show perpendicular distance from the line to ^, -h is units, or solve simultaneous equations.. (a) Both circles have centre ^, -h unit units. units. (a) units units and units (c) XY is the sum of the radii. The circles touch each other at a single point, ^0, h.. Perpendicular distance from centre ^00, h to the line is equal to the radius units; perpendicular distance from centre ^-, h to the line is equal to the radius units.. (a) ^, h, ^-, -h (c) Z ^-8, h (d) m # m - # z - ` + ZXY 90c 8. (a) units Eercises.. (a) 0 (c) (d) (e) 0 (f) (g) (h) (i) 8 (j) 8. (a) - - (c) - (d) - 8 (e) - (f) - (g) - (h) - 8 (i) - 0 (j) -. (a) (i) (0, ) (ii) - (i) (0, ) (ii) - (c) (i) (0, ) (ii) - (d) (i) (0, 9) (ii) - 9 (e) (i) (0, 0) (ii) - 0 (f) (i) (0, ) (ii) - (g) (i) (0, ) (ii) - (h) (i) c(0, m (ii) - (i) (i) c0, m (ii) - (j) (i) c0, m (ii) -. (a) (i) (0, ) (ii) (i) (0, ) (ii) (c) (i) (0, ) (ii) (d) (i) (0, ) (ii) (e) (i) (0, ) (ii) (f) (i) (0, ) (ii) (g) (i) (0, 8) (ii) 8 (h) (i) (0, 0) (ii) 0 (i) (i) c0, - m (ii) (j) (i) c0, - m (ii). (a) 8 (c) - (d) 8 (e)! (f)! (g) (h). (a) Focus ^0, h, directri -, focal length Focus ^0, h, directri -, focal length (c) Focus ^0, -h, directri, focal length (d) Focus d0, n, directri -, focal length (e) Focus d0, - n, directri, focal length (f) Focus d0, n, directri -, focal length 8 8 8

74 ANSWERS ^, h 9. X d-, - n 8 0. ^, -h and ^-, -h; 8 units. (a) - (c) units. (a) Substitute the point into the equation (c) d, - n. (a) ^0, h does not lie on the line (c) (d) Substitute ^0, h into the equation of the circle.. (a) Substitute Q into the equation of the parabola. _ q - i - q + aq 0 (c) Equation of latus rectum is a. Solving with a gives two endpoints A^-a, ah, B^a, ah. Length of AB a. Eercises.. (a) 8 0 (c) (d) (e) (f) (g) 8 (h) (i) (j). (a) - - (c) - 0 (d) - (e) - 8 (f) - 8 (g) - (h) - 0 (i) - (j) - 8. (a) (i) (, 0) (ii) - (i) (, 0) (ii) - (c) (i) (, 0) (ii) - (d) (i) (, 0) (ii) - (e) (i) (, 0) (ii) - (f) (i) (8, 0) (ii) - 8 (g) (i) (, 0) (ii) - (h) (i) (9, 0) (ii) - 9 (i) (i) c,0m (ii) - (j) (i) c,0m (ii) -. (a) (i) (, 0) (ii) (i) (, 0) (ii) (c) (i) (, 0) (ii) (d) (i) (, 0) (ii) (e) (i) (, 0) (ii) (f) (i) (, 0) (ii) (g) (i) (, 0) (ii) (h) (i) c-,0m (ii) (i) (i) c-,0m (ii) (j) (i) c-,0m (ii). (a) 0 (c) - (d) (e)! (f)! 8 (g) (h). (a) Focus ^0, h, directri -, focal length Focus ^0, h, directri -, focal length (c) Focus ^-, 0h, directri, focal length (d) Focus d, 0n, directri -, focal length (e) Focus d-, 0n, directri, focal length (f) Focus d, 0n, directri -, focal length. (latus rectum) 8., ^, h, ^, -h 9. ^9, -h, ^8, 8h 0. (a) d-, - n (c) 0 units (d) units (e). units Eercises.. (a) ] - g 8^ + h ] - g ^ + h (c) + ] - g ^ h (d) ] - g -^ - h (e) ] - g 8^ + h (f) ] + g -^ - h (g) ] - g -^ - h (h) ] + 9g ^ + h (i) ] + g -^ - h (j) ] - g 8^ + h. (a) ^ - h ] + g ^ - h 8] + g (c) ^ + h ] + g (d) ^ - 0h -] - 9g (e) ^ + h -] - g (f) ^ - h 8] + g (g) ^ + h -] - g (h) ^ + h ] + g (i) ^ - h -0] - g (j) ^ + h -8] - g. (a) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n) (o) (p) (q) (r) (s) (t) (a) (i) (, ) (ii) - (i) (, ) (ii) - (c) (i) (, 0) (ii) - (d) (i) (, ) (ii) - (e) (i) (, ) (ii) - (f) (i) (, ) (ii) (g) (i) (, 0) (ii) (h) (i) (, 0) (ii) (i) (i) (, ) (ii) (j) (i) (, ) (ii). (a) (i) (0, ) (ii) - (i) (, ) (ii) - (c) (i) (0, ) (ii) - (d) (i) (, ) (ii) - (e) (i) (, ) (ii) - (f) (i) (, ) (ii) (g) (i) (, ) (ii) (h) (i) (, ) (ii) (i) (i) (, ) (ii) 9 (j) (i) c- 0,-m (ii) ,

75 80 Maths In Focus Mathematics Etension Preliminar Course (a) Verte ^-, h, focus ^-, h, directri - Verte ^, h, focus ^, h, directri - (c) Verte ^, -h, focus ^, -h, directri 0 (d) Verte ^, h, focus ^, h, directri - (e) Verte ^0, -h, focus ^, -h, directri - (f) Verte ^-, 0h, focus ^-, 0h, directri - Eercises.8. (a) 0. Verte ^-, h, focus ^-, -h, directri, ais -, maimum value or (a) d0, n, 8. (a) (c) d-, - 8 n, m Eercises.. m. m -. m -. m d d , (d) , , M d, n , P ^-8, h. Q ^, 0.h , , ^8, -0. h ; show the point lies on the parabola b substituting it into the equation of the parabola (e) , R ^, 0h. (a) Substitute P into the equation of the parabola + p - p - p 0 (c) Substitute ^0, h into the equation of the normal. 0 + p - p - p 0 0 p + p pp ( + ) Since p! 0, p + 0

76 ANSWERS 8 (f). (a) (c) + + (d) - (e). (a) t, t t, t (c) - t, - t (d) 8t, t (e) - 8t, - 9t (f) 0t, t t t t (g) - t, - (h), t t t (i), (j) - t, - 8. (a) 0 (c) (d) - 8 (e) - 8 (f) a (g) - (h) (i) a. (a) Substitute _ t, -t i into the equation P ^-, -h (c) (a) Q ^-8, h ^, 0h, - 8. P ^, - h ; (a) Eercises.9. t + n (a) (i) (ii) - ] t + ng + tn 0 p + q (i) (ii) - ^p + qh + pq 0 m + n (c) (i) (ii) - ] m + ng + mn 0 p + q (d) (i) (ii) - ^p + qh + pq 0 a + b (e) (i) (ii) - ] a + bg + ab 0 p q (f) (i) - + (ii) + ^p + qh - pq 0 a b (g) (i) - + (ii) + ] a + bg - ab 0 (h) (i) p + q (ii) - ^p + qh - pq 0 s t (i) (i) - + (ii) + ] s + tg - st 0 (j) (i) p + q (ii) - ^p + qh - pq 0. (a) (i) p (ii) - (iii) - p + p 0 p (iv) + p p + p (i) q (ii) - (iii) - q + p 0 q (iv) + q q + q (c) (i) t (ii) - (iii) - t + t 0 t (iv) + t t + t (d) (i) n (ii) - (iii) - n + n 0 n (iv) + n n + 0n (e) (i) p (ii) - (iii) - p + p 0 p (iv) + p p + p (f) (i) k (ii) k (iii) + k - k 0 (iv) - k k + 8k (g) (i) q (ii) - (iii) - q - q 0 q (iv) + q -q - q (h) (i) t (ii) t (iii) + t - t 0 (iv) - t t + t (i) (i) m (ii) - (iii) - m - m 0 m (iv) + m -m - m (j) (i) a (ii) a (iv) - a 8a + a (iii) + a - 8a 0. (a) (i) ^p + q, pqh (ii) - pq ^p + qh, p + pq + q + A (i) ^p + qh, pqa (ii) 8- pq ^ p + qh, _ p + pq + q + ib (c) (i) ] a + bg, ab@ (ii) - ab ] a + bg, ^a + ab + b + ha (d) (i) ] s + tg, st@ (ii) - st ] s + t g, ^s + st + t + ha (e) (i) ] t + wg, tw@ (ii) - tw ] t + wg, ^t + tw + w + ha (f) (i) ^p + qh, -pqa (ii) 8- pq ^ p + qh, - _ p + pq + q + ib (g) (i) ] m + ng, -mn@ (ii) - mn ] m + ng, - ^m + mn + n + ha (h) (i) 0^p + qh, -0pqA (ii) 8-0 pq ^ p + qh, - 0_ p + pq + q + ib (i) (i) ] h + kg, -hk@ (ii) - hk ] h + kg, - ^h + hk + k + ha (j) (i) - ^p + qh, -pqa (ii) 8pq ^ p + qh, - _ p + pq + q + ib. (a) (i) + _ i (ii) - - _ - i (i) + _ i (ii) - - _ - i

77 8 Maths In Focus Mathematics Etension Preliminar Course 8 (c) (i) 8 + _ i (ii) - - _ - i (d) (i) + _ i (ii) - - _ - i 0 (e) (i) 0 + _ i (ii) - - _ - i (f) (i) - + _ i (ii) - _ - i (g) (i) - + _ i (ii) - _ - i (h) (i) - + _ i (ii) - _ - i (i) (i) - + _ i (ii) - _ - i (j) (i) - + _ i (ii) - _ - i. (a) 8_ + i _ + i (c) _ + i (d) _ + i (e) 0_ + i (f) - _ + i (g) - _ + i (h) - _ + i (i) - 8_ + i (j) - 8_ + i. (a) - p + ap 0 a _ ] t + rg + tr d - d 9 9t At e-9t, - o d 9t -d - n d 9 t For normal, mm - ` m - t The equation is given b - m( - ) 9t ` + - ( + 9t) t t + 9t - ( + 9t) - -8t + t + 9t + 8t t at + at Substitute focus ^0, -h into equation Equation of chord - ^p + qh + apq 0 Substitute ^0, ah into equation a - ( p + q) 0 + apq 0 a + apq 0 apq -a pq - i. ^-, -h. Equation of tangent at P : - p + ap 0 () Equation of tangent at Q : - q + aq 0 () ] g- ] g : - p + q + ap - aq 0 q ( - p) - aq ( - p) 0 ( q - p) - a( q + p)( q - p) 0 - a( q + p) 0 a( q + p) Substitute in (): - pa( q + p) + ap 0 - apq - ap + ap 0 - apq 0 apq. (a) Substitute ^0, h into equation.. (a) For proof, see no. 9 above N _ 0, ap + ai 8. (a) N c-, - m 9. (a) ^ -, h c, m 0. (a) F ^0, h (c) Q ^-, h (d) P: - - 0; Q: (e) mm # - -, ` tangents at P, Q are perpendicular (f) R ^-9, -h (g) directri: - a -, ` R lies on directri. P ^-, -. h mm pq - (since pq -for focal chord) ` tangents are perpendicular. Tangents intersect at a^p + qh, apq@ i.e. apq - a(since pq -for focal chord) Directri: -a ` tangents meet on the directri. a d d a At P _ 0, 0 i, d 0 d a

78 ANSWERS 8 The equation is given b - m( - ) 0 ` - ( - ) 0 0 a a - a ( - ) a ( since a ) a + a 0 0 a ( + ) 0 0 Eercises.0. (a) Q d, n (c) (, ) d (d) d At P ( 8, 8): d 8 - d ` m - At q c, m : d d ` m mm - # - So the tangents are perpendicular.. (a) - p + p 0 p + (c) R _ 0, -p i and F ^0, h FR p + PF. (a) - t + t 0 Y _ 0, -t i (c) F ^0, h TF FY ^t + h. (a) + q - q 0 R _ 0, q i (c) F ^0, -h FR FQ _ q + i So triangle FQR is isosceles. ` + FQR + FRQ (base angles of isosceles triangle). (a) Focus (0, ) Substitute into equation: LHS ] 0g+ ] g-9 0 RHS So it is a focal chord. (c) Directri - Point of intersection ^-8, -h So the point lies on the directri.. a^ - ah. - p + p 0; - q + q 0; - 8. ^ - h 9. a^ - ah 0. (a) - a a^ - ah. - ^ + h p q. (a) PO has gradient ; QO has gradient p q mm # - ` pq - a^ - ah (c) a^ - ah is a parabola in the form ( - h) a - k 0 ^ h where ^h, kh is the verte and a 0 is the focal length a ` verte is ^0, ah and focal length is. a - a or ad - a n. (a) T a ^p + qh, apq@ - a. (a) 9a^ - ah Test ourself Centre ^, h, radius. (a) ^, -h ^, -h. (a) ^88, h (a) ^0, -h (a) (i) ^, h (ii) ^, h units , , (a) 9. (a) - 0. (a) d90, n -. Sub ^0, h: LHS # 0 - # + 0. d, -n a 0. (a) R ^0, -h (c) F ^0, h FP FR RHS

79 8 Maths In Focus Mathematics Etension Preliminar Course. (a) - ^p + qh + apq 0 Sub ^0, ah : a - ( p + q) # 0 + apq 0 a + apq 0 apq -a pq (a) ^, 9h and ^-, h 0. - a Challenge eercise. (a) Midpoint of AB lies on line; mm -. (a) Put 0 into equation. -. d, -n. (a) ; mm - (c) X ^, 0. h (d) ; focus ^0, h lies on the line. ^0, 0h. (a) - - 0; Point lies on line ap + aq ap + aq 8. (a) N f, p a^ + ah (a) T ^, -0h P a ] t + sg, ats@ (c) or m t and m s m - m tan i + mm t - s tan c + ts t - s + ts t - s + ts + ts t - s + s t - ts t( - s) + s t - s t - s - + ts - - ts t - s s - t + ts t( + s) s - t + s Practice assessment task set. m, m cm. Centre ^-, h, radius. (a) - (c) 9. Focus ^0, -h, directri 8. - or - 9. k -. (a) ] + g + ^ - h 8; centre ^-, h ; radius 8 -. (a) units mm from the verte ; AFE + CBE ( et. + equal to opp. interior + in cclic quadrilateral) + CBE 80c -+ EDC ( opp. + s supplementar in cclic quadrilateral) ` + AFE 80c -+ EDC These are supplementar cointerior angles. ` AF CD , Verte ^-, -h, focus ^-, -. h. 0,. k. cm b $ -. i c 8. +, circle centre ^0, 0h and radius

80 ANSWERS BCD 90c( + in semicircle) + DAB 90c(similarl) + DBC + DAE ( + s in same segment) + BDC 90c - + DBC ( + sumof DBDC) `+ BDC 90c - + DAE `+ DAE 90c - + BDC a, b, c 0., a _ p - i. (a) - p + ap 0 R f, p (c) p + _ p - i + a - ap ap 9. AC BC and CD CE (given) AC BC ` CD CE + ACB + ECD (verticall opposite angles) ` since two sides are in proportion and their included angles are equal, Δ ABC is similar to Δ CDE. cm ,. a 0 D b - ac - ( - )( - 9) - 0 Since a 0 and D 0, for all. 8 ( - )( + ) + ( + ) ] 0 + g ( + ). sec cosec. Obtuse+ AOC + ADC ( + at centre double + at circumference) Refle+ AOC + ABC (similarl) Obtuse+ AOC + refle+ AOC 0c ( + of revolution) ` + ADC + + ABC 0c + ADC + + ABC 80c It can be proved similarl that + BAD + + BCD 80c b drawing BO and DO. ` opposite angles in a cclic quadrilateral are supplementar. Centre ^-, h, radius 8. Let+ DBA and+ EBC Then+ EDB and+ DEB ( alternate + s, DE < AC) + FDE 80c - ( + FDB straight + ) + GED 80c - ( + GEB straight + ) + FGB + DBA ( + s in alternate segment) + GFB + EBC ( similarl) ` + FDE 80c -+ FGB and + GED 80c -+ GFB Since opposite angles are supplementar, FGED is a cclic quadrilateral. 9. a 0 D b - ac ]-g - ( )( ) - 0 Since a 0 and D 0, for all 0. k (a) km c. a, b - 8, c -.,. (a) - ^-, h (c) i 9c. c 8. T ^0 and a perfect squareh k #. Let ABCD be a cclic quadrilateral of circle, centre O. Join AO and CO d, - n #

81 8 Maths In Focus Mathematics Etension Preliminar Course..9 cm,. cm units m. 8. 0, 0, 0, , -. c. + ACB 90c ( + in semicircle) `+ DCA 90c ( + DCB straight + ) ` AD is a diameter of the circle.,,, 8. -, or -, 9. ] a - bg ^a + ab + b h units. tan i. 8] + g ( - ) + ( - ) ( - ) ( ) Focus (, ), directri 8. p - - 9p (a) - ^p + qh + apq 0 a^ - ah (c) Concave upward parabola, verte (0, a ) 8. (c) 8. (d) (a) 8. (c) 8. (a) 8. (c) 88. (a) 89. (a), (d) 90. (c) Chapter : Polnomials Eercises.. (a) (c) (d) (e) (f) 0 (g). (a) (c) -. (a) - (c) (d) (e). (a) (c) - (d) 0. (a)! - (c) -, (d) (e) 0. (a) Pl] g ; Pl] g 0; (c) Pl] g ; (d) Pl] g ; (e) Pl] g 8; 0. (a),, (g) 8. (a) a 0 b 0 (c) c - (d) a - (e) a 9. (a) -, - (c) (d) (e) 0. (a) D b - ac ` f ] g has no zeros 9 (c) - (d) 9 (e), -. (a) 0 (c) (d) 0 (e) (f) (g). -,. 0,. Pl] g b - ac So Pl ( ) has no real roots. Ql] g - + b - ac 0 So Ql] g has equal roots Eercises ] + g] - 0g ] - g] - g ] - g^ + + h ] + g] - g ^ + h] - 8g + ] + g ] - g^ + + h ^ + h] - g + ] 0 + g ] + g^ h ^ - h^ - h+ ] - g ] + g^ - + h ] - gd + n ^ - h^ - - h+ ]- - g ] + g^ h ] + g] + g ] - g^ h ^ - h^ + + h+ ] + g ^ + h] - g + ]- + g ^ + + h] - g + ] + g ^ + - h^ - + h+ ] g ] + g^ h - Eercises.. (a) - (c) - (d) 9 (e) 0 (f) (g) (h) (i) (j) -. (a) k 8 k (c) k 99 (d) k 8 9 (e) k!. (a) 0 Yes (c) ] - g^ - - h (d) f] g ] - g] - g] + g

82 ANSWERS 8. (a) P ]- g ` + is a factor P] g ] + g ] - g. a -, b -. a - 8. (a) P ] g 0! 0 ` - is not a factor of P] g k a -, b - 9. (a) a, b f] g ] + g^ h (c) g ]- g 0 (d) f ] g ] + g] + g Eercises.. (a) - 0. (a) ] + g] - g ] + g] - g (c) ] - g] + g] - g (d) ] + g] - g] + g (e) ] - g] - g] - g (f) ] + g] - 9g] - g (g) ] - g] - g (h) ] + g] + g (i) ] - g] + g (j) ] + g] - g] + g. (a) P] g ] - g] + g] - g -,, (c) Yes. (a) Dividing f] g b ] + g] - g gives f ] g ] + g] - g^ + + h ` ] + g] - g is a factor of f] g f] g ] + g] - g] + g] + g. P] g ] + g] - g] + g -. (a) P] - g P] g 0 P] g ] - g] + g] - g. (a) P] ug ] u- g] u- g,. (a) f^ph ^p - h^p + h^p - h 0, -,. (a) Pk ] g ] k- g] k+ g 0c, 0c, 0c (c) 8. (a) f] ug ] u - g] u - g] u - 9g 0,, 9. -, -,- 0. i 0c, 90c, 0c, 0c, 0c, 0c. (a) a, b, c, d - a, b -, c 8, d - (c) a, b 0, c -, d (d) a, b, c, d - (e) a, b 0, c -, d 8 (f) a, b, c -, d - (g) a, b -, c - 9, d - (h) a -, b, c -, d - (i) a -, b, c, d - (j) a -, b - 0, c -, d - 0 (d). P ] g - -. a, b -, c -. P] g P ( ) has degree. Suppose P ( ) has zeros, a, a, a and a. Then _ - a a a a i_ i_ i_ i is a factor of P ( ). So P] g _ - a - a - a - a Q i_ i_ i_ ] i g. ` P ( ) has at least degree But P ( ) onl has degree. So it cannot have zeros. -

83 88 Maths In Focus Mathematics Etension Preliminar Course (e) 0 (d) (i) A ] g ] - g] + g (ii) (a) (i) P ] g ] - g] + g (ii) (e) (i) P ] g - ] - g] + g (ii) - - (i) f ] g - ] - g] + g (ii). (a) 0,, (c) (i) P ] g ] + g] + g (ii). (a) P ] g P ] g ] - g] - g] + g (c) - - -

84 ANSWERS 89. (a) (e) (f) (c) (g) - (d) (h) - -

85 80 Maths In Focus Mathematics Etension Preliminar Course (i) (c) P ] g ] + g] - g 0 Pl( ) Pl ( ) ] g - ( ) (a) ] + g Dividing b gives ^ h] - g so ] + g is a factor f ] g ] - g] + g (c) f ]- g ]- - g] - + g 0 fl( ) fl (- ) ] - g + ] - g + 8( -)- 0 (j). (a) P] g ] - kg Q ] g where Q ( ) has degree n - P] kg ] k- kg Qk ] g 0 Pl ( ) uv l + vu l Ql( ) ] - kg + ] - kg Q( ) Pl () k Ql() k ] k - kg + ] k - kg Q() k 0-8. (a) Eercises.. (a), double root 0,,, single roots (c) 0, double root,, single root (d) -, single root,, double root (e) -, triple root (f) 0,, single roots,, double root (g) -,, double roots (h) 0, triple root,, double root (i), triple root, -, single root (j), triple root. (a) (i) Positive (ii) Even (i) Negative (ii) Odd (c) (i) Negative (ii) Even (d) (i) Negative (ii) Odd (e) (i) Positive (ii) Odd (f) (i) Positive (ii) Even (g) (i) Positive (ii) Odd (h) (i) Negative (ii) Even (i) (i) Positive (ii) Odd (j) (i) Positive (ii) Even. P ] g ] + g Yes, unique. (a) P] g k ] - g Not unique P ] g ] - g. (a) ] - g Dividing b gives ^ h] + g so ] - g is a factor P ] g ] + g] - g

86 ANSWERS 8 (c) 0. - (d).. (e) -. 9.

87 8 Maths In Focus Mathematics Etension Preliminar Course.. Odd function with positive leading coefficient starts negative and turns around at the double root. It then becomes positive as becomes ver large so it must cross the -ais again. So there is another root at k - - k Even function with negative leading coefficient is negative at both ends. The triple root has a point of infleion so the curve must cross the -ais to turn negative again. So there is another root at k -. - k 9. Odd function with positive leading coefficient starts negative and turns around at both the double roots. It then becomes positive as becomes ver large so it must cross the -ais again. So there is another root at k - k

88 ANSWERS 8 0. Odd function with negative leading coefficient starts positive and turns around at the double root. It then becomes negative as becomes ver large so it must cross the -ais again. So there is another root at k Test ourself. p] g ] + g] - g] + g] - g. (a) 9 (c) (d) 9. P( ) ( - )( - )( + ) (a) p] g ] - g] + g] + g] + g. (a) - (c) -, 0, (d) k. Eercises.. (a) (i) (ii) 8 (i) - (ii) - (c) (i) - (ii) (d) (i) (ii) - (e) (i) - (ii) 0. (a) (i) - (ii) - (iii) - 8 (i) (ii) (iii) (c) (i) (ii) (iii) - (d) (i) - (ii) 0 (iii) - (e) (i) 0 (ii) (iii). (a) (i) - (ii) - (iii) (iv) (i) (ii) - (iii) - (iv) - (c) (i) (ii) - (iii) - (iv) - (d) (i) (ii) - (iii) - (iv) - (e) (i) (ii) 0 (iii) 0 (iv) (e) 00. (a) - (c) (e) -. (a) - - (c). (a) - (c) - (d) - (d). k - 8. a + b, ab - 9. a + b, ab - 0. (a) k 0 k (c) k ± (d) k -, (e) k 0. m - 9. a -, b (a) a - 8. p (- ) ]-g - ] - g + ( -)- -! ,! 0. a, b - 8, c 0. -intercepts -,, ; -intercept ( - )( ) +. 0 c, 90 c, 80 c, 0 c, 00c. k..,.. (a) P ] g 0 a + b + c, abc -. a ; a + b -. (a) p 8, q , 9.,! 0.!, -,. k - 8.

89 8 Maths In Focus Mathematics Etension Preliminar Course 9. Pa ( ) Aa ( )] a- ag 0 Pl( ) A( ) ] - ag + Al( ) ] - ag Pl( a) A( a) ] a - ag + Al( a) ] a - ag f ] g - ] g + ] g - 0. (a) f ] g - ] g - ] g + 0 fl] g - - fl ( ) ] g - ( ) - 0 (c) Double root at (d) f] g ] + g] - g.. (a) a, b -, c, d. (a) P] g ] + g Q ] g Challenge eercise. P ] g ] - g] + g ^ + + h. (a) Pb ( ) ] b- bg Qb ( ) 0 Pl( ) ] - bg Ql( ) + Q( ) ] - bg Pl( b) ] b - bg Ql( b) + Q( b) ] b - bg 0 a -, b -. i 0c, c, 0c, 0c, 80c, c, 0c, 00c, 0c a -. (a) ^, 8h. (a) a -. (a) -. i 90 c, 0 c, 0c 8. a If - a is a factor of P] g Then P ( ) ( - aq ) ( ) ` Pa ( ) ( a- aqa ) ( ) 0 0. ^-, -h, ^-, h. P ] g - ] + g ] -g. a a

90 ANSWERS 8 Chapter : Permutations and combinations. (a) Eercises # 0. (a) %. (a). 0. (a) 0 0 (c) (a) 9. (a) (c) 99 (c) (d). (a). (a) (a) 8 (c) 9 9. (a) (c) (a) (c) (d) (e). (a) 8. (a) (a) (c). (a) (c) (d) (a) (d) 9 8! 8 # # #...# # 0. (a)! # # # 8 # # #! # 0 # 9 #...# #! # # # # # # 0 # 9 # 8 # n! nn ] - g] n- g... ] r+ gr] r- g..... (c) r! rr ] - g] r- g..... n #] n - g# ] n - g#...#] r + g nn ( - )( n- ) #...#( r+ ) n! nn ] - g] n- g... ] n- r+ g] n- rg..... (d) ] n - rg! ] n - rg] n - r - g] n - r - g..... nn ] - g] n- g... ] n- r+ g Eercises.. 9%. 0. Eercises # # (a) 8 8. Yes (a) Yes (a) Eercises.. 0. (a) (c) (d) 80 ( e) 0 (f) 0 (g) (h) (i) (j) !. (a) 0 ] - g! 0! (d) ] 0 - g!! (f) 0 ] - g!! (h) 800 ] - 8g!! (j) 0 ] - g!! 0 ] - g! 9! (e) 0 80 ] 9 - g! 8! (g) 0 0 ] 8 - g! 9! (i) 9 ] 9 - g! 8! (c) ] 8 - g!. (a) 0 00 (c) (d) (a) 8 (c). (a) 0 (c) (d) 8. (a). (a) 0 (c) 8. (a) 0 8 (c) 9 (d) (e) 0 8. (a) (a) (c) (a) (c) (d) 00 (e) 0 (f) 00 (g) 0 (h) 0 (i) (j) (a) 00 (c) 0 0 (d) (e) (a) 0 (c) 00 (d) 880 (e) 8 800

91 8 Maths In Focus Mathematics Etension Preliminar Course. (a) (c) 0 0 (d) 8 00 (e) (a) 0 0. (a) (a) (c) 8 0. (a) (c) 8. (a) 0 0 (c) 80 (d) 9. (a) (c) (a) 0!!8!!! (c) 0. (a) 0 8 (c) (d)... (a) (c) 00. (a) 0 0. (a) 0 0 (c) 9 8. (a)! ] -! g (c)! ] - g! (d)! ] - g! (e) ] - g] - g! 8! 8 P ] 8 - g! 9. (a)!! 8! '!! 8! #!! 8!!! 8! 8 P ] 8 - g!!! 8! '!! 8! #!! 8!!! 8 8 P P `!! n! n Pr ] n - rg! r! r! n! ' r! ] n - rg! n! # ] n - rg! r! n! ] n - rg!! r n! n Pn- r ( n - n - r? )! ] n - rg! ] n - rg! 0. ` n n! ' ( n - r)! ] n - n + rg! n! # r! ] n - rg! n! ] n - rg!! r n P P r n- r r! ] n - rg! n n + ] n + g! P r ] n + - rg! n n! n! P + r P + r r r - ] n - rg! ^ n - r -? h! n! n! + r ] n - rg! ^ n - r -? h! n! rn! + ] n - rg! ] n - r + g! ] n + - rgn! rn! + ] n + - rg] n - rg! ] n + - rg! ] n + - rgn! rn! + ] n + - rg! ] n + - rg! n $ n! + n! - rn! + rn! ] n + - rg! nn! + n! ] n + - rg! ] n + gn! ] n + - rg! ] n + g! ] n + - rg! n + ` P P r P n n + r r r - Eercises. 9!. (a) ] 9 - g!! 8! (c) ] 8 - g!!! (e) ] - g!!! 9 ] - g!! 0! (d) 0 ] 0 - g!!. (a) (i) (ii) (iii) (iv) (v) (i) n C (ii) n C 0 n. (a) 8 8 (c) (d) 00 (e) 8 0. (a) Number of arrangements R R R R R B B B B B R R R B R B B B R B R B R B R B R B R B 0

92 ANSWERS (a) (a) (a) 9 (c). (a) 00 9 (c) 9 (d) 00 (e) (f) (g). $0. (a) 0 9 (c) (d) (a) 00 (i) 0 (ii) 88 (iii) (iv) 9. (a).8 # (c) 8 (d) (e) 0. (a) 9 00 (c) (d) 9 0 (e). (a) 0 (c) 0 0 (d) 0 (e) 00. (a) (c) 80 (d) 0 90 (e) 0. (a) 0 (c). (a) 9 9 (c) `! C ] - g!!!!!! C ] - g!!!!! C C 9. C C + C ` C C + C!. b l ] - g!!!!!! b l ] - g!!!!! ` b l b l 0 0!. b l ] 0 - g!! 0!!! 9 9 9! 9! b l + b l + ] 9 - g!! ] 9 - g!! 9! 9! +!!!! # 9! # 9! + #!! #!! # 9! # 9! +!!!! # 9! + # 9!!! 0 # 9!!! 0!!! ` b l b l + b l n n! 8. b l r ] n - rg!! r n n! b l n - r ( n - n - r? )!] n - rg! n! ] n - n + rg! ] n - rg! n! r! ] n - rg! n n ` b l b l r n - r n n! 9. P r ] n - rg! n n! r! C r! # r ] n - rg!! r n! ] n - rg! n n ` P r! C r r n n! 0. b l k ] n - kg! k! n - n n! n! k ] - g ] - g b l b l k + ] n - - ] k - gg! ] k - g! ] n - - kg! k! ] n - g! ] n - g! + ] n - kg! ] k - g! ] n - - kg! k! kn ] - g! ] n - kg] n - g! + kn ] - kg! ] k- g! ] n - kg] n - - kg! k! kn ] - g! ] n - kg] n - g! + ] n - kg! k! ] n - kg! k! ] k + n - kg] n - g! ] n - kg! k! nn ] - g! ] n - kg! k! n! ] n - kg! k! n b l k

93 88 Maths In Focus Mathematics Etension Preliminar Course Test ourself. (a) (a). (a). (a) 0 0 (c). (a) (c). 9% # (a) (c) 80 0 (d) 8. (a) (c) 8 0 (d) (e) 8 0 n n!. b l k ] n - kg! k!..08 # 0. (a) (a) (c) (a) (a) n n! c m 0 ] n - 0g! 0! n! n! 0! n n! c m n ] n - ng! n! n! 0! n! n n ` c m c m 0 n Challenge eercise. (a) 0 (c) 0. (a) n n!. b l k ] n - kg! k! n - n n! n! k ] - g ] - g b l b l k + ( n - -k -? )!] k - g! ] n - - kg! k! ] n - g! ] n - g! + ] n - kg! ] k - g! ] n - k - g! k! kn ] - g! ] n - kg] n - g! + kn ] - kg! ] k- g! ] n - kg] n - k - g! k! kn ] - g! ] n - kg] n - g! + ] n - kg! k! ] n - kg! k! kn ] - g! + ] n- kg] n- g! ] n - kg! k! ] n - g! k + n - k? ] n - kg! k! ] n - g! n ] n - kg! k! n! ] n - kg! k! n n n ` b l k b - l k - + b - l k. (a) ] n -! g. (a) 9. n n! P r ] n - rg! n n! r! C r! r ] n - rg!! r n! ] n - rg! n n ` P r! C r r ] n - k + g! k! 8. (a) (c) 0. (a) 90 0 (d) 0 9. (a) (c) 00 (d) 9 0 (e) 0 (f) (a) Practice assessment task set.. P] g ] - g] + g] + g.. (a) (c) ,. ^9, 0h. (a) - (c) (d) ; circle centre ^0, 0h radius 9. 00c; cl 0. Distance from centre ^0, 0h to line is a + b + c d a + b 0 0 radius ` line is tangent. k -. c ( + s in alternate segment) ( 80c- c) ' c ( + sum in isosceles D). 0. -,. P ] g ] - g Q ] g Pl( ) ] - g Ql( ) + ( - ) Q( ) P( ) ] - g Q( ) 0 Pl( ) ] - g Ql( ) + ( - ) Q( ) 0. (a) (c) ab -, a + b m 0. c8l

94 ANSWERS 89. (a) P ] g ] - g] - g. Domain: all real ; range: $ -. + ACB + ECD ^ verticall opposite anglesh + ABC + CED ( alternate angles AB ED) AC CD ^ givenh ` b AAS, DABC / DCDE. m ] - g. c 8l. $., # d-, n 0. (a) (c) Q ^-0, 0h (d) c l. Domain: all real!! ; range: all real. (a) sin ^a - bh cos c (c) ^ + h. (a) 9. m c 8l. X ^.,.h 8. P ]- g - P ] g ] + g^ - + h - on division ` P ]-g istheremainder d. (a) abc + acd + bcd + abd - 0 (c) - a. P ] g ] - g Q ] g P] g ] - g Q] g 0Q ] g 0 Pl] g ] - gq] g+ ] - g Ql] g Pl( ) ( - ) Q( ) + ] - g Ql( ) 0 ( ) Q( ) + 0Ql( ) Radius ; a, b -, c 9 9. (a) 8. m c l 0. (a)! P ( ) ^ - h ( + ) 0 Pl( ) ^ - h ( + ) + ^ - h $ Pl( ) ( ) ^ - h ( + ) + ^ - h $ ( ) 0..,.. 0c 8. (a) R d, n 8 t + t - t + t + 9. ^ + t h 0. f ] g - -- f ] g ] g -] g -] g So - is a factor of f] g a, b -, c , cm ; 0 cm. -, --.. (a) 0c, 0c, 0c, 00c 0c, 90c, 0c (c) 0c