Mathematical Methods CAS 3/4 Answers

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(1)P1: FXS/ABE. P2: FXS. 0521665175ans-1.xml. CUAT018-EVANS. November 3, 2005. 16:7. Answers Chapter 1. c –2 –1 0 1 2 3 4 5 6. Exercise 1A. d. 1 a {7, 9} b {7, 9} c {2, 3, 5, 7, 9, 11, 15, 19, 23} d {2, 3, 5, 11) e {2} f {2, 7, 9} g {2, 3, 5, 7} h {7} i {7, 9, 15, 19, 23} j (3, ∞) 2 a {a, e} b {a, b, c, d, e, i, o, u} c {b, c, d} d {i, o, u} 3 a {6} b {2, 4, 8, 10} c {1, 3, 5, 7, 9} d {1, 2, 3, 4, 5, 7, 8, 9, 10} e {1, 2, 3, 4, 5, 7, 8, 9, 10} f {5, 7} g {5, 7} h {6} √ 4a [−3, 1) b(−4, 5] c (− 2, 0) 1 √ e (−∞, −3) d −√ , 3 2 f (0, ∞) g (−∞, 0) h [−2, ∞) 5 a (−2, 3) b [−4, 1) c [−1, 5] d (−3, 2] 6a –3 –2 –1 0 1. –8. –6. –4. –2. Exercise 1B 1 a Domain = R b Domain = (−∞, 2] c Domain = (−2, 3) d Domain = (−3, 1) e Domain = [−4, 0] f Domain = R. Range = [−2, ∞) Range = R Range = [0, 9) Range = (−6, 2) Range = [0, 4] Range = (−∞, 2). 3 1 0. x. Domain = R Range = [1, ∞). 0 1. 2 3. y. c. 0. –3. 3. x. –3. Domain = [−3, 3] Range = [−3, 3]. b 2 3. y. b. y. 2a. 2. –4 –3 –2 –1 0 1. 0. y. d. c (4, 4). 2. d. –4 –3 –2 –1 0. e. f 0. 1 2 3. y. x. 0. 8. Domain = [0, ∞) Range = [0, ∞). Domain = R + ∪ {0} Range = (−∞, 2]. e. –2 –1. (1, 2). x. 0. –4 –3 –2 –1 0 1. f. y (4, 18). 4 5. 5. 7a –3 –2 –1 0 1 2 3 4 5 6. 0. b. x 5. Domain = [0, 5] Range = [0, 5]. –3 –2 –1 0 1 2 3 4 5 6. 682. 2 0. x 4. Domain = [0, 4] Range = [2, 18].

(2) P1: FXS/ABE. P2: FXS. 0521665175ans-1.xml. CUAT018-EVANS. November 3, 2005. 16:7. 683. Answers. 4 0 –1 (–1, –5). –. x –2 2. –2. 2 0. –5. (6, 7) (2, 3). x. Range = [3, ∞). 0. 0. x (2, –1). Range = (−∞, −1]. –2 3. x. (–4, –7). y. (3, 11) 2 0. y. f. (6, 17). (3, 4) –1 1 0. x. Range = (−∞, 11). Range = [−7, ∞) e. y. d. 1. 1 2. x. Domain = R Domain = [−1, 2] Range = (−∞, 4] Range = [−5, 4] 3 a Not a function Domain = {−1, 1, 2, 3} Range = {1, 2, 3, 4} b A function Domain = {−2, −1, 0, 1, 2} Range = {−4, −1, 0, 3, 5} c Not a function Domain = {−2, −1, 2, 4} Range = {−2, 1, 2, 4, 6} d A function Domain = {−1, 0, 1, 2, 3} Range = {4} f Not a function e A function Domain = {2} Domain = R Range = Z Range = {4} g A function h Not a function Domain = R Domain = R Range = R Range = R i Not a function Domain = [−4, 4] Range = [−4, 4] 4 a g(−2) = 10, g(4) = 46 b [−2, ∞) 5 a f (−1) = −2, f (2) = 16, f (−3) = 6 b g(−1) = −10, g(2) = 14, g(3) = 54 c i f (−2x) = 8x 2 − 8x ii f (x − 2) = 2x 2 − 4x iii g(−2x) = −16x 3 − 4x − 6 iv g(x + 2) = 2x 3 + 12x 2 + 26x + 14 v g(x 2 ) = 2x 6 + 2x 2 − 6 3 c− 6a3 b7 2 2 7 a x = −3 b x > −3 cx= 3 8 a f : R → R, f (x) = 2x + 3 4 b f : R → R, f (x) = − x + 4 3 c f : [0, ∞) → R, f (x) = 2x − 3 d f : R → R, f (x) = x 2 − 9 e f : [0, 2] → R, f (x) = 5x − 3 y y b 9a. 0. y. c. y. h. (2, 4). 1 3. x. x. –1 0 (–2, –7). Range = [−7, 17]. Range = (−∞, 4] y. g. h. (–5, 14). y (4, 19). (–1, 2) 0. x. Range = [2, 14]. 0 –1 0.2 (–2, –11). x. Range = (−11, 19). 10 a f (2) = −3, f (−3) = 37, f (−2) = 21 b g(−2) = 7, g(1) = 1, g(−3) = 9 c i f (a) = 2a 2 − 6a + 1 ii f (a + 2) = 2a 2 + 2a − 3 iii g(−a) = 3 + 2a iv g(2a) = 3 − 2(2a) = 3 − 4a v f (5 − a) = 21 − 14a + 2a 2 vi f (2a) = 8a 2 − 12a + 1 vii g(a) + f (a) = 2a 2 − 8a + 4 viii g(a) − f (a)= 2 + 4a −2a 2 2 2 2 , −1 b − , 11 a 3 3 3 1 2 c 0, − d (−∞, −1) ∪ ,∞ 3. 3 2 2 e −∞, − ∪ ,∞ 3 3.

(3) 1 f − ,0 3 12 a f (−2) = 2 b f (2) = 6 c f (−a) = a 2 − a d f (a) + f (−a) = 2a 2 e f (a) − f (−a) = 2a f a 4 + a2 a+2 13 a {2} b {x : x > 2} c 3 8 13 d − e {1} f 3 18 4 7 bk =6 ck =− 14 a k = 3 3 2 1 fk =− dk =9 ek = 9 3 6 1 1 b d1 e −1, 2 c± 15 a 5 5 3. Answers. y. g.

(4) P1: FXS/ABE. P2: FXS. 0521665175ans-1.xml. Answers. 684. CUAT018-EVANS. November 3, 2005. 16:7. Essential Mathematical Methods 3 & 4 CAS y. 13 a. Exercise 1C. (–3, 8). 1 One-to-one functions are b, d and f. 2 a Functions are i, iii, iv, vi, vii and viii. b One-to-one functions are iii and vii. √ 3y= √ x + 2, x ≥ −2; range = R + ∪ {0} y = − x + 2, x ≥ −2; range = R − ∪ {0} y 4a. 2 x. 0. b g1 (x) = x 2 + 2, x ≥ 0 g2 (x) = x 2 + 2, x < 0 5 a Domain = R Range = R b Domain = [0, ∞) Range = [0, ∞) c Domain = R Range = [−2, ∞) d Domain = [−4, 4] Range = [0, 4] e Domain = R\{0} Range = R\{0} f Domain = R Range = (−∞, 4] g Domain = [3, ∞) Range = [0, ∞) 6 a Domain = R Range = R b Domain = R Range = [−2, ∞) c Domain = [−3, 3] Range = [0, 3] d Domain = R\{1} Range = √ √ R\{0} 7 a R\(3) b (−∞, − 3] ∪ [ 3, ∞) cR d [4, 11] e R\{−1} f (−∞, −1] ∪ [2, ∞) g R\{−1,

(5) 2} 1 i 0, h (−∞, −2) ∪ [1, ∞) 3 j [−5, 5] k [3, 12] 8 a Even b Odd c Neither d Even e Odd f Neither y 9a. –1. x. 0. 1. 2. 8 5 x. 0. –3. b Range = [5, ∞) 14 a f (−4) = −8 b f (0) = 0 1 c f (4) = 4 ⎧ ⎨ 1 , a>0 d f (a + 3) = a + 3 ⎩ 2(a + 3), a ≤ 0 ⎧ 3 1 ⎪ ⎨ , a> 2 e f (2a) = 2a ⎪ ⎩4a, a ≤ 3 2 ⎧ ⎨ 1 , a>6 f f (a − 3) = a − 3 ⎩ 2(a − 3), a ≤ 6 √ 15 a f (0) = 4√ b f (3) = 2 c f (8) = 7 √ a, a ≥ 0 d f (a + 1) = 4, a<0 √ a − 2, a ≥ 2 e f (a − 1) = 4, a<2 y 16. –2. x. 0. –1. Range = [−1, ∞). 1 –1 –2. ⎧ −x − 4, ⎪ ⎪ ⎪ ⎨1 17 y = 2 x − 1, ⎪ ⎪ 1 ⎪ ⎩ − x + 2, 2. x < −2 −2 ≤ x ≤ 3 x >3. –1. Exercise 1D. –2. b Range = [−2, ∞) 10 Domain = (−3, 0] ∪ [1, 3) Range = [−2, 3) 11 Domain = [−5, 4] Range = [−4, 0) ∪ [2, 5] y 12 a (2, 10). (–4, 9) 6. –4. –3. 0. 3. –5. 0. 5. b. c2 f4 (−3, 3) (−∞, −5] ∪ [5, ∞) [1, 3]. 1 2 3 x. –4. b8 e −2. c. 5. 0. 1a8 d −2 2a. 2. b Domain = (−∞, 2] Range = [5, 10] ∪ {−4}. d. (−1, 5) –1. 5. (−∞, −8] ∪ [2, ∞). e –8 –3. f –3 –2 –1. 2. [−3, −1].

(6) P1: FXS/ABE. P2: FXS. 0521665175ans-1.xml. CUAT018-EVANS. November 3, 2005. 16:7. 685. Answers b. 5. 0 –5 –1 –1. Range = [1, ∞). (–4, –1) –5 –3 0. x. Range = (−∞, 2]. y. d. y (1, 2). 3. x. Range = [−1, ∞). 1 –1 0 3. 1 (3x + 2)2 Domain = R + 3 b g(h(x)) = 2 + 2 x Domain = R\{0} 1 d5 c 25 5 a ran f = [−4, ∞), ran g = [0, ∞) b f ◦ g(x) = x − 4, ran = [−4, ∞) c ran f ⊂ dom g 1 6 a f ◦ g(x) = x, domain = R\ , 2 1 range = R\ 2 b g ◦ f (x) = x, domain = R\{0], range = R\{0} 7 a ran f = [−2, ∞) ⊂ dom g = R + ∪ {0} b f ◦ g(x) = x − 2, x ≥ 0. 4 a h(g(x)) =. (–3, 2) x. 0 (4, 1). c. y. x. Range = (−∞, 2]. Exercise 1E 1 a ( f + g)(x) = 4x + 2 ( f g)(x) = 3x 2 + 6x dom = R b ( f + g)(x) = 1 ( f g)(x) = x 2 − x 4 dom = (0, 2] x +1 c ( f + g)(x) = √ x ( f g)(x) = 1 dom = [1, ∞) √ d ( f + g)(x) =√x 2 + 4 − x 2 ( f g)(x) = x 4 − x dom = [0, 4] 2 a i Even ii Odd iii Even iv Odd 1 b ( f + h)(x) = x 2 + 1 + 2 even; x (gk)(x) = 1 even; 1 ( f h)(x) = 1 + 2 even; x ( f + g)(x) = x 2 + x + 1 neither; 1 (g + k)(x) = x + odd; x ( f g)(x) = x 3 + x odd. y. 0 –1. x 1. 2. –2. 8 a f ◦ g(x) = 4 − |x| and g ◦ f (x) = |4 − x|. f (g(x)) = 4x − 1, g( f (x)) = 4x − 2 f (g(x)) = 8x + 5, g( f (x)) = 8x + 3 f (g(x)) = 4x − 7, g( f (x)) = 4x − 5 f (g(x)) = 2x 2 − 1, g( f (x)) = (2x − 1)2 f (g(x)) = 2(x − 5)2 + 1, g( f (x)) = 2x 2 − 4 f f (g(x)) = 2|x| + 1, g( f (x)) = |2x + 1| 2 a f ◦ h(x) = 6x + 3 b h( f (x)) = 6x − 1 c f ◦ h(2) = 15 d 11 e 21 f −7 g3 3 a 9x 2 + 12x + 3 b 3x 2 + 6x + 1 c 120 d 46 e3 fl. y. y 4. –4. g o f(x) = |4 – x| f o g(x) = 4 – |x|. x. 0. 4. 4 x. 0. 4. b f ◦ g(x) = 9 − |x|2 = 9 − x 2 and g ◦ f (x) = |9 − x 2 | y. y f o g(x) = 9 –. –3. g o f(x) = |9 – x2|. x2. 9. 9 0. x. 3. c f ◦ g(x) =. Exercise 1F 1a b c d e. Answers. y. 3a. –3. x. 0. 3. 1 1 1 and g ◦ f (x) = = |x| x |x|. y. y f o g(x) = 1. g o f(x) = 1. |x|. 0. |x|. x. 0. 9 a f ◦ g is not defined since ran g = [−1, ∞) ⊂ dom f = (−∞, 3] b g ∗ : [−2, 2] → R, g ∗ (x) = x 2 − 1 f ◦ g ∗ : {x: − 2 ≤ x ≤ 2} → R, f ◦ g ∗ (x) = 4 − x 2 10 a f ◦ g is not defined since ran g ⊂ dom f b g1 : {x: x < 3} → R, where g1 (x) = 3 − x. x.

(7) P1: FXS/ABE. P2: FXS. 0521665175ans-1.xml. Answers. 686. CUAT018-EVANS. November 3, 2005. 16:7. Essential Mathematical Methods 3 & 4 CAS 11. Domain. Range. f. R. [0, ∞). g. (−∞, 3]. [0, ∞). a ran g ⊂ domain f, ∴ f ◦ g exists b ran f ⊂ domain g, ∴ g ◦ f does not exist 12 a S = [−2, 2] b ran f = [0, 2], ran g = [1, ∞) c ran f = [0, 2] ⊂ dom g = R, ∴ g ◦ f is defined ran g = [1, ∞) ⊂ dom f = [−2, 2], ∴ f ◦ g is not defined 13 a ∈ [2, 3]. √ f g −1 (x) = −1 + x, dom g −1 = (0, 16), −1 ran g = (−1, 3) g g −1 (x) = x 2 + 1, dom g −1 = R + ∪ {0}, ran g −1 = [1, ∞) √ h h −1 (x) = 4 − x 2 , dom h −1 = [0, 2], −1 ran h = [0, 2] y a 6 y = 2x + 4 (0, 4) y = (–2, 0) 0. Exercise 1G b. y 3 3 2 0. 3 2. x. 3. 3–x. y= 2. y = 3 – 2x. f. −1. (x) = 3 − 2x, dom = R, ran = R. c. y. y = (x – 2)2. y = 2 + √x (0, 2) 0. x. (2, 0). √ f −1 (x) = x + 2, dom = R + ∪ {0}, ran = [2, ∞) d. y y = √x + 1. 1 0. –1. x. (4, 0) (0, –2). x −4 , dom = R, ran = R 2. y=. x b f −1 (x) = 1 a f −1 (x) = x + 4 2 4x 4x + 2 −1 −1 c f (x) = d f (x) = 3 3 1 −1 2 a f (x) = (x + 4), domain = [−8, 8], 2 range = [−2, 6] 1 b g −1 (x) = 9 − , domain = (−∞, 0), x range = (9, √∞) c h −1 (x) = x − 2, domain = [2, ∞), range = R + ∪ {0} 1 d f −1 (x) = (x + 2), domain = [−17, 28], 5 range = [−3, √ 6] e g −1 (x) = x + 1, domain = (0, ∞), range = (1, ∞) f h −1 (x) = x 2 , domain = (0, ∞), range = (0, √ ∞) 3 a g −1 (x) = x + 1 − 1, domain = [−1, ∞), range = [−1, ∞) y b. (x – 4) 2. x 1. 2. –1. 1 , dom f −1 = [−3, 3] 2 x 5 a f −1 (x) = , dom f −1 = [−2, 6], 2 ran f −1 =[−1, 3] x +4 , dom f −1 = [−4, ∞), b f −1 (x) = 2 −1 ran f = [0, ∞) c {(6, 1), (4, 2), (8, 3), (11, 5)}, dom = {6, 4, 8, 11}, ran = {1, 2, 3, 5} d h −1 (x) = −x 2 , dom h −1 = R + , ran h −1 = √ R− −1 e f (x) = 3 x − 1, dom f −1 = R, ran f −1 = R. y = (x – 1)2. (0, 1). 3. x. 0 (1, 0). √ f −1 (x) = x + 1, dom = R + ∪ {0}, ran = [1, ∞). 4 f −1 (2) =. e. y y = (x – 2)2 (0, 4) (0, 2) (4, 0) 0. (2, 0). x. y = –√x + 2. √ f −1 (x) = 2 − x + dom = R ∪ {0}, ran = (−∞, 2].

(8) P2: FXS CUAT018-EVANS. November 3, 2005. 16:7. Answers. Exercise 1H . 1 y=x. 4 if 0 ≤ x ≤ 2 4 + 2(x − 2) if x > 2 4 if 0 ≤ x ≤ 2 = 2x if x > 2. 1 f (x) = x. 0. y. f. −1. 687. 1 (x) = , dom = R + , ran = R + x. 8. y. g. 4 y = 12 x 1 √x. y=. 2. 4. 2 V (x) = 4x(10 − x)(18 − x) Domain = (0, 10) 3 a A(x) = −x 2 + 92x − 720 b 12 < x < 60 c A. x. 0. 1 f −1 (x) = √ , dom = R + , ran = R + x h. x. 0. y. (46, 1396). y = 2x + 4 4. (60, 1200). 1 y = (x – 4) 2. 0. –2. (12, 240). x. 4 –2. −1. 7a. h (x) = 2x + 4, dom = R, ran = R y y b (1, 2) (3, 3). (0, 1) x. 0 (0, 0). (3, 4) (4, 3) (2, 1) x. 0 (1, 0) y. d. y. c. x. 0. d Maximum area = 1396 m2 occurs when x = 46 and y = 34 4 a i S = 2x 2 + 6xh 3V ii S = 2x 2 + x b Maximal domain = (0, ∞) c Maximum value of S = 1508 m2 1 1 3 ,b = , c = 45, d = − , e = 75 5aa= 30 15 2 b S 7 75, 2. 1. 3 2. 0. –4. 0. 2. e. x. 3. y. 1. x 3 45, 2. –4. f. 0. y. 3 3 –3. 0. (1, 1) x (0, 0). x. –3. (–1, –1). h. y. g. y.

(9). t. 7 2 6 a C = 1.20 for 0 < m ≤ 20 = 2.00 for 20 < m ≤ 50 = 3.00 for 50 < m ≤ 150 b C ($) c Range = 0,. 3 0. x. 0. x. 2. y = –2. 1. 8aC bB cD dA 9 a A = (−∞, 3] √ b b = 0, g −1 (x) = 1 − x, x ∈ [−3, 1]. Domain = (0, 150] Range = {1.20, 2.00, 3.00}. m(g). Answers. y. f. 60 80 10 0 12 0 14 0 15 0. 0521665175ans-1.xml. 20 40. P1: FXS/ABE.

(10) P1: FXS/ABE. P2: FXS. 0521665175ans-1.xml. Answers. 688. CUAT018-EVANS. November 3, 2005. 16:7. Essential Mathematical Methods 3 & 4 CAS. Multiple-choice questions 1E 6C. 2B 7D. 3E 8B. 4C 9B. (5, 4). 5E 10 B. 3 2. Short-answer questions (technology-free) y. 1a. f(x) = x2 + 1 (0, 1) 0. x. Domain = R. Range = [1, ∞). y. b. f(x) = 2x – 6. x. 0. y. 2a. y = g(x) x. 0. –3.

(11). 3 ,4 b Range of g =

(12) 2. 3 −1 ,4 → R cg : 2 g −1 (x) = 2x − 3 Range = [0, 5].

(13) 3 Domain = ,4 2 7 d {5} e 2 1 3a b {11} 5 y. 4. 3. –6. (2, 3). Domain = R. Range = R. y. c. 0 5. x2 + y2 = 25. 0. –5. x. 5. –5. Domain = [−5, 5]. Range = [−5, 5]. y. d. y> – 2x + 1 1 x. 0. –1 2. x 1 2 (0, –1). √ √ 5 a R\{3} b R\[− 5, 5] d [−5, 5] e [5, 15]. Range = R. y y<x–3 3 0 –3. Domain = R. c R\{1, −2} f R\{2}. 6 ( f + g)(x) = x 2 + 5x + 1 ( f g)(x) = (x − 3)(x + 2)2 7 ( f + g) : [1, 5] → R, ( f + g)(x) = x 2 + 1 ( f g) : [1, 5] → R, ( f g)(x) = 2x(x − 1)2 √ 8 f −1 : [8, ∞) → R, f −1 (x) = x + 1 9 a ( f + g)(x) = −x 2 + 2x + 3 c {−1, 3} b ( f g)(x) = −x 2 (2x + 3) y 10 y=x. 0, 4 3. Domain = R e. 1 c − 10. (–4, 0). (2, 2) 0. 4, 0 3. x. (0, –4) x. Extended-response questions Range = R. 1 a C1 = 64 + 0.25x C2 = 89.

(14) P1: FXS/ABE. P2: FXS. 0521665175ans-1.xml. CUAT018-EVANS. November 3, 2005. 16:7. 689. Answers. C2. 100 80 60 40 20. C1. 0. 20 40 60 80 100 120 x (km). c x > 100 km 2 a S = 6x 2 √ 2 3s 3a A= 4 √ 4 a d(x) = 9 − x 2 ,. 2. b S = 6V 3 √ 2 3h bA= 3 b dom = [0, 3] ran = [0, 3]. d 3 x. 3. 0. 160x x + 80 → R, 6 a V1 : (0, 12) 5 S(x) =. h2 h 4 √ b V2 : (0, 6) → R, V2 (r ) = 2r 2 36 − r 2 7a domain range f R R g R R V1 (h) = 36 −. ran f = dom g, ∴ g ◦ f exists g ◦ f (x) = 2 + (1 + x)3 b g ◦ f is one-to-one and therefore a function. ∴ (g ◦ f )−1 is defined, (g ◦ f )−1 (10) = 1 8a y. 0. –2. 2. → R, f −1 (x) =. 13 a i f (2) = 3 f ( f (2)) = 2 f ( f ( f (2))) = 3 ii f ( f (x)) = x −x − 3 b f ( f (x)) = , f ( f ( f (x))) = x, x −1 i.e. f ( f (x)) = f −1 (x). Chapter 2 Exercise 2A. (3, 3). 2. a . b − xd c xc − a 2−x b i f −1 : R\{1} → R, f −1 (x) = 3x − 3 3 ii f −1 : R\ → R, 2 3x + 2 f −1 (x) = 2x − 3 1−x iii f −1 : R\{−1} → R, f −1 (x) = x +1 1 −x iv f −1 : R\{−1} → R, f −1 (x) = x +1 c If a, b, c, d ∈ R\{0}, f = f −1 when a = −d. 11 a i YB = r ii ZB = r iii AZ = x − r iv CY = 3 − r √ x + 3 − x2 + 9 br = 2 ci r =1 ii 1.25 q 12 b f (x) = x 3x + 8 c i f −1 (x) = = f (x) √x − 3 ii x = 3 ± 17 10 a f : R\. –4. b i −3 ii 3 c S = (−∞, 0] 2 4x − 4 if x d f ◦ h(x) = 2x if x 2 2x − 8 if x h ◦ f (x) = 2x if x ⎧ 2 3t ⎪ ⎨ ,0 ≤ t ≤ 1 2 9 A(t) = ⎪ ⎩ 3t − 3 , t > 1 2 Domain = [0, ∞] Range = [0, ∞]. 1 2 7 7 17 g j f h 21 i2 5 2 9 2 a x = 12, y = 8 b x = 5, y = −8 c x = 3, y = 1 d x = 2, y = 1 e x = 17, y = −19 f x = 10, y = 6 3 80 km 4 96 km 5 Width = 6 cm, length = 10 cm 6 John scored 4, David 8 7 a w = 20n + 800 b $1400 c 41 units 8 a V = 15t + 250 b 1150 litres c 5 hours, 16 minutes and 40 seconds 9 a V = 10 000 − 10t b 9400 litres c 16 hours and 40 minutes 10 a C = 25t + 100 b i $150 ii $162.50 c i 11 hours ii 12 hours 1 a 10. x. <1 ≥1 <2 ≥2. b1. c4. d 28. e8. Answers. b C ($).

(15) P1: FXS/ABE. P2: FXS. 0521665175ans-1.xml. Answers. 690. CUAT018-EVANS. November 3, 2005. 16:7. Essential Mathematical Methods 3 & 4 CAS. Exercise 2B m−n a 5 dx= p−q g x = 3a. bc a ab = 1−b a 2 − b2 = 2ab 1 = 3a − b 5a =− 3. b b−a m+n ex= n−m. cx =−. h x = −mn. ix. bx=. 1ax =. fx. p−q 3ab lx kx= p+q b−a 2 2 2 p + p t +t mx= nx q( p + t) c − ad d − bc , y= 4ax = 1 − ab 1 − ab a 2 + ab + b2 ab bx= , y= a+b a+b t −s t +s ,y = cx= 2a 2b d x = a + b, y = a − b e x = c, y = −a f x = a + 1, y = a − 1 2a 2 5 a s = a(2a + 1) bs = 1−a a a2 + a + 1 ds = cs = (a − 1)2 a(a + 1) 3a fs = e s = 3a 3 (3a + 1) a+2 1 5a 2 g s = 2a 2 − 1 + 2 hs = 2 a a +6 jx=. Exercise 2C 1 1 a 3, 7 2 y 2a 0. 0. 0. x 3. e. 6C =. x. 6. –3. 0. x 3 y. f. y. 0 x. –4. y x − =1 3 4 d. y. x. y. 0 0. x. 0. –4. 4. x. x y − =1 2 4. 14 3. 0. 6. x. 0. –4. 2. 4 2. 3. 2. y. d. y. c. 2. c. 4 3 2 1. 1. –1 –2 –3. b −3x − 5 = y d y = 2x + 1 x y b + =1 4 6 y x d − =1 6 2 y b. y x + =1 4 2. 3 1 5 , c b − , −2 2 2 2 y b x. 3 a 2x − 6 = y c 5 = 3y − 4x y x 4a − =1 2 3 y x c− − =1 4 3 y 5a. x 6. –3. y x − =1 6 3. 11 n + 2, $57 200. 7 a C = 5n + 175 b Yes c $175 √ √ b 2 ≈ 1.414 8 a 5 ≈ 2.236 √ √ c 29 ≈ 5.385 d 2 82 ≈ 18.111 √ f5 e 20 ≈ 4.472 9 a i y = 2x + 4 ii 2y + x = 13 b i y = −2x + 7 ii 2y − x = 4 10 y = 2x − 3 11 y = −5 or 3 12 a i 5y + 2x = 13 ii 5y + 4x = 11 b i 2y − 5x = 11 ii 4y − 5x = 17 13 a 32.01◦ b 153.43◦ c 56.31◦ d 120.96◦ 14 45◦ 15 a = −12 or a = 8 16 a y = 3x − 6 b (2, 0) 17 k = 5 and h = 4 or k = −2 and h = −3 4 18 a a + 2 b 5 1 b (5, 7) 19 a m = √ 2√ c AB = 13, AC = 2 13 20 a 3y − x = 22 b (14, 12) c (16, 6) d 80 square units 21 a (2, 3) b y + 5x = 13 c i 2y = 3x − 13 ii (3, −2) iii (1, 8). Exercise 2D. 2. 1a4−T b i 90T c i T =1. ii 70(4 − T ).

(16) P1: FXS/ABE. P2: FXS. 0521665175ans-1.xml. CUAT018-EVANS. November 3, 2005. 16:7. 691. Answers. (litres). 5400. 0. 45 t (days). e 45 days. f 120 litres per day. 3 a C A = 0.4n + 1 C B = 0.6n. b Cost. CB. ($). CA. 1 0. n. cn=5 dD D = 1 – 0.2n. 1 0. 5. n. The difference in charges against kilometres travelled. 9 b 24 622 m 4ay= x 4 855 27 5393 cy=− x+ d 26 26 108 4 5 a i −4 ii 9 10 4 ii y = −4x + 30 bi y= x+ 9 3 c Equation of AC: y = x, equation of BD: y = 4 d (4, 4). Exercise 2E

(17). 1 a 0 −1 6 4. d 4 0 g.

(18) 3 0.

(19) 6 5 −3 −2.

(20). b −1 0. −2 −2. e 2k 3k. k 2k.

(21). c −10 12.

(22). f −2 −4 15 10. j2. 2 a −1 −4.

(23) 1 3. h1 1 k −3 2 ⎡2 7 b⎢ ⎣ 1 7. i 1. . −1 −.

(24) −6 7.

(25) 2 −1 −3 2. l2. 1⎤ 14 ⎥ c 1 ⎦ 3 0 14. 0 1 k. 1.

(26). 1 1 0 −1 3 A = 2 2 , B = −3 1 0 −1 ⎡ 1 1⎤

(27). 5 1 ⎢ 2⎥ , (AB)−1 = ⎣ 2 AB = −3 −5 ⎦ −3 −1 2 2 ⎡ 1 1⎤ 1 ⎢ 2 −1 2⎥ −1 −1 A−1 B−1 = 2 , B A = ⎣ −3 −5 ⎦, 3 −1 2 2 (AB)−1 = B−1 A−1 ⎡ ⎤ 1 5 −7

(28) 3 0 7 − 2 ⎥ c⎢ 4a 2 2 b 1 −8 ⎣ 112 −21 ⎦ 1 −2 2 2 ⎤ ⎡ ⎤ ⎡ −11 17 −3 11 8⎥ b ⎢ 16 16 ⎥ 5a⎢ ⎣ 18 3⎦ 7 ⎦ ⎣ −1 4 4 16 16 ⎤ ⎡ ⎤ ⎡ 16 14 42 5 7 7 b ⎣ 0 10 48⎦ 7 a ⎣−6 11 12⎦ 24 13 30 14 2 0 ⎤ ⎡ ⎤ ⎡ 9 18 15 24 24 16 d ⎣−18 30 18⎦ c ⎣66 16 6⎦ 36 3 0 0 22 16 −1. ⎡. ⎤ 3 −1 ⎡ −1 ⎥ 1 ⎢2 2 ⎢ ⎥ f ⎣−6 1 3 ⎥ e ⎢−3 ⎣1 1⎦ 10 0 − 2 2 ⎡ ⎤ ⎡ 9 −1 4 1 g ⎣−6 h ⎣−6 8 −6⎦ 18 8 −1 0 ⎡ ⎤ ⎡ −3 −3 7 7 ⎢ 8 ⎥ ⎢ 8 48 24 ⎢ ⎥ ⎢ 4 −11 −8 ⎥ ⎢ 4 i⎢ ⎢ ⎥ ⎢ ⎢ 3 18 9⎥ j ⎢ 3 ⎣ −5 4 5 ⎦ ⎣ −5 18 18 27 27 ⎤ ⎡ 17 −3 −23 8 54 22⎥ 8a⎢ ⎥ ⎢24 95 ⎣ 9 52 58 1⎦ 1 13 10 1 ⎤ ⎡ 10 10 14 8 ⎥ b⎢ ⎢16 26 24 1⎥ ⎣63 50 80 13⎦ 70 74 92 55 ⎤ ⎡ 9 18 15 −18 3⎥ c⎢ ⎥ ⎢30 18 36 ⎣ 0 12 3 18⎦ 0 0 3 3. ⎤ 5 3 9 0⎦ 0 0 ⎤ 9 11 13 24⎦ 4 0 7 48 −11 18 4 27. ⎤ 7 24 ⎥ ⎥ −8 ⎥ ⎥ 9⎥ 5⎦ 27. Answers. ii 90 km on freeway, 210 km on country roads 2 a L = −120t + 5400 b 5400 litres d [0, 45] c V.

(29) P1: FXS/ABE. P2: FXS. 0521665175ans-1.xml. Answers. 692. CUAT018-EVANS. November 3, 2005. 16:7. Essential Mathematical Methods 3 & 4 CAS  −1. −1   12 8   −2   0   9  d    7 1    36 8    37 −1 36 8   3 5 4 −7 8 5 10 1  e −1 −2 −1 5  0 −7 −7 1 . 3 6 f −2 0   1 9 a 0 0. 11 24 −1 9 7 72 1 72. 1 12 2 9 −7 36 −1 36.  4 3 −8 4 8 1  −8 −3 4 −14 −15 1  11 .  −7 .  24     2    10 a  9     13     72   43  72. 2 b −12 2  −1 .  2     −2    b 3    5      6   11  6. Exercise 2G 1ax cx 2ay bx. = 2, y = 3, z = 1 b x = −3, y = 5, z = 2 = 5, y = 0, z = 7 d x = 6, y = 5, z = 1 = 4z − 2 = 8 − 5, y = 4 − 2, z = . 3 a −y + 5z = 15, −y + 5z = 15 b The two equations are the same. c y = 5 − 15 d x = 43 − 13 4 a x = − 1, y = , z = 5 b x = + 3, y = 3, z = 14 − 3 10 − 3 cx= ,y = ,z = 6 6 26 − 3t −3(t + 2) ,x= , 5 z = t, y = 4 4 t −2 ;t∈R w= 2 x = −4, y = −12, z = 14 6 a x = 1, y = 2, z = 3 −(3 + 5) −5 , y= , z= bx= 3 3 2 − 3t c z = t, y = −2(t − 1), x = 2  8−12a 1 2a−8 3a−4 − + 13a(a−2) 13 13a(a−2) 13a(a−2) 7a  −2(a+1) 1−2a  13a−2 13a(a−2) 13a(a−2)  13a(a−2) 16 13a(a−2). 16−10a 13a(a−2). 1  13.   . 3a−8 13a(a−2). b a = 0, 2 c a = 0 26 − t t , z = t; t ∈ R dx= , y= 2 8 8 p = 3; z = t, y = 5 − 4t, x = 3(t − 1). Exercise 2F −3 1 ,y= 2 2 37 7 −31 51 dx= ,y = ,y = cx= 10 5 38 38 2 Their graphs are parallel straight lines that do not coincide. 3 x = t + 6, y = t where t ∈ R 4m =9 5 a m = −5 bm =3 6 a i m = −2 ii m = 4 4 2(m + 4) bx= ,y= m+2 m+2 −3 7 a x = 2, y = 0 k = 2 −3 bk = 2 8 a b ∈ R\{10} b b = 10, c = 8 c b = 10, c = 8 1 a x = 4, y = −3. bx=. Multiple-choice questions 1E 6A. 2E 7A. 3D 8C. 4C 9B. 5B 10 C. Short-answer questions (technology-free) 7 5 2 a x = −2, y = 2. 1 a −8 3a. b. c 30. b x = −44, y = −39 b. y. d7 y. 0, 5 3. 0. x 5, 0 2. 0 (0, –6). x (6, 0).

(30) P1: FXS/ABE. P2: FXS. 0521665175ans-1.xml. CUAT018-EVANS. November 3, 2005. 16:7. Answers. (0, 3). (2, 0). 0. Exercise 3A x. 1 x4 Maximal domain = R\{0}, range = R + Asymptotes at x = 0, y = 0. 1 a f (x) =. 4 a y = −2x + 5. b y = 2x + 2. 1 1 cy= x+ 2 2 √ 5 13. d y = −2x + 3. y. b. Extended-response questions 1a. x. 0. P $. 1 2 a i f (x) = √ x Maximal domain = R + , range = R + Asymptotes at x = 0, y = 0. 100 (100, 50) 0. N. 1 b P = − N + 100 2 c i $56 2 a p = 1448t + 7656 p b. ii. y. ii N = 80. x. 0 3. b i f (x) = x 5 Maximal domain = R, range = R. (8, 19240) 7656 0. t. c 22 136 5 3a y = x −4 3 d 15 4ay=. 31 4 x+ 7 14. d 1448 people per year 5 66 82 c b , 3 7 7 629 square units e 14 59 b 14 65 d square units 28 √ √ b 269 c 269. √ c 65 1 5 a 1, − 2 10 33 4 13 ey= x− dy=− x+ 13 26 10 5 7 15 f ,− g (26, −33) 2 4 1 2 6 a 125 litres b x = 291 , y = 208 3 3 7 Brad 20, Flynn 10, Elizabeth 15. y. ii. 0. x. 3. c i f (x) = x − 5 Maximal domain = R\{0}, range = R\{0} Asymptotes at x = 0, y = 0 ii. y. 0. x. Answers. Chapter 3. y. c. 693.

(31) P1: FXS/ABE. P2: FXS. 0521665175ans-1.xml. Answers. 694. CUAT018-EVANS. November 3, 2005. 16:7. Essential Mathematical Methods 3 & 4 CAS y. e. 1 d i f (x) = √ 3 x Maximal domain = R\{0}, range = R\{0} Asymptotes at x = 0, y = 0. y=. 1 25x2. y. ii. y=. 1, 1. 25. x. 0 y. 3. y= (1, 3). x. 0. (1, 16). 0. 3 2x. y=. 3 1, 2. y= 0. 16 x2. 3 x. (1, 1). 1 x x. (–1, –1). √ 5. –1, –. e i f (x) = x Maximal domain = R, range = R. 3 2. (–1, –3). 4. y. ii. y. f. y. f(x) = 3√x. x. 0. (1, 3) 0. 3 a {x: 0 < x < 1} b {x: 0 < x < 1} 4 a Odd b Even c Odd d Odd e Even f Odd. Exercise 3B y. b. y. 1a. 1,. (1, 4) x. 0. y. 4 x2. 2 x2. (1, 2) x. x. 0 y. d y = 32 x. (1, 4) 0. x. y y=. y. c. x. (1, 2) 0. y = 12 4x 0. y = 22. (–1, 2). b. y. 2a. y=. y. d y = √3x (3, 3) 1, 1 3 x. 0. x. 0 1 y= 2x. y=4 x. c. 1 2. x. (1, 3) 0. x. x. Range = R + 5 a Dilation of factor 5 from the x-axis b Dilation of factor 4 from the x-axis 1 c Dilation of factor from the y-axis 5 2 6 a i y = 4|x| ii y = |x| 3 1 iii y = 2|x| iv y = |x| 5 √ 2 √ b i y=43x ii y = × 3 x 3 √ x 3 iv y = 3 iii y = 2x 5 4 2 c i y= 3 ii y = 3 x 3x 1 125 iii y = 3 iv y = 3 8x x 4 2 d i y= 4 ii y = 4 x 3x 1 625 iii y = iv y = 4 4 16x x 2 4 ii y = √ e i y= √ 3 33x x 1 5 iii y = √ iv y = 3 3 2x x 2 2 2 ii y = (x) 3 f i y = 4(x) 3 3 x 2 2 3 3 iv y = iii y = (2x) 5 2 g i y= 4 ii y = 3 3 3x 4 x4 3 5 4 1 iv y = iii y = 3 x (2x) 4. x.

(32) P1: FXS/ABE. P2: FXS. 0521665175ans-1.xml. CUAT018-EVANS. November 3, 2005. 16:7. Answers. y. 1a. y = √–x. x y = –√x. 0. dom = [0, ∞) 2 Reflection in the y-axis 3 a i y = −|x| ii √ 3 bi y=− x ii −1 ii ci y= 3 x −1 ii di y= 4 x −1 ei y= √ ii 3 x. gi y=. ii y =. 3 4. x 4 a Even e Odd. y = |x| √ y=−3x −1 y= 3 x 1 y= 4 x −1 y= √ 3 x. b Odd f Even. (2, 0). Domain = [2, ∞), range = R + ∪ {0} y. e. y= 1. x–1. x. 0 (0, –1) x=1. Domain = R\{1}, range = R\{0} y 1, 0 4. 1 3 (−x) 4. 0. c Odd d Even g Neither odd nor even. y = –4 1 y= x –4. y. g. y. x = –2 0, 1 2. y= 1 +3 x y= 1 x+2. x. 0. Domain = R\{−2}, range = R\{0}. Domain = R\{0}, range = R\{3}. y. h. y. b. y=. –. x. 0. y=3. 1 – ,0 3. x. Domain = R\{0}, range = R\{−4}. Exercise 3D 1a. x. 0. f. ii y = x 3. −1. x. dom = (−∞, 0]. 2. 2. f i y = −(x) 3. y = √x – 2. y. b. 0. y. 1 ,0 √3. y= 1 x–3. 1 –3 x2 1 ,0 √3. 0. x. 0 0, – 1 3. x. x=3. y = –3. Domain = R\{0}, range = (−3, ∞). Domain = R\{3}, range = R\{0} y. i. y. c y=. 1 (x + 2)2. 0, 1 9. 0, 1 4 0. 1 (x – 3)2. x. 0. x. x=3. x = –2. Domain = R\{−2}, range = R. y=. +. Domain = R\{3}, range = R +. Answers. d. Exercise 3C. 695.

(33) P1: FXS/ABE. P2: FXS. 0521665175ans-1.xml. Answers. 696. CUAT018-EVANS. November 3, 2005. 16:7. Essential Mathematical Methods 3 & 4 CAS y. j y=. y. c x = –3. 1 (x + 4)2. 0, 1. 1. 0, 3. 16. x. 0. x. 0 y = f(x + 3). x = –4. Domain = R\{−4}, range = R + k. y. d. y. y = f(x) – 3. y= 1 +1. 1, 0 3. x–1. y=1. x. 0 y = –3. x. 0 x=1. Domain = R\{1}, range = R\{1}. y. e. y. l. x = –1 (0, 1). y= 1 +2 x–2. x. 0 0, 1. x. 0. y=2. y = f(x + 1). 1 2. 1 ,0. 1 2. y. f. x=2. Domain = R\{2}, range = R\{2} m. y = f(x) – 1. y. (1, 0) y= –1 2. 1 –4 x2. 0 y = –1. 1 2. x. 0 y = –4. –4. Domain = R\{0}, range = (−4, ∞) 2a. y y = f(x – 1) 0. x. (0, –1) x=1. b. y y = f(x) + 1 y=1 ( –1, 0). 0. x. x. 3 a Translation with mapping (x, y) → (x − 5, y) b Translation with mapping (x, y) → (x, y + 2) c Translation with mapping (x, y) → (x, y + 4) 4 a i y = |x − 7| + 1 ii y = |x + 2| − 6 iii y = |x − 2| − 3 iv y = |x + 1| + 4 √ 3 b i y=√ x −7+1 3 ii y = √ x +2−6 3 iii y = √ x −2−3 iv y = 3 x + 1 + 4 1 c i y= +1 (x − 7)3 1 ii y = −6 (x + 2)3 1 iii y = −3 (x − 2)3 1 +4 iv y = (x + 1)3.

(34) P1: FXS/ABE. P2: FXS. 0521665175ans-1.xml. CUAT018-EVANS. November 3, 2005. 16:7. Answers vi y =. −6. d i y=. −3. ii y =. +4. iii y =. +1. iv y =. −6. v y=. −3. vi y =. +4. e i y=. f i y = (x − 7) 3 + 1. ii y =. ii y = iii y = iv y =. (x (x (x (x. e i y= √ 3 ii y = √ 3 iii y = √ 3 iv y = √ 3. x −7 1 x +2 1 x −2 1 x +1. 2. ii y = (x + iii y = (x −. 2 2) 3 2 2) 3 2 1) 3. −6 −3. iv y = (x + +4 1 +1 g i y= 3 (x − 7) 4 1 −6 ii y = 3 (x + 2) 4 1 iii y = −3 3 (x − 2) 4 1 iv y = +4 3 (x + 1) 4. iii y = iv y = v y= vi y =. −2 +8 (x − 3)3 −2 −4 (x − 3)4 −2 +4 (x − 3)4 −2 −4 (x − 3)4 −2 −8 (x − 3)4 −2 +8 (x − 3)4 −2 +8 (x − 3)4 −2 −4 √ 3 x −3 −2 +4 √ 3 x −3 −2 −4 √ 3 x −3 −2 −8 √ 3 x −3 −2 +8 √ 3 x −3 −2 +8 √ 3 x −3. Answers. 1 − 7)4 1 + 2)4 1 − 2)4 1 + 1)4 1. +1. d i y=. 697. 2. f i y = −2(x − 3) 3 − 4 2. ii y = −2(x − 3) 3 + 4 2. iii y = −2(x − 3) 3 − 4 2. iv y = −2(x − 3) 3 − 8 2. Exercise 3E 1a i ii iii iv v vi b i ii iii iv v vi. y y y y y y y y y y y y. c i y ii y iii y iv y v y. = −2|x − 3| − 4 = −2|x − 3| + 4 = −2|x − 3| − 4 = −2|x − 3| − 8 = −2|x − 3| + 8 = −2|x − 3| + 8 √ = −2 3 x − 3 − 4 √ = −2 3 x − 3 + 4 √ = −2 3 x − 3 − 4 √ = −2 3 x − 3 − 8 √ = −2 3 x − 3 + 8 √ = −2 3 x − 3 + 8 −2 −4 = (x − 3)3 −2 +4 = (x − 3)3 −2 = −4 (x − 3)3 −2 = −8 (x − 3)3 −2 = +8 (x − 3)3. v y = −2(x − 3) 3 + 8 2. vi y = −2(x − 3) 3 + 8 −2 −4 g i y= 3 (x − 3) 4 −2 ii y = +4 3 (x − 3) 4 −2 −4 iii y = 3 (x − 3) 4 −2 −8 iv y = 3 (x − 3) 4 −2 +8 v y= 3 (x − 3) 4 −2 +8 vi y = 3 (x − 3) 4 y 3 2. y. (5, 4). 0. x. 3. –8 –3. (8, –2). (–4, –3). 0. x.

(35) P1: FXS/ABE. P2: FXS. 0521665175ans-1.xml. Answers. 698. CUAT018-EVANS. November 3, 2005. 16:7. Essential Mathematical Methods 3 & 4 CAS 4a. b y = −2|x + 2| + 1. y. y. b x = –1. (–2, 1) –3 2. –5 2. (0, 1) (1, 0). x. 0. 0. –3. x. y = –1. 5a. y= 2 –1. 1. b y = (1 − 2x) 3 − 2. y. x+1. Range = R\{−1} –7 2. c. 0. –1 1 , –2 2. y= 0, 3 4. 1 a A dilation of factor 2 from the x-axis, followed by a translation of 1 unit in the positive direction of the x-axis and 3 units in the positive direction of the y-axis. b A reflection in the x-axis and a dilation of factor 3 from the x-axis (in either order), followed by a translation of 4 units in the negative direction of the x-axis and 7 units in the negative direction of the y-axis. c A reflection in the y-axis and a dilation of factor 4 from the x-axis (in either order), followed by a translation of 1 unit in the positive direction of the x-axis and 5 units in the negative direction of the y-axis. d A reflection in the x-axis, followed by a translation of 1 unit in the negative direction of the x-axis and 2 units in the positive direction of the y-axis. e A reflection in the y-axis and a dilation of factor 2 from the x-axis (in either order), followed by a translation of 3 units in the positive direction of the y-axis. f A translation of 3 units in the negative direction of the x-axis and 4 units in the negative direction of the y-axis, followed by a reflection in either axis and a dilation of factor 1 from the x-axis (in either order). 2 y 2a. x. x=2. Range = R + y. d. y= (0, 1) (1 – √2, 0). (1 + √2, 0) 0. y = –1. x=1. Range = (−1, ∞) y. e. x=3 0, 1 3. x. 0 y = –1. x–3. Range = R\{0} y. f. x+2. x–1. x. 2 3. –1 , 0. y=3 0. x = –2 x=1. Range = R\{0}. 2 –1 (x – 1)2. y = –1 + 3. y= 3. 0. 3 (x – 2)2. 0. Exercise 3F. (0, –3). y. x. Range = R\{3}. x 1 0, 2 2. x.

(36) P1: FXS/ABE. P2: FXS. 0521665175ans-1.xml. CUAT018-EVANS. November 3, 2005. 16:7. Answers y. f. x. 0. (–2, –2). 4. (0, –5√2 – 2). 106 27. 3. x. 0. 3–. 1 3. √2. y = –5√x + 2 – 2. ,0. Range = (−∞, −2] y. g. Range = R\{4} y. 3a. x=2 0, 3 2. 3. x. 0. x. 0. y = –√x – 3 y = –3. x–2. Range = R\{0}. Range = R − ∪ {0}. y. h. y. b. x = –2 x. 0 y = –4. (3, 2) x. 0 (7, 0). 0, –4. y = –√x – 3 + 2. 2 –4 (x + 2)2. y=–. Range = (−∞, 2]. Range = (−∞, −4). y. c. y. i (0, √6). y = √2(x + 3). 3 ,0 10. x. 0. –3. 1 2. 0. x. y = –5 y= 3 –5 2x. Range = R + ∪ {0}. Range = R\{−5}. y. d. y=. 1 2x – 3. 5. x. 0 0, – 1 3. x=. 3 2. –. Range = R\{0}. –2. x. y. k (0, 5√2). 0. 1 2. Range = R\{5}. y. e. y. j. y = 5√x + 2 x. 0. 11 5 0. +. Range = R ∪ {0}. 3. Range = [5, ∞). x. Answers. y. g. 699.

(37) P1: FXS/ABE. P2: FXS. 0521665175ans-1.xml. Answers. 700. CUAT018-EVANS. November 3, 2005. 16:7. Essential Mathematical Methods 3 & 4 CAS. 7. y. b. y. 4a. 2 0 –7 –2. y=. x. 0. 1. x. (0, 3). Range = [2, ∞). (4, 1) 0 y = –3√x – 4 + 1. Range = (−∞, −6] y. 5a. 1 4 ,0 9. y = 5√2x + 4 + 1. (–2, 1) 0. x. Range = [1, ∞). Exercise 3G. x+1. y=3 (0, 2) 0. x. –2,0 3. x = –1 y. b y = 4x – 5. 2x + 1. y=2 0. x. (0, –5) 1 x=–. 5,0 4. 2. 6a. (0, 11) x. Range = (−∞, 1]. y = 3x + 2. y. f. y. –6 –7. x. 0. x=1. e. x. 0. y y= 2 +4 x–3. y=4 0 0, 3. x. Range = (1, ∞). y –1. (1, 2). y=1. Range = R\{2}. y = 2√x – 1 + 2. 2 +1 (x – 1)2. 0. Range = R. y. d. 3. 7 – √2. –2. c. y. c. x. 1 2. 2 ,0. 1 3. x=3. Range = R\{4} y. b. x=3. 1 a (1, −6) d (−1, −2)

(38) 1 0 2 0 −2

(39). −1 0 3a 0 1

(40). 1 0 d 0 2

(41)

(42). 1 0 x 4 0 3 y. b (2, −2) e (−2, 1). c (1, 2).

(43)

(44). 0 1 0 −1 c 1 0 −1 0

(45). 3 0 e 0 1

(46)

(47) x 2 = + 1 y − 1 y x = x − 2; y = 3 2 x x2 x x − +2 = − +4 5y=2 16 4 8 2 6 y = −2(x 3 + 2x) −x 3 x 7y= +3 −4 8y= 3 2 2 3 9 y = x −9 2 1 11 10 y = x + 6 2 x +1 3 y−4 x +1 11 = −2 +6 2 4 4 2−y x +1 2 x +1 3 12 +6 +2 = −2 2 3 3. b. 0, 5 1 3. y=4 0. (4, 0). y= 4 +4 3–x. Range = R\{4}. x. Exercise 3H 1 b = −1, B = 2, A = 1 2 A = 2, B = 3 3 A = 2, B = −1 4 b = 2, B = −3, A = 8 5 a = −2, b = 1 6 a = 3, b = −5.

(48) P1: FXS/ABE. P2: FXS. 0521665175ans-1.xml. CUAT018-EVANS. November 3, 2005. 16:7. Answers. 1y. Exercise 3J y. 2. y = √x + x. y = x + √x + 2. y=x. (2, 4). (2, 2). y = √x. (1, 1). b f −1 : R\{0} →R, 1 f −1 (x) = + 3 x y. y. y=x. (1, 2). 1 a f −1 : [2, ∞) →R, f −1 (x) = (x − 2)2. (0, √2). y = √x + 2. x. 0. 0. x. 0. y=3. x. 2. 0, –. (–2, –2). y=x. y. (1, 1). (4, 2). y = x – √x 0. x. 0. x. (1, 0) y = – √x. (1, –1). d f −1 : R\{1} → R, f −1 (x) = y y. 4. (1, 2). 2. (1, 1). (–1, 1) y=. 1 x2. 0. y=2. 0. 1 1 y= x + 2 x. –1. 3 +2 x −1. – 1, 0. 1 y= x. local min. at –2, – 1 4. 0. c f −1 : [4, ∞) → R, f −1 (x) = (x − 4)2 + 2. y. 3. 1 3. (0, –1). x. x=1 x. e f. −1. : R\{−1} → R, f −1 (x) = y. (–1, –1). 5 +1 x +1. x = –1 (0, 6). y=1. x. (–6, 0) 0 y. 5a. f f −1 : [1, ∞) → R, f −1 (x) = 2 − (x − 1)2. y = √x + 2. y. (–2, 4). (1, 2). (0, √2). 1 + √2. (0, 0) (–2, 0). 2ay=. y. by cy. y = √2 – x (–2, 2). (0, 2√2) y = √x + 2 √2. 0. dy. (2, 2). ey 2. x. fy. x. 3 , domain = R\{0} x = (x + 4)3 − 2, domain = R = (2 − x)2 , domain = (−∞, 2] 3 , domain = R\{1} = x −1 2 = 3 + 6, domain = R\{5} 5−x 1 + 1, domain = (2, ∞) = 4 (x − 2) 3. y = –2x. b. –2. 0. x y = √x + 2 – 2x. x. Answers. Exercise 3I. 701.

(49) P1: FXS/ABE. P2: FXS. 0521665175ans-1.xml. Answers. 702. CUAT018-EVANS. November 3, 2005. 16:7. Essential Mathematical Methods 3 & 4 CAS. Multiple-choice questions. y. 3a 3. 1C 6B. 2 1 –3 –2. 0 1. 2. 3. y 4 3 2 1 4. x. 1, 0 3 0. Inverse is not a function.. c. –3 –2 –1 0 1 2 3 4 –1 –2 –3. 4. 1 3. 4. 5. 6. –4 –3 –2 –1 0 1 2 3 4 –1 –2 –3 x –4. e. 1 2. Range = (4, ∞). y x=1 (0, 2). x. y=1. 0. (2, 0). x. Range = R\{1} y. 3a. y. b y = 2√x – 3 + 1. (2 – √3, 0). (3, 1). 0. x. 0. (2 + √3, 0) y = –1. 0, – y. c. 1 4. x=2. x=2 x. 0 0, –. 5. y. y = –1. 7 4. 4y=−. 1 2 1 x +3 c f −1 : R\ → R, f −1 (x) = 2 2x − 1. x. x=2. x=1. Inverse is not a function. x +1 −1 −1 4 a f : R\{1} → R f (x) = x −1 −1 2 b f −1 : R + ∪ {0} → R f (x) = x + 2 2 2x +3 c f −1 : R\ → R f −1 (x) = 3 3x − 2 5 domain range 1 1 a Since f R\ R\ 2 2 ran f = dom f, f ◦ f is defined. 1 b f ◦ f (x) = x, x ∈ R\ 2 y. x. y=4 0. Range = R\{−3}. y. Inverse is a function.. 0. x. y = –3. (0, –5). 3 2. y. d. y. 0. 4 3 2 1. 5. Range = (0, ∞). Range = R\{−3}. x. 6. 2. x. 0. 5 ,0 3. Inverse is not a function. e d y. 1. x. x=2. 4 3 2 1. 0. 5D 10 A. y = –3. y. c. 4A 9A. 1 a Domain = R\{0}, range = R\{2} 2 b Domain = , ∞ , range = (−∞, 3] 3 c Domain = R\{2}, range = (3, ∞) d Domain = R\{2}, range = R\{4} e Domain = [2, ∞), range = [−5, ∞) y b y 2a. –3. Inverse is a function.. –3 –2 –1 0 1 2 3 –1 –2 –3. 3D 8C. Short-answer questions (technology-free). x. –2. b. 2E 7B. 6 + 9, a = −6, b = 9 x x +2 6 f −1 (x) = f –1(x) = (x – 4)2 + 2 1−x y=x. (6, 6) (2, 4) (4, 2) 0. x. x.

(50) P1: FXS/ABE. P2: FXS. 0521665175ans-1.xml. CUAT018-EVANS. November 3, 2005. 16:7. 703. Answers. 1 a R\{−2} b Dilation of factor 24 from the x-axis Translation of 2 units in the negative direction of the x-axis Translation of 6 units in the negative direction of the y-axis c (0, 6), (2, 0) 24 d i g −1 (x) = −2 x +6 −1 ii Domain of g√ = range of g = (−6, ∞) iii x = −4 + 2 7 y iv x = –2. 4n m c i (x, y) → x + , − 2 y + n 2 m 4n m 2 +n ii y = − 2 x − m 2 4n 3m 2 iii y = − 2 x − +n m 2 4 4 b 4 a R\ 3 3 4 1 3 d x = 6.015 c f −1 (x) = + 3 3 x − 6 y e y=. 3 (3x – 4)2. +6 y=x. y=6 4 x= 3. x = –6 y=x. y=. y=. 4 3. 1 3. 4 3 + x–6 3. 6. x. 0. 2. x=6. y = g –1(x) 02. x. 6. y = –2. 5 a f (x) =. y = g(x). y = –6. 50 20 − x. y 5 2 0. 2 a [−3, ∞). 1 from the y-axis 2 Dilation of factor 2 from the x-axis Reflection in the x-axis Translation of 3 units in the negative direction of the x-axis Translation of 4 units in the positive direction √ of the y-axis c (0, 4 − 2 6), (−1, 0) (x − 4)2 d f −1 (x) = −3 8 e Domain of f −1 = range of f = (−∞, 4] √ √ √ f x = 8 − 6 2 or x = 2 2 or x = −2 2 y g. b. Dilation of factor. (–3, 4) y = f –1(x). y=x. y = f(x). (–2√2, 2√2) x. 0 (8 – 6√2, 8 – 6√2) (4, –3). c g −1 (x) =. y. b ii 0. 20. iii iv b i ii iii. 3 ii (x, y) → (x, −y) 125 (x, y) → (x + 25, y + 15) 3 y + 15) (x, y) → (x + 25, − 125 3 2 (x − 25) + 15 y=− 125 (x, y) → (x + 50, y) 3 y=− (x − 75)2 + 15 125. x. 20x 50 + x. x. –50. 6 a i y = f −1 (x − 5) + 3 ii y = f −1 (x −3) + 5 x x iv y = 3 f −1 iii y = 5 f −1 5 3 x −b −1 b y = cf + d. The graph of a y = f (x) has undergone the following sequence of transformations: a reflection in the line y = x, then a dilation of factor c from the x-axis and factor a from the y-axis, and a translation of b units in the positive direction of the x-axis and d units in the positive direction of the y-axis.. (2√2, –2√2). 3a i. 20. Chapter 4 Exercise 4A 1 A = 1, B = 3 2 a A = 1, B = −2, C = 6 3 b A = 4, B = − , C = 5 2 c A = 1, B = −3, C = 5. Answers. Extended-response questions.

(51) P1: FXS/ABE. P2: FXS. 0521665175ans-1.xml. Answers. 704. CUAT018-EVANS. November 3, 2005. 16:7. Essential Mathematical Methods 3 & 4 CAS 33 x −3 b 5x 4 + 8x 3 − 8x 2 + 6x − 6 26(x − 2) c x 2 − 9x + 27 − x2 − 2 4 a −16 ba=4 5 a 28 b0 c f (x) = (x + 2)(3x + 1)(2x − 3) 11 6bk = 2 7 a a = 3, b = 8 b (2x − 1), (x − 1) −92 8a= ,b = 9 9 9 81 10 b 6x − 4 11 b x 2 − 3, x 2 + x + 2 1 c ,2 12 a 2, −4, 2, 3 b 0, 2 2 d −2, 2 e 0, −2, 2 f 0, −3, 3 3 a 2x 2 − x + 12 +. 1 3 1 1 h 1, −2 i 1, −2, , g 1, −2, − , 3 2 4 3 13 a (−1, 0), (0, 0), (2, 0) b (−2, 0), (0, 6), (1, 0), (3, 0) c (−1, 0), (0, 6), (2, 0), (3, 0) 1 d − , 0 , (0, 2), (1, 0), (2, 0) 2 e (−2, 0), (−1, 0), (0, −2), (1, 0) 2 f (−1, 0), − , 0 , (0, −6), (3, 0) 3 2 g (−4, 0), (0, −16), − , 0 , (2, 0) 5 1 1 h − , 0 , (0, 1), , 0 , (1, 0) 2 3 3 i (−2, 0), − , 0 , (0, −30), (5, 0) 2 14 p = 1, q = −6 15 −33 16 a (x − 9)(x − 13)(x + 11) b (x + 11)(x − 9)(x − 11) c (x + 11)(2x − 9)(x − 11) d (x + 11)(2x − 13)(2x − 9) 17 a (x − 1)(x + 1)(x − 7)(x + 6) b (x − 3)(x + 4)(x 2 + 3x + 9) 18 a (x − 9)(x − 5)(2x 2 + 3x + 9) b (x + 5)(x + 9)(x 2 − x + 9) c (x − 3)(x + 5)(x 2 + x + 9) d (x − 4)(x − 3)(x + 5)(x + 6). y. 2a. b y = 2(x –. y. 1)2 y = 2(x – 1)2 – 2. (0, 2). (2, 0). x. 0 (1, 0). 0 (1, –2). y. c. y = –2(x – 1)2 (1, 0). x. 0 (0, –2). d. y. (–1, 4) y = 4 – 2(x + 1)2 (0, 2) x. 0. (–1 + √2, 0). (–1 – √2, 0). y. e. y=4+2 x+. –. 1,4 2. 0, 4. 1 2. x. 0. y. f. y = 2(x + 1)2 – 1 (0, 1) x. 0 (–1, –1) –1 – 1 , 0 √2. –1 +. 1,0 √2. y. g. y = 3(x – 2)2 – 4. Exercise 4B 1 a Crosses the x-axis b Does not cross the x-axis c Just touches the x-axis d Crosses the x-axis e Does not cross the x-axis f Does not cross the x-axis. 1 2. 2 – 2√3, 0 3. (0, 8) x. 0 (2, –4) 2+. 2√3, 0 3. 2. x.

(52) P1: FXS/ABE. P2: FXS. 0521665175ans-1.xml. CUAT018-EVANS. November 3, 2005. 16:7. Answers y. b y = (x +. 1)2. –1. 8. x. (0, 0) (–2, 0) (–1, –1). i. 3 0. 1. √5. 0. 0. 1 1 2 4. –3 –2. 3 2. 2. −4. 1 4. 0. x 2. x. –6. y. e –3.56. 0.56 0. x. –2. – 3 , –4 1 2. 4. y. f –3.12. 1.12 0. x. –7 (–1, –9) y. g. –0.095 –1 0. (1, 1). 1. 0. –1. (0, – 2) (–1 + √2 , 0). 1 1 Min. = −4 , range = −4 , ∞ 4 4 b f (x) = (x − 3)2 − 1 Min. = −1, range = [−1, ∞) c f (x) = 2(x + 2)2 − 14 Min. = −14, range = [−14, ∞) d f (x) = 4(x + 1)2 − 11 Min. = −11, range = [−11, ∞) 5 2 25 e f (x) = 2 x − − 4 8 25 25 Min. = − , range = − , ∞ 8 8 1 2 22 f f (x) = −3 x + + 3 3

(53) 22 22 Max. = , range = −∞, 3 3 9 2 169 g f (x) = −2 x − + 4 8

(54) 169 169 Max. = , range = −∞, 8 8 y 4a 1. y 2 ,. x. 3 a f (x) = x +. 3. d. (–1 – √2 , 0) y = 2(x + 1)2 – 4. . 2. –6. y. (–1, – 4). 1. x. (0, –1). j. x. 0. 1 ,0 √5. ,0. (2, 2). 2. y = 5x2 – 1. –. y. c. y. x. 2 4 (3, –1). –6. (1, –6). 2 2.095. x. Answers. y. h. 705.

(55) P1: FXS/ABE. P2: FXS. 0521665175ans-1.xml. Answers. 706. CUAT018-EVANS. November 3, 2005. 16:7. Essential Mathematical Methods 3 & 4 CAS y. h. y. c (1, 1). 1. 1.71. 0.29 0. (0, 1). (–1, 0). x. 1. 2. x. 0. –1 y. i. y. d 0.3 x. 0. 0. 1 4 3, 0. –0.11. (0, –4). –1.09. x. (–0.6, –0.6). j. y (–1.08, 0.604). –2.09. 0 –0.08. y. e x. (1, 0). –0.1. (–1, –8). 5aB bD 6aC bB cD 7 a m > 3 or m < 0 9 a k < −5 or k > 0. dA bm =3 b k = −5. 0 (0, –7). y. f. Exercise 4C. (2, 0). 4 2 x b y = −x 2 25 2 c y = x + 2x d y = 2x − x 2 2 e y = x − 5x + 4 f y = x 2 − 4x − 5 2 g y = x − 2x − 1 h y = x 2 − 4x + 6 1 2 1 2 y = − x + x + 1, y = x 2 + x − 5 8 8 3 b = 2, B = 4, A = 1 1a y =4−. (0, –4). (1, –2). y. g. (0, 4) (1, 2). y. 0. (1, 2) 0. x. 0. Exercise 4D 1a. (2, 0). y. y. 0. x. (2, –4) 2+. (–1, 2) 0. x. x. h b. x. x. (0, –28). 1 4 3, 0 3.

(56) P1: FXS/ABE. P2: FXS. 0521665175ans-1.xml. CUAT018-EVANS. November 3, 2005. 16:7. 707. Answers y. c. (0.435, 0). (0, 10). (1, 2). Answers. y. i. x. 0. 0. (3.565, 0). x. (2, –6) y. j. y. d 1 –2 3 + 1, 0. x. 0. (2.159, 0) (0, 161). (0, –2) (1, –4). x. 0 (3, –1). y. k. (3.841, 0) y. e x. 0. –1. 1. (–4, 1). –31. (–1, –32). l. x. 0. (–5, 0). (–3, 0). y. (0, –255) 0. x. 2. y. f. (1, –2) –4. x. 0 (2, –3). 2 a = −3, h = 0, k = 4 (0, –51). 3 a = 16, h = −1, k = 7 4 a y = 3x. b y = (x + 1) + 1. 3. 3. c y = −(x − 2) − 3 3. d y = 2(x + 1)3 − 2 5a. ey=. y. x3 27. Exercise 4E 1a. (0.096, 0). +. 1. 3. 6. –6. –3. 1. x. – (0, 1) 0. (1.904, 0). x. b. (1, –2). x. –. y. b. +. c (–2, 0). d (0, –32). 3. –1. 5. x. –. x. 0. +. + –. 1 2. 4. 5. x.

(57) P1: FXS/ABE. P2: FXS. 0521665175ans-1.xml. Answers. 708. CUAT018-EVANS. November 3, 2005. 16:7. Essential Mathematical Methods 3 & 4 CAS y. 2a. y. y = f(x). y = f (2x) 8.00. 1. y = f (x) 6.00. x. –0.62 0. 1. 1.62. y=f. 4.00. 4 , –0.19 3. 2.00 (2.73, 0.75). (–0.73, 0.75). y. b. 0. –2.00 –1.00. y = f(x + 2) y = f(x) y = f(x – 2). –3. –2. y. (2, 1). 1. – 9 + 1, 81. x. 0. –1. 1.00 2.00 3.00 4.00. Turning points for y = f (2x) are at (−0.18, 0.75) and (0.68, 0.75). 2a. (–2, 1). 2. 3. 2. 9 + 1, 81 2 4. 4. 3.62 8. 10 , –0.19 3. –2 , –0.19 3. For clarity the graph of y = 3 f (x) is shown on separate axes.. x 0 1. –2. 4. y. b. y. 9 81 , 2 2. – y = 3f(x). 9 81 , 2 2. 3 y = f(x) 1. 0. –3. x. 3. x. 0. y. c 4 , –0.56 3. – 9 – 1, 81 2. Exercise 4F. 9 – 1, 81 2 2. 2. 16. y. 1a. x. –1 0. –4. 2. y = f(x) y. d. (–0.37, 0.75). 1. (1.37, 0.75). – 9, 0. 9, 0 2. 2. x. 0 y = f(x). 0, – 81. y = f(x – 2). 4. e. y – 9 , 85. 9 , 85 2 4. 2 4. 0. x. 0. y. b. x 2. x (1.63, 0.75). (3.37, 0.75). Graphs of dilations shown on separate axes for clarity.. (0, 1) 0. x. x.

(58) P1: FXS/ABE. P2: FXS. 0521665175ans-1.xml. CUAT018-EVANS. November 3, 2005. 16:7. Answers. 709. 11 (x + 5)(x + 2)(x − 6) 60 5 = (x + 1)(x − 3)2 9 y = x3 + x + 1 y = x3 − x + 1 y = 2x 3 − x 2 + x − 2 y = (2x + 1)(x − 1)(x − 2) 1 y = x(x 2 + 2) 4 y = x 2 (x + 1). 1y= 2y 3a b c 4a b c. d y = x 3 + 2x 2 − x − 2 e y = (x + 2)(x − 3)2 5 a y = −2x 3 − 25x 2 + 48x + 135 b y = 2x 3 − 30x 2 + 40x + 13 6 a y = −2x 4 + 22x 3 − 10x 2 − 37x + 40 b y = x 4 − x 3 + x 2 + 2x + 8 31 4 5 3 157 2 5 11 cy= x + x − x − x+ 36 4 36 4 2. Exercise 4H √.

(59) −1 1 1 − 4k 2 , k∈ , \{0} 2k 2 2 1 b x = 4a or 3a or 0 c x = 0 or x = a 3 √ k ± k 2 − 4k dx= , k > 4 or k < 0 2 √ f x = ±a e x = 0 or x = ± a g x = a or x = b √ 1 h x = a or x = a 3 or x = ± a if a > 0 1 b+c 3 7 2ax = b x = (a) 3 a c 13 dx= −b c x = (a − c)n a 3 b 1 ex= f x = (c + d) 3 a 1 1 b , , (0, 0) 3 a (0, 0), (1, 1). 2 2 √. √ √ 3 − 13 3 + 13 √ , 13 + 4 ; , 4 − 13 c 2 2 1ax =. −1 ±. 4 a (13, 3), (3, 13) b (10, 5), (5, 10) c (11, 8), (−8, −11) d (9, 4), (4, 9) e (9, 5), (−5, −9) 5 a (17, 11), (11, 17) b (37, 14), (14, 37) c (14, 9), (−9, −14) 6 (2, 4), (0, 0) √. √ 5+5 5+5 7 , , 2 2. √ √ 5− 5 5− 5 , 2 2. √ √. −130 − 80 2 60 − 64 2 8 , , 41 41 √. √ 80 2 − 130 64 2 + 60 , 41 41. √. √ 1 + 21 −1 + 21 , , 9 2 2. √ √. 1 − 21 −1 + 21 , 2 2 √ √. 4 −6 5 3 5 10 ,2 11 , 9 5 5 1 12 −2, 13 (3, 2), (0, −1) 2 −8 −15 27 10 , b , , (5, 9) 14 a (3, 2), 4 2 3 5 −12 , −10 c (6, 4), 5 2 15 c − ac + b = 0 √ √ 16 (−1 √ − 161, √ 1 − 161), ( 161 − 1, 161 + 1) √ √ 4m + 25 + 5 4m + 25 + 5 17 a x = ,y= 2m 2√ √ 5 − 4m + 25 5 − 4m + 25 ,y= or x = 2 2m −25 2 5 bm = , − , 4 5 2 −25 cm < and m = 0 4. Multiple-choice questions 1E 6A. 2D 7C. 3E 8E. 4C 9C. 5E 10 C. Short-answer questions (technology-free) y. 1a. h(x) = 3(x – 1)2 + 2 (0, 5) (1, 2). x. 0 y. b. y = (x – 1)2 – 9. (–2, 0). 0. (0, –8). x (4, 0) (1, –9). Answers. Exercise 4G.

(60) P1: FXS/ABE. P2: FXS. 0521665175ans-1.xml. Answers. 710. CUAT018-EVANS. November 3, 2005. 16:7. Essential Mathematical Methods 3 & 4 CAS y. c. y. c y = x2 – x + 6. (0, 6). 1 3 ,5 2 4. 0. x. 0. (–3, 0) (–2, –1) x. (0, –9). d. y. y. d y = x2 – x – 6. (0, 26) (–2, 0) 0. x. (3, 0). (–2, 0) x. 0 (–3, –1). (0, –6) 1 25 ,– 2 4. y. e. e y=. 2x2. y. –x+5 1, 1 2. (0, 2). (0, 5) 1 , 39 4 8. 0. x. 0. x (1, 0). y. f. 5a y=. 2x2. + x. –x–1. 0. –. –2 –1 –1 2. x. 0. 1. b. + x 0. 1, – 9 4 8. (0, –1). 2. –1. 4 1 4 1 2 y = x2 − ; a = , b = − 3 3 3 3 √ 1 3 (1 ± 31) 3 y 4a. – 1. 3. c. + x 0 –4. –. –2 –1. d. +. 0. x –. –3. –1. (3, 0) x. 0 (0, –18). b. (1, –16). y. 2 3. 6a8 b0 c0 7 y = (x − 7)(x + 3)(x + 2) y 8a y = f(x – 1). (–1, 8). (0, 7) 0 (1, 0). x. 0. 3 2 –83 2 , 4 3 3. x.

(61) P1: FXS/ABE. P2: FXS. 0521665175ans-1.xml. CUAT018-EVANS. November 3, 2005. 16:7. Answers. y = f(x + 1). x. 0. –2. 1. –8 2 –83 , 3 34 y. c. y = f(2x). (6, 144π). 1 – ,0 2. 0 –2. d. x. 1. 4,. 224π 3. (3, 45π). 5 –83 , 6 34. 64π 2, 3. y. 0. y = f(x) + 2. (–1, 2). 6. d x = 5. The depth is 5 cm. h2 5 a r = 25 − 4 b V. x. 0. x. 3. (5.77, 302.3). 3. 2 –8 1 , 4 +2 3. 3. 9 y = −x 3 + 7x 2 − 11x + 6. 0. Extended-response questions 4 1ak = 3375 c i Rnew. b 11.852 mL/min. 10. h. c V = 96cm √ d h = 2, r =√2 6, i.e. height = 2 cm and radius = 2 6 cm, or h = 8.85 and r = 2.33 6 a V = (84 − 2x)(40 − 2x)x b (0, 20) c y 3. (15, 40). (8.54, 13 098.71) y = V(x). 0. 20. t. ii 23.704 mL/min d i Rout 40 0. t. 20 (35, –20). ii 11.852 mL/minute out 2 a i 2916 m3 ii 0 m3 b V (m3) (0, 2916). 0. c 3.96 hours. 9 t (hours). 0. 20. x. d i x = 2, V = 5760 ii x = 6, V = 12 096 iii x = 8, V = 13 056 iv x = 10, V = 12 800 e x = 13.50 or x = 4.18 f 13 098.71 cm3 7 a i A = 2x(16 − x 2 ) ii (0, 4) b i 42 ii x = 0.82 or x = 3.53 c i V = 2x 2 (16 − x 2 ) ii x = 2.06 or x = 3.43 8 a A = x 2 + yx 2 b i y = 100 − x 100 ii A = 100x − x 2 iii 0, 2 c x = 12.43 2 x 100 d i V = 100 − x , x ∈ 0, 50 2 ii 248.5 m3 iii x = 18.84. Answers. 3 a V = x(96 − 4x)(48 − 2x) = 8x(24 − x)2 b i 0 < x < 24 ii x ≈ 8, maximum volume ≈ 16 500 cm3 c When x = 10, V = 15 680 cm3 . d Max. volume = 14 440 cm3 e Min. volume = 9720 cm3 1 4 V = x 2 (18 − x) 3 a i 64 cm3 ii 45cm3 iii 224 cm3 3 3 b 144cm3 c V. y. b. 711.

(62) P1: FXS/ABE. P2: FXS. 0521665175ans-1.xml. Answers. 712. CUAT018-EVANS. November 3, 2005. 16:7. Essential Mathematical Methods 3 & 4 CAS 1 2 17 1 x3 − x + x 12 000 200 120 b x = 20 29 2 1 1 3 x + x − x dy=− 6000 3000 20 y e i. 9ay=. y. 2a. y = 2 x +1 – 2. 2 –1. 0. –1. x 1. –2 (40, 3). Range = (−2, ∞) b. 0. 40. y. x. 80. ii Second section of graph is formed by a reflection of the graph of y = f (x), x ∈ [0, 40], in the line x = 40.. (0, 4). Chapter 5. Range = R +. Exercise 5A 1a. y=0 x. 0. y. c. y y = 3x y = 2x y=0 (0, 1) x. 0. x. 0 –1. Range = (−1, ∞) y. d. y. b y = 3–x. 5 (0, 1). y = 2–x. y=0 x. 0. 2 1 0. y. c. x 1. 2. Range = (1, ∞) e y = 5x. (0, 1) 0. 0, 2. y=0 x. 1 4. 2. y = –5x. (0, –1). y. x. 0 y. d. Range = (2, ∞) y. f. (0, 1). y=0 x. 0 (0, –1). (0, 6). y = (1.5)x. y = –(1.5)x. 2 0. Range = (2, ∞). x.

(63) P1: FXS/ABE. P2: FXS. 0521665175ans-1.xml. CUAT018-EVANS. November 3, 2005. 16:7. Answers y. h. y. y = f(x – 2) (0, 1). (2, 1). y=0 x. 0. 0, 1 0. Range = R + b. y. y. c. 1. 1 0. x. y = 10 x –1. x. 0. Range = (1, ∞). Range = (−∞, 1). y. y = –1. 3 y=0 x. 0. Range = (−1, ∞). 2 0. Range = R +. x. 0. y. e 1. y. 5a. (0, 2). d. y=0 x. 4. x. b. y x. y = 10 10 + 1. Range = (2, ∞). (10, 11). y. f. (0, 2). x. 0 –1. y. Range = (1, ∞). (0, 2). y=1 x. 0 y. c. y=2 0. 0. y. d. y = f(–x) + 2. (0, 3). y = 2(10 x) – 20. y = f(x) + 1. (0, 2) y = 0 x. 0. y. c. y. b y = f(x + 1). (–1, 1). x. 0. Range = (−1, ∞) 4a. y=1. (0, –18). 0. x. x. y = –20. Range = (−20, ∞). x y = –1. (0, –2). (1, 0). y. d. y = –f(x) – 1. e. y. y = f(3x) 0. f. y. y=1. x. y=f 2. (1, 8) (0, 1) y = 0 x. (0, 1) 0. 0. (2, 2) y=0 x. y = 1 – 10 –x. Range = (−∞, 1). y. g. e. y y = 10 x + 1 + 3. y = 2 f(x – 1) + 1. (0, 13) y=3. (–1, 4). (0, 2). y=1 0. x. x. 0. Range = (3, ∞). x. Answers. 3a. 713.

(64) P1: FXS/ABE. P2: FXS. 0521665175ans-1.xml. Answers. 714. CUAT018-EVANS. November 3, 2005. 16:7. Essential Mathematical Methods 3 & 4 CAS y. f. Exercise 5B. x. y = 2 10 10 + 4 (10, 24). y. 1a. y=4. (0, 2). y=1. x. 0. Range = (4, ∞). y = 1 – ex. Range = (−∞, 1). y. c. d. (100, 408.024). (0, 1). x. 0. 0. b i $408.02 c 239 days 7 36 days 8a i. y. y y = e x–1 – 2. ii $1274.70 d ii 302 days. (0, e –1 – 2) 0. y = 5 x y = 3x y = 2x. x y = –2. Range = (−2, ∞) f. y. (0,1). y = 2ex x. 0. ii x < 0 b i. iii x > 0 1 3. y=. x. 0. x. y y=. x. 1 2. (0, 2). iv x = 0. 1 5. x. Range = (0, ∞) y. g (0,1). y = 2(1 + ex) x. 0. ii x > 0 c i a>1. iii x < 0. (0, 4). iv x = 0. y=2. ii a = 1. y. (0, 1) 0. x. 0 y. y = ax. Range = (2, ∞). y=1 x. 1 0. x. h. y. iii 0 < a < 1 y=2. y 0 (0, 1) 0. y = ax x. y = 2(1 – e –x). Range = (−∞, 2). x. Range = (0, ∞). Range = (−∞, 1) e. y y = e –2x. y=1. y = 1 – e –x. x. x. 0. Range = (1, ∞). 0. y=1. x. 0. 6 a C1. y=. y. b y = ex + 1. (0, 6). x.

(65) P1: FXS/ABE. P2: FXS. 0521665175ans-1.xml. CUAT018-EVANS. November 3, 2005. 16:7. 715. Answers. 1 d 4 5 g 2 3a1. y = 2e –x + 1. (0, 3). y=1. x. 0. e −2. Range = (1, ∞) j. y. i6 4 a 1, 2 e 1, 2 i −1, 2. (1, 2). (0, 2e –1). h6 b1 f4 3 j 5 b1 f 0, 1 j −1, 0. c4 f3. 3 2 10 g− 3 1 k± 2 c2 g 2, 4 c−. i4 d3 h−. 3 2. d1 h 0, 1. Exercise 5D. Range = (0, ∞) y. y. l. y=2 (0, 3e – 2) x. 0. x. (0, –1) 0. y = –2. Range = (−2, ∞) Range = (−∞, 2) 2 a x = 1.146 or x = −1.841 b x = −0.443 c x = −0.703 d x = 1.857 or x = 4.536 y 3 a, b i y = f(x) 1. y = f (x – 2) e –2 x. 0 y. ii, iii y = f (–x). y = f (x) y=f x. 3. 1. x. 0. Exercise 5C 1 a 6x 6 y 9. b. x. 0. k. 1 2 3 e 5. 2a4. b 3x 6. d 18x 8 y 4. e 16. g 24x 5 y 10. h 2x y 2. 6y 2 x2 5x 28 f 6 y. c. i x 2 y2. 1a3 b −4 c −3 d6 e6 f −7 2 a loge 6 b loge 4 c loge 106 (= 6 loge 10) d loge 7 1 e loge (= − loge 60) 60 f loge u 3v 6 (= 3 loge uv 2 ) g 7 loge x = loge x 7. h loge 1 = 0. 3 a x = 100 b x = 16 cx =6 d x = 64 e x = e3 − 5 ≈ 15.086 1 g x = −1 fx= 2 1 −3 i x = 36 h x = 10 = 1000 4 a x = 15 bx =5 cx =4 1 d x = 1(x = − is not an allowable solution) 2 3 ex= 2 5 a log10 27 b log2 4 (= 2) a 1 a 1 2 c log10 = log10 2 b b. 10a d log10 1 b3 1 e log10 (= −3 log10 2) 8 1 6a1 b1 c2 d3 e0 2 c0 7 a −x b 2 log2 x 3e 8ax =4 ≈ 0.7814 bx= 5 + 2e √ −1 + 1 + 12e 9ax = , i.e. x ≈ 0.7997 6 b x = loge 2 ≈ 0.6931. Answers. y. i.

(66) P1: FXS/ABE. P2: FXS. 0521665175ans-1.xml. Answers. 716. CUAT018-EVANS. November 3, 2005. 16:7. Essential Mathematical Methods 3 & 4 CAS 10. 1 ,2 4. y. f. x=2. 3 2 8 11 N = = 3 27. (3, 0) 0. y. 12 a. x. 2. x=3. Domain = (2, ∞), range = R y. g 0. 3. x. 4. x = –1. Domain = (3, ∞), range = R. (0, 1). y. b. x. 0. –1. x = –3. (1.72, 0). Domain = (−1, ∞), range = R. (4.39, 0). y. h. x=2. 0. x. –0.9. –3. (0, 0.69). 0. Domain = (−3, ∞), range = R. x. 1. y. c. Domain = (−∞, 2), range = R. x = –1. y. i. x= 4 3. –1 0. (0.65, 0) (0, –1). x (0, 0.39). d. (0.43, 0). . y x= 2. Domain =. 3. y = log2 2x. x. (0.79, 0). 4 , range = R −∞, 3. y. 13 a. 0 2. x. 0. Domain = (−1, ∞), range = R. 3. Domain =. 0. 2 , ∞ , range = R 3. y. e. y. b 0 (0, –1.4). x. Domain = R +. x = –2. –2 –1. 1, 0 2. x=5. x. y = log10 (x – 5) 0. (6, 0). Domain = (−2, ∞), range = R Domain = (5, ∞). x.

(67) P1: FXS/ABE. P2: FXS. 0521665175ans-1.xml. CUAT018-EVANS. November 3, 2005. 16:7. Answers y. i y = –log10 x. Answers. y. c. 717. – 1, 0 3. (1, 0). x. 0. x. 0 y = 4 log2 (–3x). Domain = R −. Domain = R +. y. j. y. d. x=2. y = 2 log2 (2 – x) – 6. y = log10 (–x). Domain = (−∞, 2) 14 a x = 1.557 b x = 1.189 y 15 b i, ii. Domain = R − y. e. x. (–6, 0) 0 (0, –4). x. 0. (–1, 0). x=5 y = f (–x). y = log10 (5 – x). y = f (x). (0, log10 5) x. 0. (–1, 0). (4, 0). x. (1, 0). y = –f(x). Domain = (−∞, 5) f. 0. y. iii, iv. y. y = f(3x) y = f (x). 1 3. 1 ,0 4. 0. x. 0. (1, 0) 3 y=f. x. x 3. y = 2 log2 2x + 2. Domain = R +. Exercise 5E. y. g. y = –2 log2 (3x) 1,0 3. x. 0. 3 a = 200, b = 500. 4 a = 250, b =. 5 a = 3, b = 5. 6 a = 2, b =. 7 a = 2, b = 4. Domain = R + h. 1 a = 2, b = 4 14 14 2a= ,b= (a ≈ 8.148, e−1 1−e b ≈ −8.148). y. 9 b = 1, a =. x = –5. 10 a = 1 –5100, 0. 0. y = log10 (–x – 5) + 2. Domain = (−∞, −5). 1 loge 5 3. 1 loge 5 3 8 a = 2, b = 3. 2 , c = 8 (a ≈ 2.885) loge 2. 2 ,b = 4 loge 2. x. Exercise 5F 1 a 2.58 e −2.32 i 2.89 m −6.21. b −0.32 f −0.68 j −1.70 n 2.38. c 2.18 g −2.15 k −4.42 o 2.80. d 1.16 h −1.38 l 5.76.

(68) P1: FXS/ABE. P2: FXS. 0521665175ans-1.xml. Answers. 718. CUAT018-EVANS. November 3, 2005. 16:7. Essential Mathematical Methods 3 & 4 CAS 2 a x < 2.81 b x > 1.63 c x < −0.68 d x ≤ 3.89 e x ≥ 0.57 loge 5 3ax = ≈ 2.3219 loge 2 loge 7 bx= ≈ 1.7712 loge 3 loge 8 c x = 0 or x = ≈ 1.8928 loge 3 4 a 10 b3 c8 5 a 2.9656 b 5.8329 c 2.0850 d 2.8551 e −4.6674 f −0.8273 6 0.544 b 549.3 2 1 9 a 9u c bu+ u 2 2 10 625 11 p. g f −1 (x) = 10x − 1; domain = R, range = (−1, ∞) 1 h f −1 (x) = loge x + 1; 2 domain = (0, ∞), range = R y 4 a, c f –1. y=x. y=1. f x. 0 x=1. b. f −1 (x) = − loge (1 − x), domain = (−∞, 1). 5 a, c. y. Exercise 5G. y=x. f. x = –3 (0, 2). 0 (2, 0) f –1. 1. y y=x. y = –3. (0, 4) y=3. (4, 0) x=3. b. x. 0. y = f –1(x). −x. f (x) = e + 3 f −1 (x) = − loge (x − 3). 1 x +3 loge , 2 5 domain = (−3, ∞) f −1 (x) =. y. 6 a, c. f –1. y. 2. x. 1 0, 2 loge 0.6. y = f (x). f x=1 y = f –1(x) y=x. (0, 2). 0, e. –. 1 2. 0. y = f(x) y=x. y=1 0. (2, 0). x. f (x) = loge (x − 1) f −1 (x) = e x + 1 1 3 a f −1 (x) = e x ; domain = R, 2 range = R + 1 x−1 b f −1 (x) = e 3 ; 2 domain = R, range = R + c f −1 (x) = loge (x − 2); domain = (2, ∞), range = R d f −1 (x) = loge (x) − 2; domain = R + , range = R 1 e f −1 (x) = (e x − 1); domain R, 2 1 range = − , ∞ 2 1 x f f −1 (x) = e 4 − 2 ; domain R, 3 2 range = − , ∞ 3. b. f −1 (x) = e. 7ax =e cn=. –. 1 2,. x 0. x−1 2 , range. y−5 2. loge. e. y. a loge x. = R+ 1 P b x = − loge 6 A d x = log10. y. 1 5−y ex= e 3 2⎛ y ⎞ log e 1⎜ 6 ⎟ fn= ⎝ ⎠ 2 loge x 1 y (e + 1) 2 5 h x = loge 5 −y x +4 −1 8 a f (x) = loge 2 b (0.895, 0.895), (−3.962, −3.962) gx=. x−4. 9 a f −1 (x) = e 2 − 3 b (8.964, 8.964), (−2.969, −2.969). 5.

(69) P1: FXS/ABE. P2: FXS. 0521665175ans-1.xml. CUAT018-EVANS. November 3, 2005. 16:7. Answers y = ex. y. y. b. y=x. y = loge x. 1 0. f (x) = 10 –x + 1 (0, 2). x. 1. y=1. ii. y y=. x–3 e2. x. 0. y=x. c. y. y = 2 loge (x) + 3. h(x) =. 1 x (e – 1) 2. –3 e2. x 0. x. 0. y=–. –3 e2. iii. y. y. d. y = 10 x. 1 2. y=x y=2 (0, 1) x. 0. 1. y = log10 x. 0. (–loge 2, 0). x. 1. y. e. f(x) = loge (2x + 1). b f (x) and g(x) are inverse functions. x. 0. Exercise 5H. 1. 1 m = 0.094, d0 = 41.9237 2 k = 0.349, N0 = 50.25 3 a i N0 = 20 000 ii −0.223 b 6.2 years 4 a M0 = 10, k = 4.95 × 10−3 b 7.07 grams c 325 days. x=–2. f. y x=1. x. 0 (1 + e –1, 0). Multiple-choice questions 1C 6C. 2D 7B. 3B 8A. 4E 9C. h(x) = loge (x – 1) + 1. g. y. 5A 10 D. g(x) = –loge (x – 1) (2, 0). Short-answer questions (technology-free) y. 1a. x. 0. x=1 y. h. f (x) = –loge (1 – x). f (x) = e x – 2 (loge 2, 0) x. 0. 0. x. (0, –1) y = –2 x=1. Answers. 10 a i. 719.

(70) P1: FXS/ABE. P2: FXS. 0521665175ans-2.xml. Answers. 720. CUAT018-EVANS. November 3, 2005. 9:27. Essential Mathematical Methods 3 & 4 CAS 1 loge (x + 1) 2 x b f −1 : R → R, f −1 (x) = e 3 + 2 c f −1 : R → R, f −1 (x) = 10x − 1 d f −1 : (2, ∞) → R, f −1 (x) = log2 (x − 1) 3 a y = e2 x b y = 10x c y = 16x 3 e3 x5 ey= dy= f y = e2x−3 x 10 loge 11 loge (0.8) 4ax = bx= loge 3 loge 2 loge 3 cx= loge ( 23 ) 2 1 bx= cx= 5ax =1 3 20 d x = log10 3 or x = log10 4 2 a f −1 : (−1, ∞) → R, f −1 (x) =. 2. 6 a = 2, b = 2−3 − 1 1 287 8 loge 9 2a 3 4 11 a = loge 5, b = 5, k = 2. 6. 7 10 5 − 1. 8 k = −0.5, A0 = 100 9a x. x=8. b i 0 grams ii 2.64 grams iii 6.92 grams c 10.4 minutes 10 a k = 0.235 b 22.7◦ c 7.17 minutes 11 a N(t) 20e10. 10 R. 10e10. Extended-response questions 1 a 73.5366◦ C 2 a 770 3 a k = 22 497, = 0.22 4 a A = 65 000, p = 0.064 5a y. y=. 00 10 0. b 59.5946 b 1840 b $11 627 b $47 200. 00 t + 10. 15. b i (12.210, 22 209.62) ii t = 12.21 iii 22 210 c ii (12.21, 12.21) d c = 0.52 1 6 a iii a = or a = 1 2 iv If a = 1, e−2B = 1 and B = 0, 1 1 A ∈ R + . If a = , B = loge 2. 2 2 v A = 20 000 n b. 20 000. t. 2 loge 10 loge (0.1) 1 = ≈ 6.644 1 loge 2 loge 2 2 After 6.65 hours the population is 18 000. 7 a 75 b 2.37 c 0.646 c. (0, 20) 0. t. 50 70. b i N (10) = 147.78 ii N (40) = 59 619.16 iii N (60) = 20e10 = 440 529.32 iv N (80) = 220 274.66 c i 25 days ii 35 days. Exercise 6A t. 0. y = 10e10. Chapter 6. y = 8000 + 3 × 2t 0. t. 0. 5 18 17 d 9 2 a 60◦ d 140◦ 3 a 45.84◦ d 226.89◦ 4 a 0.65 d 2.13 1a. 34 45 7 e 3 b 150◦ e 630◦ b 93.97◦ e 239.50◦ b 1.29 e 5.93 b. 25 18 49 f 18 c 240◦ f 252◦ c 143.24◦ f 340.91◦ c 2.01 f 2.31 c. Exercise 6B 1 a 0, 1 b −1, 0 d 1, 0 e 0, −1 2a0 b Not defined d0 e Not defined 3 a −1 b0 d0 e1 4 a 0.99 b 0.52 d 0.92 e −0.67 g −0.99 h 0.44 j −2.57 k 0.95. c −1, 0 f 0, 1 c Not defined c −1 f1 c −0.87 f −0.23 i −34.23 l 0.75.

(71) P1: FXS/ABE. P2: FXS. 0521665175ans-2.xml. CUAT018-EVANS. November 3, 2005. 9:27. Answers d. 1 a 0.52 b −0.68 e −0.52 f 0.68 2 a 0.64, 2.5 c 3.61, 5.816 3 a 17.46, 162.54 c 233.13, 306.87. c 0.52 d 0.4 g −0.4 h −0.68 b 0.64, 5.64 d 1.77, 4.51 b 66.42, 293.58 d 120, 240. e f g h. Exercise 6D 1 0.6 3 61 7. i. 2 0.6. 3 −0.7. 4 0.3. 7 −0.3. 8 0.6. 9 −0.6 10 −0.3. 5 −0.3. Exercise 6E √. 1 1 3 ,− ,−√ 2 2 3 1 1 c √ , √ ,1 2 2 1 1 e √ , √ ,1 2 2. 1a. 1 1 b −√ ,−√ ,1 2 2 √ 1 √ 3 ,− , 3 d− 2 2 √ 3 1 1 ,−√ f− , 2 2 3 √ 1 3 d c− 2 2 1 1 h g −√ 2 2 √ 1 3 k l 2 2 1 1 o√ p− 2 2. 1 1 b− 2a √ 2 2 1 1 f −√ e− 2 √ 2 1 3 i j 2 2 1 1 m√ n− 2 2 b 60◦ , 120◦ 3 a 60◦ , 300◦ d 120◦ , 240◦ e 60◦ , 120◦ 5 b − or 4 a − or − 6 6 6 6 5 5 b 5a 6 6 d +b e +b 2 2 4 7 b 6a 3 6 3 d −b eb+ 2 3 5 3 7a , b , 4 4 4 4 7 7 11 e , , d 4 4 6 6 5 7 13 15 , , , 8a 8 8 8 8 5 7 17 19 29 b , , , , , 18 18 18 18 18 5 13 17 c , , , 12 12 12 12. 7 3 5 17 23 , , , , , 12 12 4 4 12 12 5 13 17 25 29 , , , , , 18 18 18 18 18 18 3 9 11 , , , 8 8 8 8 5 7 17 19 29 31 , , , , , 18 18 18 18 18 18 5 7 13 5 7 23 , , , , , 12 12 12 4 4 12 2 4 5 , , , 3 3 3 3. c 210◦ , 330◦ f 150◦ , 210◦ −5 5 c or 6 6 2 c 3. Exercise 6F 1 a i 2 2 bi 3 ci . ii 3 1 ii 2 ii 3. g i 4 y 2a 2. π 3. 0. θ. 2π 3. –2. Amplitude = 2 2 Period = 3 y. b 2. 7 c 6. π. π. 4. 2. 3π 4. π. θ. –2. Amplitude = 2 Period = y. c 3 0. 31 18. ii 2 1 ii 2 ii 2. d i 6 ei 2 f i 2. 0. fb+ 5 7 c , 6 6 2 4 f , 3 3. ii 3. –3. 3π 3π 2. 9π 2. Amplitude = 3 Period = 6. 6π. θ. Answers. Exercise 6C. 721.

(72) P1: FXS/ABE. P2: FXS. 0521665175ans-2.xml. CUAT018-EVANS. 722. November 3, 2005. 9:27. Answers. Essential Mathematical Methods 3 & 4 CAS y. d. Exercise 6G. 1 3. y. 1a. 0 –1 3. π. π. 4. 2. θ. π. 3π 4. 2. Amplitude =. Period = . 1 3. y. e. π 8. –3. π 4. θ. π 2. 3π 8. θ. 7π 3. y. b 1. Amplitude = 3 Period = 2. π 2. 0. y. 3. 4π 3. –2 Period = 2 Amplitude = 2 Range = [−2, 2]. 3. 0. π 3. 0. θ. π. 5 –5π 6. –π 2. –π. –2π 3. π 2. 0 –π 3. π 6. –π 6. π 3. –1. 5π 6 2π 3. π. x. Period = Amplitude = 1 Range = [−1, 1]. –5. y. c. y. 4 1 2. –π. –π 2. 3. 0 –. π 2. π. 3π 2. 2π. x. 1 2. 0. π 3. –2. x 6 y = 2 sin 3 x 1 7 y = cos 2 3 1 x 8 y = sin 2 2. θ. –3. 2 5π 3 2π π 4π 2π 3 3. 3π 4. Period = Amplitude = 3 Range = [−3, 3]. y. 5. π 4. 0. –π 4. x. y. d √3 0. π 2. 5π 6. –√3. 2 3 √ Amplitude = 3√ Range = [− 3, 3] Period =. 7π 6. θ.

(73) P1: FXS/ABE. P2: FXS. 0521665175ans-2.xml. CUAT018-EVANS. November 3, 2005. 9:27. Answers. 5 4 3 2 1. 3 2 1 0 –1. π 6. π 3. 2 3 Amplitude = 2 Range = [−1, 3]. x. π 2. 2π 3. y. 2a. 2 1. b. 0 –1 –2 –3 –4. x. π. π 2. c 3a. b. Period = Amplitude = 3 Range = [−4, 2] y. g 4. c. 3 2 1 0 –1. π 6. 7π 6. θ. Period = √ Amplitude =√ 2 √ Range = [− 2 + 2, 2 + 2]. d. y. h. π 2. 3π 2. 0 –1. Period =. f. y. i. 7. θ. π. Period = Amplitude = 3 Range = [−1, 5] 1 1 y = cos x− 2 3 4 y = 2 cos x − 4 1 y = − cos x − 3 3 Dilation of factor 3 from the x-axis 1 Dilation of factor from the y-axis 2 Reflection in the x-axis Dilation of factor 3 from the x-axis 1 Dilation of factor from the y-axis 2 Reflection in the x-axis Translation of units in the positive 3 direction of the x-axis Dilation of factor 3 from the x-axis 1 Dilation of factor from the y-axis 2 Translation of units in the positive 3 direction of the x-axis Translation of 2 units in the positive direction of the y-axis Dilation of factor 2 from the x-axis 1 Dilation of factor from the y-axis 2 Reflection in the x-axis Translation of units in the positive 3 direction of the x-axis Translation of 5 units in the positive direction of the y-axis. 3. Exercise 6H 0 –1. π. π. x. 1a. y. 2. Period = Amplitude = 4 Range = [−1, 7]. 3 2 1 0 –1. 2π π 3. . Intercepts:. 4π 3. x 2π. 4 2 , 0 , , 0 3 3. Answers. y. e. 723.

(74) P1: FXS/ABE. P2: FXS. 0521665175ans-2.xml. Answers. 724. CUAT018-EVANS. November 3, 2005. 9:27. Essential Mathematical Methods 3 & 4 CAS y. b. y. d (0, 2 – √3 ). 1 0 –1 –2 –3. π 2. π. 3π 2. 2π. (0, –2 – √3). (0, √2 – 1) 0. (0, –√2 – 1). Intercepts:. π 4. π. y 1 + √2 (0, 2). (2π, 2). x. 3π 2. 7 , 0 , , 0 4 4. 0. . π 2. 1 – √2. π. . Intercepts: (, 0),. y. 2a –11π –2π 6. –7π 6. –π 2. π 6. –π 6. π 2. 5π 6. 3π 2. 2π. x. 0 (2π, –2). –2 (–2π, –2) –4 y. b –23π 12 –2π. 3 –15π 12. π 12. –7π 12 –π. 9π 12. 17π 12. 0. 2π. 1 – √2. –1. (–2π, 1 – √2). x. (2π, 1 – √2) y. c –2π. –π. π. 0. 2π. x. –1 (2π, –3). –3. (–2π, –3). –5 y. d 3 (–2π, 1) –2π –11π 6. x. Intercepts: (0, 0), (2, 0) e. y 7π 4 2π. 2π. –4. 11 , 0 , , 0 , Intercepts: 12 12 13 23 , 0 , , 0 12 12 . c. π. 0. x. 1 –π –7π 6. 0 –1. (2π, 1) π 6. π 5π 6. 2π. x. x 2π. 3 ,0 2.

(75) P1: FXS/ABE. P2: FXS. 0521665175ans-2.xml. CUAT018-EVANS. November 3, 2005. 9:27. Answers e. 725. –23π 12. –19π –15π 12 12. –11π 12. –7π 12. –π 4. π 12. 2. 5π 12. 9π 12. 13π 12. 17π 12. 21π 12. 0 π. –π. –2π. –√2 –2. (–2π, –√2). Answers. y. 2π (2π, –√2). x. y. f (–2π, √2). 2. (2π, √2). √2. 0. –21π 12. –17π –13π 12 12. x. π. –π. –2π. –9π 12. –5π 12. –2. –π 12. 3π 12. 7π 12. 2π. 11π 12. 15π 12. 19π 12. 23π 12. y. g 2. 0. –π. –2π. x. π. 2π. y. h –π. –2π (–2π, –1). –π 2. x. π. 0. 2π (2π, –1). 3π 2. –1 –2. y. 3. Exercise 6I y. 1. 1. y = sin θ + 2cos θ 2 –4. –2 –1. 0. –2. 2. 1. –4 –3. 4. 2. 3. 0. x. x –1. 1 y = cos2θ – sin θ 2. –2. y. 4. y. 2. 4. y = 2cos2θ + 3sin2θ 2 –2. y = 3 cos θ + sin 2θ. 2. –1. 1 0 –2 –4. 2. 0 x. –4 –3 –2 –1 –2. x 1. 2. 3. 4.

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