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389 Answers 389Exercise 12K 1 2t – 3 2 2Ο€ 3 – 4 _ 3 4 48Ο€ 5 18 6 a Let x = width of garden. x + 2y = 80 A = xy = 2y b 20 m Γ— 40 m, 800 m2 7 a 2Ο€ r2 + 2Ο€rh = 600Ο€ β‡’ h = 300 – r2 ________ r V = Ο€ r2h = Ο€r (300 βˆ’ r2) = 300Ο€r βˆ’ Ο€r3 b 2000Ο€ cm3 8 a Let ΞΈ = angle of sector . Ο€r2 Γ— ΞΈ ____ 360 ...
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390 Answers 390 Full worked solutions are available in SolutionBank. Online b R = xy + Ο€ __ 2 ( x __ 2 ) 2 = x (1 – x __ 2 – Ο€x ___ 4 ) + Ο€x2 ____ 8 = x – x2 __ 2 – Ο€x2 ____ 4 + Ο€x2 ____ 8 = x __ 8 (8 – 4x – Ο€x) c 2 _____ 4 + Ο€ m2 (0.280 m2) 28 a Ο€x...
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391 Answers 3917 a 1 _ 3 x3 + 2x βˆ’ 1 __ x + c b 1 _ 2 x2 + 8 _ 3 x 3 _ 2 + 4x + c c 2 x 1 _ 2 + 4 _ 3 x 3 _ 2 + c 8 a 3 _ 5 x 5 _ 3 βˆ’ 2 ___ x 2 + c b βˆ’ 1 __ x 2 βˆ’ 1 _ x + 3x + c c 1 __ x x4 βˆ’ 1 _ 3 x3 + 3 _ 2 x...
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392 Answers 392 Full worked solutions are available in SolutionBank. Online b A has coordinates ( βˆ’5 + √ ___ 10 _________ 3 , βˆ’80 + 37 √ ___ 10 _____________ 27 ) The roots at 1 correspond to point B. The root βˆ’5 βˆ’ √ ___ 10 _________ 3 gives a point on the curve to the left of βˆ’2 be...
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393 Answers 3936 k = 3, a = 2 7 a As x increases, y decreases b p = 1.2, q = 0.2 Challenge Oy y = 2x – 2 + 5 x21 4 Exercise 14B 1 a 2.7183 b 54.5982 c 0.0000 d 1.2214 2 a Oy x1y = ex e = 2.71828… e3 = 20.08553… 3 a Oy x2y = ex + 1 y = 1 b y xy = 4e–2x O4 c y x y = –3–1y = 2ex – 3 O d y x4 y = 4 – ex3 e y xy...
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394 Answers 394 Full worked solutions are available in SolutionBank. Online 2 a 30 000 b 38 221 c tP (thousands) 30 10094 OP = 20 + et 50 d Model predicts population of the country to be over 200 million, this is highly unlikely and by 2500 new factors are likely to affect population growth. Model not valid fo...
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395 Answers 395Exercise 14G 1 a ln 6 b 1 _ 2 ln 11 c 3 – ln 20 d 1 _ 4 ln ( 1 _ 3 ) e 1 _ 2 ln 3 – 3 f 5 – ln 19 2 a e2 b e _ 4 c 1 _ 2 e4 – 3 _ 2 d 1 _ 6 ( e 5 _ 2 + 2) e 18 – e 1 __ 2 f 2, 5 3 a ln 2, ln 6 b 1 _ 2 ln 2, 0 c e3, eβˆ’5 d l...
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396 Answers 396 Full worked solutions are available in SolutionBank. Online Mixed exercise 1 a Oy x1y = 2–x y = 0 b Oy x4y = 5ex – 1 y = –1 c Oy x1y = ln x x = 0 2 a 2 loga p + loga q b loga p = 4, loga q = 1 3 a 1 _ 4 p b 3 _ 4 p + 1 4 a 2.26 b 1.27 c 7.02 5 a 4x – 2x + 1 – 15 = 0 22x – 2 Γ— 2x...
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397 Answers 397Challenge a y = 9x = 32x, log3(y) = 2x b y2 = (9x)2 = 92x, log9(y2) = 2x c x = βˆ’ 1 __ 3 or x = βˆ’2 Review exercise 3 1 βˆ’4.5 2 √ __ 7 3 a All equal to √ ____ 145 b (x βˆ’ 1)2 + (y + 3)2 = 145 4 a βˆ’2i βˆ’ 8j b | ⟢ AB | = | ⟢ AC | = √ ___ 85 c | ⟢ BC | = √ ___ 68 cos ...
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398 Answers 398 Full worked solutions are available in SolutionBank. Online Practice paper 1 a 1 __ 3 b 5 √ __ 2 2 y = 2 __ 3 x + 8 __ 3 3 a error 1: = βˆ’ 3 ___ √ __ x = βˆ’3 x βˆ’ 1 _ 2 , not βˆ’3 x 1 _ 2 error 2: [ x5 __ 5 βˆ’ 2 x 3 __ 2 + 2x] 2 1 = ( 3...
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399 Index 399elimination 39–40 equations circle 117–119 normal 268–269 solving using logarithms 324–325 straight line 90, 93–95 tangent 268–269 equilater al triangles 208 exhaustion, proof by 150 expanding brack ets 4–6, 163–164 exponential functions 312–318 gradient functions of 314–316 graphs ...
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or 30Β°, 45Β° and 60Β° 208 value f or multiples of 90Β° 192, 204 cos ΞΈ = k 213–215 cos (ΞΈ + Ξ±) = k 218 cos nΞΈ = k 217 cosβˆ’1 ΞΈ 213 cosine rule 174–176, 187–189 counter-e xamples 150–151 critical v alues 49 cube roots 10 cubic functions 60–62 cubic graphs 60–62 compared with quadra tic graphs 68 curves ...
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400 Index 400tan ΞΈ = p 213–215 tan (ΞΈ + Ξ±) = p 218 tan nΞΈ = p 217 tanβˆ’1 ΞΈ 213 tangents 43, 123–125, 256 equations 268–269 gradients 256, 268–269 theorems 146 transf ormations 71–80 to functions 79–80, 194–197 translations 71–73 tra peziums, areas 303 triangle law 231 triangles areas 100–101, 185–186,...
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lising denominators 13–14 real n umbers 7, 232key points summaries algebr aic expressions 17 algebr aic methods 156 binomial expansion 172 circles 135–136 differentiation 286 equations 58 exponentials and lo garithms 336–337 graphs 84 inequalities 58 integration 310 quadratics 37 straight l...
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