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AnswersChapter 11.1Exercise 1Ae 6 f 1_ 2 g 21 h 10 xNB ( 1 x__) = 4 2x so (c) is y = 4 2x 44 a 625 b 9 c 7 d 21.4Exercise 1Dd log 6 ( 6881 ) e log 10 1202 a log 2 8 = 3 b log 6 36 = 2c log 12 144 = 2 d log 8 2 = 1_ 3 e log 10 10 = 116 3 a 3 log a x 1 4 log a y 1 log a ze 6 1_ 3 f ___ 1125 g 1 h 66b 5 log a x 2 2 log a yi 6 12564 j 9_ 4 k 5_ 6 l __ 64 49 c 2 1 2 log a xd log a x 1 1_ 2 log a y 2 log a ze 1_ 2 1 1_ 2 log a1.57 2 6√ __2 3 5√ __2 2 5 3√ ___10 6 √ __3 Exercise 1Ea 1_ 2, 512 e __ 116 , 1_ 4 ___ 5 5 17 √ _______ 11 11 18 √ _____ 2 2 d 1.66___ 5 5 20 __ 1 21 __ 1 4 a 1_ 2, 512 b 116 , 1_ 4 c 2.522 4____ 13 13 23 __ 1 1.6 3Exercise 1F1 y1y = (4) y = 6 xy = 4 x(b)(a)1(c)1 a x 7 b 6x 5 c 2p 2 d 3x 22e k 5 f y 10 g 5x 8 h p 2i 2a 3 j 2p 27 k 6a 29 l 3a 2 b 22m 27x 8 n 24x 11 o 63a 12 p 32y 6q 4a 6 r 6a 122 a x 5 b x 22 c x 4d x 3 e x 5 f 12x 0 = 12g 3x 1_ 2 h 5x i 6 x 213 a 65 b 69 c 3 d 11.2Exercise 1B1 2√ __4 4√ __7 √ __3 8 6 √ __5 9 7√ __2 10 12√ __7 11 23√ __7 12 9 √ __5 13 23√ __5 14 2 15 19√ __3 √16 __19 √ __22 √ ___1.3Exercise 1C1 a log 4 256 5 4 b log 3 ( 1_ 9) 5 22c log 10 1 000 000 5 6 d log 11 11 5 12 a 2 4 = 16 b 5 2 = 25c 9 1_ 2 = 3 d 5 21 = 0.2e 10 5 = 100 0003 a 3 b 2 c 7 d 15 a 1.30 b 0.602 c 3.85 d 20.1056 a 1.04 b 1.55 c 20.523 d 3.001 a log 2 21 b log 2 9 c log 5 801 a 2.460 b 3.465 c 4.248d 0.458 e 0.7742 a 1.27 b 2.09 c 0.7213 a 6.23 b 2.10 c 0.431Further Pure Mathematics© Pearson Education Ltd 20101

Chapter 33.1Exercise 3A21 a x 2 1 5x 1 3 b x 2 1 x2 9c x 2 2 3x 1 7 d x 2 2 3x 1 2e x 2 2 3x – 22 a 6x 2 1 3x 1 2 b 3x 2 1 2x2 2c 2x 2 2 2x 2 7 d 23x 2 1 5x 2 7e 25x 2 1 3x 1 52x 3 + 6xb 2x3 + 5x 2 2x 2_________________ 2 5x + 1 = x2x 2 1+ 3x 2 13x 3 + 2x 2 2 3x 2 2 = (3x 1 2)(x 2 2 1) 3x3 + 2x 2 2x 2_________________ 2 3x 2 2 = x(3x + 2)2 1+3x 23 a 2x 3 + 5x 2 2 5x + 1 = (2x 2 1)(x 2 + 3x 2 1)c 6x 3 + x 2 2 7x 1 2 = (3x 2 1)(2x 2 1 x 2 2) 6x3 + x 2 22x 2________________ 2 7x 1 2 = 2x3x 2 12 1 x 2 222x 2d 4x 3 + 4x 2 1 5x 1 12 = (2x 1 3)(2x 2 2 x 1 4) 4x3 + 4x 2 16x 2___________________ 1 5x 1 12= 2x2x 1 32 2 x 1 416x 2e 2x 3 + 7x 2 1 7x 1 2 = (2x 1 1)(x 2 1 3x 1 2)1x 2_________________ 2x3 + 7x 2 1 7x 1 22x 2 1= x 2 1 3x 1 23.3Exercise 3C1 a 27 b 26 c 0 d 12 273 184 306 297 8 827 8 a = 5, b = 289 p 5 8, q 5 33.4Exercise 3D1 a x 5 5, y 5 6 or x 5 6, y 5 5b x 5 0, y 5 1 or x 5 4_ 5 , y 5 3_ 5 c x 5 21, y 5 23 or x 5 1, y 5 3d x 5 4 1_ 2 , y 5 41_ 2or x 5 6, y 5 3e a 5 1, b 5 5 or a 5 3, b 5 21f u 5 1 1_ 2, v 5 4 or u 5 2, v 5 32 (211, 215) and (3, 21)3 (21 1_ 6 , 241_ 2) and (2, 5)4 a x = 21 1_ 2 , y = 53_ 4or x = 3, y = 21b x = 3, y = 1_ 2 or x = 61_ 3 , y = 225_ 6 5 a x 5 3 1 √ ___13 , y 5 23 1 √ ___13 or x 5 3 2 √ ___13 ,y 5 23 2 √ ___13 b x 5 2 2 3√ __5 , y 5 3 1 2√ __5 or x 5 2 1 3√ __5 ,y 5 3 2 2√ __5 Further Pure Mathematics3.2 Exercise 3B1 (x 2 1)(x 1 3)(x 1 4)2 (x 1 1)(x 1 7)(x 2 5)3 (x 2 5)(x 2 4)(x 1 2)4 (x 2 2)(2x 2 1)(x 1 4)5 a (x 1 1)(x 2 5)(x 2 6)b (x 2 2)(x 1 1)(x 1 2)c (x 2 5)(x 1 3)(x 2 2)6 a (x 2 1)(x 1 3)(2x 1 1)b (x 2 3)(x 2 5)(2x 2 1)c (x 1 1)(x 1 2)(3x 2 1)d (x 1 2)(2x 2 1)(3x 1 1)e (x 2 2)(2x 2 5)(2x 1 3)7 28 2169 p 5 3, q 5 7© Pearson Education Ltd 20103.5Exercise 3E1 a x , 4 b x > 7 c x . 2 1_ 2 d x < 23 e x , 11 f x , 2 3_ 5 g x . 212 h x , 1 i x < ??j 8 k x . 1 1_ 7 2 a x > 3 b x , 1 c x < 23 1_ 4 d x , 18 e x . 3 f x > 4 2_ 5 g x , 4 h x . 27 i x < 2 1_ 2 j x > 3_ 4 3 a x . 2 1_ 2 b 2 , x , 4 c 21_ 2 , x , 3d No values e x 5 45

Further Pure Mathematics3.6Exercise 3F1 a 3 , x , 8 b 24 , x , 3c x , 22, x . 5 d x < 24, x > 23e 2 1_ 2 , x , 7 f x , 22, x . 21_ 2 g 1_ 2 , x , 11_ 2 h x , 1_ 3 , x . 2i 23 , x , 3 jk x , 0, x . 5 lx , 22 1_ 2 , x . 2_ 3 21 1_ 2 , x , 02 a 25 , x , 2 b x , 21, x . 1c 2 , x , 1 1_ d 23 , x , 1_ 4 3.7Exercise 3G1 23 < x , 42 y , 2 or y > 53 2y 1 x > 10 or 2y 1 x < 44 22 < 2x 2 y < 25 4x 1 3y < 12, y > 0 and y , 2x 1 46 y . 2 ___ 3x2 2 37 x > 0, y > 0, y , 2______ 3x2 1 9 and y < 2 ______ 2x3 1 6 8 y > 0, y < x + 2, y < 2x 2 2 and y < 18 2 2x91213 a Let a = no. of adults, and c = no. of children.a + c < 14 (no more than 14passengers)12a + 8c > 72(money raised must covercost of £72)c . a(more children thanadults)a > 2(at least 2 adults)c141210+864Ma = 2PNQa = c22 4+6 8 10 12 143a + 2c = 18 a + c = 14d1011NB 12a + 8c > 72 requires line 3a + 2c = 18b To find smallest sized group you need to considerpoints close to M and NM(2, 6) is 2 adults and 6 childrenPoints close to N are (3, 5) and (4, 5)So the smallest sized group is 8: 2 adults and 6 childrenor 3 adults and 5 children.c To find the maximum amount of money that can bemade you need to consider points close to P and QP(2, 12) raises 2 3 12 1 12 3 8 = £120Q(7, 7) is not in the region ( c . a) but (6, 8) is on d(6, 8) raises 6 3 12 1 8 3 8 = £136So the maximum amount available for refreshments is£64 from taking 6 adults and 8 children6© Pearson Education Ltd 2010

14 by282824+24+20163b + 2a = 4012 R+20R1612S8+844 8Q12 16 20 24 28+++10a + 14b = 1404a + 5b = 100Let a = no. of machine Ab = no. of machine B4a + 5b

c y = x(x + 1) 2 d y = x(x + 1)(3 2 x)yyey10x01 3x0xe y = x 2 (x 2 1) f y = x(1 2 x)(1 1 x)yy2 a y by270 1x1 0 1x30x03xg y = 3x(2x 2 1)(2x 1 1) h y = x(x + 1)(x 2 2)yy10 12 2x01 2i y = x(x 2 3)(x 1 3) j y = x 2 (x 2 9)yyxcy101xd27y802xFurther Pure Mathematics03 3x0 9xey4.2Exercise 4B1 a byy18012x0x0x4.3Exercise 4C1ycydyy4x0x0x0y2xx© Pearson Education Ltd 20109

2ybiyy2xy x(x 2)0yx2x20y3xx3y2xyii 1iii x(x 1 2) 5 2__ 3 xc i yFurther Pure Mathematics450y0yyyy3xxx4x8xdy x 2 1 0 1ii 3iii x 2 5 (x 1 1)(x 2 1) 2iy0y x 2 (1 x)1yxy (x 1)(x 1) 2x2x0yy3xx8xeii 2iii x 2 (1 2 x) 5 2__ 2 xiy4.4Exercise 4D1 a i yy x 2y1x04y x(x 4)x1 0 1xii 1iii x(x 2 4) 5__ 1 xy = x(x 2 2 1)ii 3iii x 2 5 x(x 2 2 1)10© Pearson Education Ltd 2010

fiyj i yy x(x 4)y x(x 2)04x21y x0xii 3y x 3giii x(x 2 4) 5 2__ 1 ii 3xiii 2x 3 5 2x(x 1 2)iy2 a yy x 2 (x 4)y (x 2) 320x04xy x(4 x)y x(x 4)b (0, 0); (4, 0); (21, 25)ii 13 a yiii x(x 2 4) 5 (x 2 2) 3h iy12.5 0y2xy x(1 x) 30xy x(2x 5)b (0, 0); (2, 18); (22, 22)y x 34 ayii 2iii 2x 3 5 2__ 2 xy (x 1)(x 1)i i yy (x 1) 3y x 21 0 1 x0xb (0, 21); (1, 0); (3, 8)y x 35 ayii 2y x 2Further Pure Mathematics0y27xxb (23, 9)© Pearson Education Ltd 201011

6 a yy x 2 2x11 ayy6x0 2 3x1 04xy x(x 2)(x 3)b (0, 0); (2, 0); (4, 8)7 a yy2xy x 2 (x 3)y x 3 3x 2 4xb (0, 0); (22, 212); (5, 30)12 a yy 14x 203xFurther Pure Mathematicsb8 aOnly 2 intersectionsy1 0 1y (x 1) 3b Only 1 intersection9 a yy0 11xxy x(x 1) 2b Graphs do not intersect10 a yy x(x 2) 2y 3x(x 1)x21 0 1 2b (0, 2); (23, 240); (5, 72)13 a y2 0 2y (x 2 1)(x 2)xy (x 2)(x 2) 2xy x 2 8b (0, 28); (1, 29); (24, 224)4.5Exercise 4E1y120 122xOxy 1 4x 24b1, since graphs only cross once12© Pearson Education Ltd 2010

2yTransformation f(x + 1) givesy1x1 O x1y = _____x + 1 3yFinally transformation f(x) + 3 givesyy = 3xy = 3 +_____ 1x + 1 O1 1 2xx = 1 Asymptotes x = 21, y = 3y455yO1 1 2yx8 y =_____ 1x 2 2 2 1 Start with y = __ 1 xTransformation f(x 2 2) givesy2xy =_____ 1x 2 2 Finally f(x) 2 1 givesyxFurther Pure Mathematics6O5150 330yx12x1y = _____x 2 2 2 1Asymptotes x = 2, y = 2119 y = 2 + _____ Vertical asymptote is x = 1x 2 1(put denominator = 0)Horizontal asymptote is y 5 2 (let x ⇒ ∞)y21x = 0, y = 13 O xy1xAsymptotes x 5 1 andy 5 210 y = ______ 3 + 2x =___________(1 + x) + 12 = 2 + _____ 11 + x 1 + x1 + x 7 y = 3 +_____ 1x + 1 Start with y = __ 1 xxy321Oxx = 0, y = 3Asymptotes x = 21, y = 2© Pearson Education Ltd 201013

4.7Exercise 4G1 a x 5 1, y 5 4.21; x 5 5, y 5 3.16b y+5⇒ 1_ 3 ex = 2 2 2x 2 + 1_ 3 ex = 4 – 2x so draw y = 4 2 2xand intersection at 0.65 or 0.7 to 1sf3 a x 5 1, y 5 21ln1 5 2; x 5 4, y 5 3.39by5+4+y = 44y = x+3211 2y = 1 + x+++3 4 5 6c e – 1_ 2 x = 0.5 ⇒ 3 + 2e – 1_ 2 x = 3 + 2 3 0.5 = 4Draw y = 4 and intersection is at 1.35d x = 22 ln ( x 2 2 _____2 )⇒ e –x_ 2 =_____ x 2 22 ⇒ 2e – x_ 2 = x 2 2 3 + 2e – 1_ 2 x = 1 + xDraw y 5 1 1x and intersection at x 2.552 a x 5 0, y 5 2 1 1_ 3= 2.33; x = 2.5, y = 6.06by987+x32y1504+13⇒ 3lnx = 0 x 2 210410130+ + + +++y = 2.51 2 3 4 y = xc lnx = 0.5y 2⇒ 2 + lnx = 2.5 so draw y = 2.5Intersection 4 1 1.60d x = e x 2 2 ++ + + +++⇒ 2 + lnx = x so draw y =x2 1Intersection 3.1 to 1sf4 a x 5 30, y 5 2.60; x 5 75, y 5 1.981b y++++2 3 4y = 2.5++20 30 40 50 60 70 80 90 x+xxy = 2+Further Pure Mathematics+6+y = 6+22+y = 25+143+ +01100++20 30 40 50 60 70 80 90 x++21y = 4 – 2x–1 1 2 3 4 5 6c e x = 12 ⇒ 2 + 1_ 3 ex = 2 + 1_ 33 12 = 6Draw line y = 6 and intersection is at 2.45d x = ln (6 2 6x)⇒ e x = 6 2 6xxc+22 + 2 cosx = 5 sin 2x⇒ 2 = 5 sin 2x 2 2 cosxDraw y = 2and intersections at 25.1 and 74.6or 25 and 75 to 2sf© Pearson Education Ltd 201015

Mixed Exercise 4H1 a yy x 2 (x 2)by1.00.8+y =x40.602x0.4+0.22x x 21 2 3 4 5 6 7+xb x 5 0, 21, 2; points (0, 0), (2, 0), (21, 23)2 a y0.20.4+Further Pure MathematicsA10Bb A(23, 22) B(2, 3)c y = x 2 + 2x 2 5yy 1 x6xxy x 2 2x 53 a y = x 2 2 2x 2 3= (x 2 3)(x + 1)1b y = x 2 42 2x 1 4= (x 2 1) 2 3+ 314 a y = 2(x 2 2 4x + 3)= (3 2 x)(x 2 1) 3b y = 2(x 2 2 4x 1 5)= 2 [ (x 2 2) 2 +1 ]= 21 2 (x 2 2) 25 a y = 1_ 2 ex 1 4(x = 0, y = 4.5)y3yy15yy4.53 x1 3 x2xxy = 4x0.60.8c ln(x 2 1) = 0⇒ 1 2 ln(x 2 1) = 1 so draw y = 1Intersection at x = 2d x = 1 + e 1 + x_ 4 ln(x 2 1) = 1 + x_ 4 +⇒ x 2 1 = e 1 2 x_ 4 ⇒ 1 2 ln(x 2 1) = 1 2 (1 1 x_ 4 ) = x_ 4 so draw y = x_ 4 and intersection at ≈ 2.5Chapter 55.1Exercise 5A1 Arithmetic sequences are a, b, c, h, l2 a 23, 2n + 3 b 32, 3n + 2c 23, 27 2 3n d 35, 4n 2 5e 10x, nx f a + 9d, a + (n 2 1)d3 a £5800 b £(3800 1 200m)4 a 22 b 40 c 39d 46 e 18 f n5.2Exercise 5B+16b y = ln(x 1 1) 1 22(x = 0, y = 2)y = 0 ⇒ ln(x + 1) = 22 0.86⇒ x + 1 = e 22 x = e 22 2 1 ≈ 20.866 a x = 5 y = 1 2 ln4 = 20.39x = 6 y = 1 2 ln5 = 20.61© Pearson Education Ltd 2010yx1 a 78, 4n 2 2 b 42, 2n 1 2c 23, 83 2 3n d 39, 2n 2 1e 227, 33 2 3n f 59, 3n 2 1g 39p, (2n 2 1)p h 271x, (9 2 4n)x2 a 30 b 29 c 32d 31 e 221 f 773 d 5 64 a = 36, d = 23, 14th term5 246 x = 5; 25, 20, 157 7 3 1_ 2 , x 5 8

Further Pure Mathematics5 a p 5 5 b 210 c 280d Subtract 5 from the previous term,4 1 2 15x 1 90x 2 2 270x 3i.e. U n11 5 U n 2 5 (or U n 5 15 2 5n)6 1 2 0.6x 1 0.15x 2 2 0.02x 3 , 0.94148, accurate to 5 dpe The square numbers (U n 5 n 2 )7 a 220x 3 b 120x 3 c 1140x 3f Multiply the previous term by 1.2, i.e. U n11 5 1.2U n 8 b = 22(or U n 5 (1.2) n21 )Arithmetic sequences are:6.2a a 5 5, d 5 6b a 5 3, d 5 3Exercise 6Bc a 5 10, d 5 251 a 1 1 6x 1 12x 2 1 8x3, valid for all x2 a 81 b 860b 1 1 x 1 x 2 1 x 3 , |x|, 13 32c 1 1 1_ 4 a £13 780 b £42 1982 x 2 1_ 8 x2 1 116 x3 , |x| , 15 a a 5 25, d 5 23 b 23810d 1 2 6x 1 24x 2 2 80x 3 , |x| , 1_ 2 6 a 26 733 b 53 467e 1 2 x 2 x 2 2 5_ 3 x3 , |x| , 1_ 3 7 a 5 b 45f 1 2 15x 1__ 7528 a d = 5 b 591 1252 x3 , |x| , 110 9 b 11k 2 9_ 3 c 1.5 d 415g 1 2 x 1 5_ 8 x2 2 516 x3 , |x| , 410 a Not geometric b Geometric r = 1.5b Geometric r = 1_ h 1 2 2x 2 √1 ..., |x| , _____ 2 2 c Geometric r = 222 d Not geometric e Geometric r = 12 |x| < 1_ 2 11 a 0.8235 (4 dp), 10x (0.7) n21b 640, 5 3 2 n213 1 1___ 3x2 2 9_ 8 x2 1__ 2716 x3 , 10.148 891 88, accurate to 6 d.p.c 24, 4 3 (21) n214 a = ±8, ±160x 3d 3128 , 3 3 (2 1_ 2 )n216 1 2___ 9x +_____27x2+ _____27x3, x = 0.01, 955.339(1875)2 8 1612 a 4092 b 19.98 (2 dp)c 50 d 3.33 (2 dp)Mixed Exercise 6C13 a 9 b 8_ 3 c Doesn’t converge d 163 1 a p 5 16 b 270 c 2189014 b 60.72 c 182.252 a 1 2 20x 1 180x 2 2 960x 3d 3.16b 0.817 04, x = 0.0115 b 200 c 333 1_ 3 d 8.95 3 102416 a 76, 60.8 b 0.8763 a n 5 8 b __ 358 c 367 d 3804 a 1 + 24x + 264x 2 + 1760x 317 a 1, 1_ 3 , 2 1_ 9 b 1.2681618 a 0.8 b 10c 1.268 241 795c 50 d 0.189 (3sf)d 0.006 45% (3 sf)19 a 2 1_ 2 b 3_ 4, 22 c 145 a 1 2 12x 1 48x 2 2 64x 3 , all xb 1 1 2x 1 4x 2 1 8x 3 , |x| , 1_ 2 Chapter 6c 1 2 2x 1 6x 2 2 18x 3 , |x| , 1_ 3 6 1 2__ x 2___ x2 2____ x3 4 32 128 6.17 1 2 3_ 2 x2 1__ 278 x4 2 13516 x6Exercise 6A8 1 +__ x 2___ x22 8 + x3 ___,_____114516 512 1 1 1 8x 1 28x 2 1 56x 39 a n 5 22, a 5 32 1 2 12x 1 60x 2 2 160x 3b 21083 1 1 5x 1 454 x2 1 15x 3c |x| , 1_ 3 18© Pearson Education Ltd 2010

Chapter 77.1Exercise 7A1ab2 253 √ ____569 23.94 a d 2 a b a 1 b 1 cc a 1 b 2 d d a 1 b 1 c 2 d7.2Exercise 7Bc1 a 2a 1 2b b a 1 b c b 2 a2 a b 2 1_ 2a b b 2 3ac 3_ 2a 2 b d 2a 2 b3 a Yes (l 2) b Yes (l 4) c Nod Yes (l 21) e Yes (l 23) f No4 a l = 1_ 2, m = 23 b l = 22, m = 1c l = 1_ 4, m = 5 d l = 22, m = 21e l = 4, m = 8 1_ 2 5 a b 2 a, 5_ 6 (b 2 a), 1_ 6 a 1 5_ 6 bbc2 1_ 6 a 1 (l 2 5_ 6 )b2ma 1 (m 2 l)bd l 5 1_ 2 , m 5 1_ 6 6 a i 2 a + b ii 2_ 3 a 2 _7 4 bb ( 2_ 3 l 2 m)a + (3_ 4 2 _7 4l 1 m)b = 0d 613 e 6 a +__71313 bf __ 1310 7 a 2a 1 b b 1_ 2 a 1 1_ 2 bc 3_ 8 a 3_ 8 b d 2 5_ 8 a 1 3_ 8 be 2a 1 kb f 5 : 3, k 5 358 a 1_ 3 ab 1_ 4 a 1 3_ 4 bc 2 __ 1 a 1 3_ 124 bd 2be 2 1_ 4 a + 1_ 4 bf 2 1_ 4 a + 9_ 4 bg 1 : 3h 2 1_ 3 a + b, 2a + 3b, ___ ›AG= 3 ___ ›EB⇒ paralleld7.3Exercise 7C1 5_ 6 a 1 1_ 6 b2 2 1_ 2 a 2 1_ 2 b + c3 ___ ›OC5 22a 1 2b, ___ ›OD5 3a 1 2b, ___ ›OE5 22a 1 b7.4Exercise 7D1 a ( 12 3 ) b ( 2116 ) c ( 221229 )2 a 3i 2 j, 4i 1 5j, 22i 1 6jb i 1 6jc 25i 1 7jd √ ___40 = 2√ ___10 e √ ___37 f √ ___74 3 a 1 __5( 4 3 ) b 1 ___13( 5212 )c ___ 125( 2724 1) d ____( √ 10 123 )4 27 or 223Mixed Exercise 7E1 m 5 3, n 5 12 m 5 22, n 5 53 a ___ ›XM( 21 3 ) ___ ›XZ5 ( 210 6 )b v( 7 3 )c ( 8 0 )1 w( 210 6 )dv 5 2_ 3 , w 5 1_ 3 4 a ___ ›AC= x + y; __ ›BE= 1_ 3 y 2 xb i __ ›BF= v( 1_ 3y – x)ii ___ ›AF= x + __ ›BF= x + v( 1_ 3y – x)iiiv = 3_ 4 5 v 1 w 5 ( 4 5 ), √ ___41 2v – w 5 ( 522 ), √ ___29 v – 2w 5 ( 127 ), √ ___50 Further Pure Mathematics© Pearson Education Ltd 201019

Further Pure Mathematics6 p 1 q = ( 5 4 ), √ ___41 3p 1 q = ( 9 2 ), √ ___85 p 2 3q = ( 2 7216 ), √ ____305 7 a Chloe ( 5 7 ); Leo ( 4 5 ); Max ( 3 2 )bChloe: 74 km, 2.9 km/hLeo: 41 km, 2.1 km/hMax: 13 km, 1.2 km/hChapter 88.1Exercise 8A1 a 22 b 21 c 3 d 1_ 3 e 2_ 3 f 5_ 4 g 1_ 2 h 2i 1_ 2 j 1_ 2 k 2 2 l 3_ 2 2 a 4 b 25 c 2 2_ 3 d 0e_ 7 5 f 2 g 2 h 22i 29 j 23 k 3_ 2 l 2 1_ 2 3 a 4x 2 y 1 3 5 0 b 3x 2 y 2 2 5 0c 6x 1 y 2 7 5 0 d 4x 2 5y 2 30 5 0e 5x 2 3y 1 6 5 0 f 7x 2 3y 5 0g 14x 2 7y 2 4 5 0 h 27x 1 9y 2 2 5 0i 18x 1 3y 1 2 5 0 j 2x 1 6y 2 3 5 0k 4x2 6y 1 5 5 0 l 6x 2 10y 1 5 5 04 y 1 5x 1 35 2x 1 5y 1 20 5 067y 1 2 1_ 2 x 1 7y 5 2_ 3 x8 (3, 0)9 ( 5_ 3, 0)10 (0, 5), (24, 0)8.2Exercise 8B1 a 1_ 2 b 1_ 6 c 3_ 5 d 2e 21 f 1_ 2 g 1_ 2 h 8i 2_ 3 j 24 k 1_ 3 l 1_ 2 m 1 n q2 2 p 2 _______q 2 p 5 q 1 p2 73 124 4 1_ 3 5 2 1_ 4 6 1_ 4 7 268 258.3Exercise 8C1 a y 5 2x 1 1 b y 5 3x 1 7c y 5 2x 2 3 d y 5 24x 2 11e y 5 1_ 2 x 1 12 f y 5 2_ 3 x 2 5g y 5 2x h y 52 1_ 2x 1 2b2 y 5 3x 2 63 y 5 2x 1 84 2x 2 3y + 24 = 05 2 1_ 5 6 y = 2_ 5 x 1 37 2x 1 3y 2 12 5 08 8_ 5 9 y 5 4_ 3x 2410 6x 1 15y 2 10 = 08.4Exercise 8D1 a Perpendicular b Parallelc Neither d Perpendiculare Perpendicular f Parallelg Parallel h Perpendiculari Perpendicular j Parallelk Neither l Perpendicular2 y 5 2 1_ 3 x3 4x 2 y 1 15 5 04 a y 5 22x 1 1_ 2 b y 5 1_ 2 xc y 5 2x 2 3 d y 5 1_ 2 x 2 85 a y 5 3x 1 11 b y 5 2 1_ 3x 1 133 c y 5 2_ 3 x 1 2 d y 5 2 3_ 2x 1__172 6 3x 1 2y 2 5 5 07 7x 2 4y 1 2 5 020© Pearson Education Ltd 2010

8.5Exercise 8E11 a (21, 1); (4, 11) in ratio 3 : 2________________ 2 3 (21) + 3 3 4 , ______________2 3 1 + 3 3 11 )551 10 2 13 3 5so c = (2, 7)4 √ __5 5 2√ ___10 6 √ ____106 b M AB =_______ 11 2 1 =___107 √ ____113 8 a√ ___53 9 3b√ __4 2 21 5 = 2 m i = 2__ 1 25 Equation of l is y 2 7 = 2 1_10 5c 11 d√ ___61 12 2e√ __2(x 2 2)5 2y 2 14 = 2x + 28.6or x 1 2y 2 16 = 0c x = 0 ⇒ y = 8 D is (0, 8)Exercise 8Fy1 a (0, 6); (4, 10) is ratio 3 : 1B_____________( 3 3 4 1 1 3 0 ,______________ 3 3 10 1 1 3 6C)= (3, 9)44Ab (1, 5); (22, 8) is ratio 1 : 2( ________________2 3 1 1 1 3 (22)x(2 3 5 1 1 3 8)L, ______________)= (0, 6)33d ΔABDc (3, 27); (22, 8) is ratio 3 : 22 3 3 1 3x 3 (22)( _________________, ________________2 3 (27) 1 3 3 8CD = √ _______2 2 + 1 2 = √ __5 )= (0, 2)55AB = √ ____________________(4 2 21) 2 + (11 2 1) 2 = √ ________5 2 + 10 2 = 5√ __5 d (22, 5); (5, 2) is ratio 4 : 3.... Area of Δ ABD = __ 1 AB.CD =__ 1 3 √ __5 3 5√ __5 2 2( ________________4 3 5 1 3 3 (22) , _____________4 3 2 1 3 3 5 )= (2, 3 2_ 777 )=___ 252 a (4, 2); (6, 8) midpoint ( ______ 4 1 62 1 8,______22 2 12 a y = )3 x 1 1_ 3 = (5, 5)b (0, 6); (12, 2) midpoint ( _______ 0 1 12 , ______ 6 1 22 2 )Chapter 9c (2, 2); (24, 6)= (6, 4)9.1midpoint ( ______ 2 2 4 1 6,______22 2 )Exercise 9A= (21, 4)d (26, 4); (6, 24) midpoint ( _______ 26 1 6 , ______ 4 2 41 7x 6 2 8x 7 3 4x 32 2 )4 1_ 2__3x2 3 5 1_ 4 x2 4_ 3 6 1_ 3 x2 2_ 3 = (0, 0)7 23x 24 8 24x 25 9 22x 23Mixed Exercise 8G10 25x 26 11 2 1_ 3 x2 4_ 3 12 2 1_ 2 x2 3_ 2 13 22x 23 14 1 15 3x 21 a y 5 23x 1 14 b (0, 14)2 a y = 2 1_ 2 x + 4 b y 5 2 1_ 2 x + 16 9x 8 17 5x 4 18 3x 23_ 2, (1, 1)3 a y 5 1_ 7x 1 127, y 5 2x 1 12 b (9, 3)9.24 a y = 2 512x + 116 b 2225 a y 5 3_ 2 x 2 Exercise 9B3_ 2 b (3, 3)6 11x 2 10y + 19 = 01 a 4x 3 2 x 22 b 2x 23 c 2x 2 3_ 2 7 a y 5 2 1_ 2 x 5 3 b y 5 1_ 4 x 1 9_ 4 2 a 0 b 11 1_ 2 8 a y = 3_ 2x 2 2 b (4, 4) c 203 a (2 1_ 2 , 261_ 4) b (4, 24) and (2, 0)9 a 2x 1 y 5 20 b y 5 1_ 3 x 1 4_ 3 c (16, 231) d ( 1_ 2 , 4), (2 1_ 2, 24)Further Pure Mathematics© Pearson Education Ltd 201021

Further Pure Mathematics4 a x 2 1_ 2 b 26x 23 c 2x 24 = 28 sin p__ 4 [ sin p__ 2= sin 90° = 1 ]d 4_ 3 x3 2 2x 2 e 26x 24 + __ 12 x2 1_ 2 = 28f 1_ 3 x2 2_ 3 2 1_ 2 x22 g 23x 22 h 3 + 6x 22 i 5x 3_ 2 + 3_ 2 x2 1_ 2 j 3x 2 2 2x + 29.4k 12x 3 1 18x 2 l 24x 2 8 + 2x 22 Exercise 9D5 a 1 b 2_ 9 c 24 d 41 a y = (1 + 2x) 4 ⇒ y′ = 4(1 + 2x) 3 3 2 = 8(1 + 2x) 3b y = (1 + x 2 ) 3 ⇒ y′ = 3(1 + x 2 ) 2 3 2x = 6x(1 1 x 2 ) 2c y = (3 +4x) 1_ 2 ⇒ y′ = 1_ 29.3(3 + 4x)2 1_ 2 3 4 =________ 2√ 3 1 4x d y = (x 2 + 2x)Exercise 9C3 ⇒ y′ = 3(x 2 + 2x) 2 3 (2x + 2)1 a y = e 2x ⇒___ dy= 6(x 1 1)(x 2 1 2x) 2dx = 2e2x 2 a y = 4e 3x2 ⇒ y′ = 4e 3x2 3 6x = 24x e 3x2 b y = e 26x ⇒___ dy8(1 1 2x) 3= 26e26xdx c y = e x + 3x 2 ⇒___ dyb y = 9e 32x ⇒ y′ = 9e 3 2 x 3 21 = 29e 3 2 x dx = ex + 6xc y = e 26x ⇒ y′ = e 26x 3 26 = 26e 26x d y = sin 2x ⇒___ dy = 2 cos 2xd y = e x2 + 2x ⇒ y′ = e x2 + 2x 3 (2x + 2)dxe y = cos 3x ⇒ ___ dy= 2(x + 1) e x2 + 2x= 23 sin 3xdxf y = 3 sin 4x + 4 cos 3x ⇒___ dy3 a y = sin(2x + 1) ⇒ y′ = cos(2x + 1) 3 2 =dx =2 cos(2x 1 1)12 cos 4x 212 sin 3xb y = cos(2x 2 + 4) ⇒ y′ = 2sin(2x 2 + 4) 3 4x ⇒2 a y = sin 5x ⇒ ___dy24x sin(2x 2 1 4)= 5 cos 5xdx c y = sin 3 x ⇒ y′ = 3 sin 2 3 cos xb y = 2 sin 1_ 2x ⇒ ___ dydx = 2 3 1_ 2 cos 1_ 2 x = cos 1_ 2 x3 sin 2 x cos xc y = sin 8x ⇒ ___dyd y = cos 2 2x ⇒ y′ = 2 cos 2x 3 (2sin 2x) 3 2= 8 cos 8x= 24 sin 2x cos 2xdxd y = 6 sin 2_ 4 a y = x (1 + 3x) 5 ⇒ y′ = (1 + 3x) 5 + x.5(1 + 3x) 4 3 33x ⇒ ___ dydx = 6 3 2_ 3 cos 2_ 3 x = 4 cos 2_ 3 x= (1 + 3x) 4 [1 + 3x + 15x]e y = 2 cos x ⇒ ___ dy = 22 sin x= (1 + 3x) 4 (1 + 18x)dxf y = 6 cos 5_ b y = 2x (1 + 3x 2 ) 3 ⇒ y′ = 2(1 + 3x) 36x ⇒ ___ dydx = 6 3 5_ 6 sin 5_ 6 x = 25 sin 5_ 6 x+ 2x 3 3(1 + 3x 2 ) 3 3 6xg y = cos 4x ⇒ ___ dy = 24 sin 4x= 2(1 + 3x 2 ) 2 [1 + 18x 2 ]dx c y = x 3 (2x + 6) 4 ⇒ y′ = 3x 2 (2x + 6) 4 h y = 4 cos ( x_ 2)⇒ ___ dydx = 24 3 1_ 2 sin x_ 2 = 22 sin x_ 2 + x 3 3 4(2x + 6) 3 3 23 y = 2e 2x ⇒ ___ dydx = = x 2 (2x + 6) 22e2x @ (0, 2)3 [6x + 18 + 8x]4 y = 3 sin x ⇒ ___ dy= 2x 2 (2x + 6) 3 (7x + 9)dx = 3 cos x @ ( p__ 3= x )5 a y = xe 2x ⇒ y′ = e 2x + x.2 e 2x = e 2x (1 1 2x)b y = (x 2 + 3) e 2x ⇒ y′ = 2x 3 e 2x + (x 2 + 3)(2e 2x )Remember x is in radians m = 3 cos ( p__ 3 )= 3_ 2 [ = e 2x (2x 2 x 2 2 3)cos p__ 3 = cos 60° = 1_ 2 ]c y = (3x 2 5) e x2 ⇒ y′ = 3 3 e x2 + (3x 2 5)e x2 3 2x5 y = 4 cos 2x ⇒ ___ dydx = 28 sin 2x @ ( x = p__ 4 )= e x2 (6x 2 2 10x 1 3)6 a y = x sin x ⇒ y′ = sin x + x cos x m = 28 sin ( 2 3 p__ 4 ) b y = sin 2x cos x⇒ y′ = 2 sin x cos x 3 cos x + sin 2 x (2sin x)22© Pearson Education Ltd 2010

cy′ = sin x (2 cos 2 x 2 sin 2 x)y = e x cos x ⇒ y′ = e x cos x 2 e x sin x= e x (cos x 2 sin x)7 a y =_____ 5x ⇒ y′x + 1+ 1) 3 5 25x 3 1= (x__________________(x + 1) 2 =___________5x + 5 2 5x(x + 1) 2 =________ 5(x 1 1) 2 bcy =_______ 2x ⇒ y′ = (3x____________________2 2) 3 2 2 2x 3 33x 2 2 (3x 2 2) 2y =_________ 3x2y′(2x 2 1) 2⇒8 a y =___ x y′e2x⇒ =bce 2x 10 a y = x 2 (3x 2 1) 3 ⇒ y′ = 2x (3x 2 1) 3bwhen x = 0m = 6e 0 + 2e 0 = 8cm = 6 3 ___ 1√ 2 3 ___ 1√ 2 = 36x 2 4 2 6x=___________ d(3x 2 2) 224= _________(3x 2 2) 2 = 2 __ p 29.5Exercise 9E1 a y 1 3x 2 6 5 0b 4y 2 3x 2 4 5 0c y 5 xe 4xe 2x________________ xe= _______x(x + 1) 2 (x + 1) 2_________________= ___________ex2 (2x 2 2 1)x 2 x 2y 2 0 = 2p (x 2 p)ory = p________________2 xpx 27 y = 2 cos 2 x ⇒ y′ = 4 cos x (2sin x) @ ( p__ 4____________x cos x 2 sin xx 2 m = 24.____ 1√ 2 . ___ 1√ = 222 _________________normal has gradient =__ 1 2cos 2 x equation of normal is:______________cos 2 xy 2 1 = __ 1 2( x 2__ p 4 )_____________________________or8y 2 8 = 4x 2 pe 4xor8y 2 4x = 8 2 p_____________________e 4x9.6=__________________________________ (2x 2 1)2 3 6x 2 3x 2 3 2(2x 2 1) 3 2(2x 2 1) 46x (2x 2 1)[2x 2 1 2 2x]= ______________________(2x 2 1) 4=_________ 26x(2x 2 1) 3 3 1 2 x 3 2e 2xe2x _________________= _______ 1 2 2x y =_____ exx + 1 ⇒ y′ = (x + 1)ex 2 e x 3 1y =___ ex2 x ⇒ y′ = x 3 2xex2 2 e x2 3 19 a y = _____ sin xcos x 2 sin x 3 1⇒ y′ = xxbce xy = _____⇒ y′cos x =y_____ sin2 x⇒ y′e 2 x == cos x 3 ex + e x sin x= ex (sin x 1 cos x)e2x 3 2 sin x cos x 2 sin 2 x 3 2e 2x= 2 sin xe2x (cos x 2 sin x)2 sin x(cos x 2 sin x)= __________________+ x 2 3 3 (3x 2 1) 2 3 3 when x = 1m = 2 3 2 3 + 1 3 9 3 2 2 = 16 + 36 = 52y = (2x 1 3)e 2x ⇒ y′ = 2(2x + 3)e 2x + 2 3 e 2xy = 3 sin 2 x ⇒ y′ = 6 sin x cos x when x = p __4y = x cos x ⇒ y′ = cos x 2 x sin x when x = p __2m = cos p __22 p __2sin p __2= 0 2 p __23 12 a 7y 1 x 2 48 5 0 b 17y 1 2x 2 212 5 03 y 5 28x 1 10, 8y 2 x 2 145 5 04 y 5 2ex 2__ e 25 y 5 1_ 3 e6 y = x sin x ⇒ y′ = sin x + x cos x @ (p, 0) m = sin p + p cos p = 2p equation of tangent:1 1_ 4 x4 1 x 2 1 c2 22x 21 + 3x + c, 1)Further Pure Mathematics© Pearson Education Ltd 201023

Further Pure Mathematics3 2x 5_ 2 2 x 3 1 c4 4_ 3 x3_ 2 2 4x 1_ 2 1 4x 1 c5 x 4 1 x 23 1 rx 1 c6 t 3 1 t 21 1 c7 2_ 3 t3 1 6t 2 1_ 2 1 t 1 c8 1_ 2 x2 1 2x 1_ 2 2 2x 2 1_ 2 1 c9 ____ p5x 5 1 2tx 2 3x21 1 c10_ p 4 t4 1 q 2 t 1 px 3 t 1 c11 a 1_ 2 x4 1 x3 1 cbcde2x 2 3 __x 1 c 4_ 3 x3 1 6x 2 1 9x 1 c 2_ 3 x3 1 1_ 2 x2 23x 1 c 4_ 5 x 5_ 2 1 2x 3_ 2 1 c12 a ∫2 sin 3x dx = 2 2 __3cos 3x 1 cb ∫3e 4x dx = 3 __4e 4x 1 cc ∫2 cos 3x dx = 2 __3sin 3x 1 cd ∫2e 2x dx = 22e 2x 1 c13 a 5e x 1 4 cos x 1___ x42 1 cb 22 cos x 2 2 sin x 1 x 2 1 Ccd9.7Exercise 9G5e x 1 4 sin x 1 2 __x 1 ce x 2 cos x 1 sin x 1 C1 10t2 48 2 32t3 a 40 1 10t b 70 m/s4 a 30 2 1t b 0 m/s5 a a 5 32b v 5 32t 1 100⇒ s 5 16t 2 1 100t 1 d; when t = 0, s = 0 ⇒ d = 0 s 5 16t 2 1 100t6 a a 5 232b v 5 160 2 32t⇒ s 5 160t 2 16t 2 1 d;t = 0, s = 384 ⇒ d = 384 s 5 384 + 160t 2 16t 2c s 5 0 ⇒ 16(t 2 2 10t 2 24) = 0i.e. 16(t 2 12)(t 1 2) = 0 passes through origin when t = 127 a 3t 2 1 8t 2 5 b 6t 1 8c v 5 6 m/s, a 5 14 m/s 28 a v 5 3t 2 2 4t 1 3 b a 5 6t 2 49 a a 5 2t 1 10 b a 5 14 m/s 2c v 5 t 2 + 10t + 5⇒ s 5__ t33 + 5t2 + 5t + d; s = 0 when t = 0 ⇒ d = 0 when t = 2 s = 8_ 3 + 20 + 10 5 322_ 3 10 a 6 2 2t b 2 m/sc v 5 24 + 6t 2 t 2⇒ s 5 24t + 3t 2 2__ t3 + d 3s = 100, t = 3 ⇒ 100 = 24 3 3 + 3 3 9 2 ___ 273 + di.e. 100 5 72 + 27 2 9 + d d = 10 s 5 10 + 24t + 3t 2 2__ t3 39.8Exercise 9H1 a 228 b 217 c 2 1_ 5 2 a 10 b 4 c 12.253 a (2 3_ 4 , 2 9_ 4 ) b (1_ 2 , 9 1_ 4 )c (2 1_ 3, 1 527 ), (1, 0) d (3, 218), (2 1_ 3e (1, 2), (21,22) f (3, 27)5 ( ___ 3p8 , ___ 1 3p√ e2 9.9Exercise 9I__ 4 ) maximum, ( 7p1 a 8 b 9 3_ 4 c 19 2_ 3 d 21e 8 __ 512 2 a i 2 ln2 ii 2pb i 8_ 3 ii __ 64, __ 1427 )___8 , 2 ___ 1√ __ 7p4e 2 ) minimum15 p3 a A(1, 3), B(3, 3) b 1 1_ 3 4 6 2_ 3 5 a (2, 12) b 13 1_ 3 6 3 3_ 8 9.10Exercise 9J1 8_ 9 p2 6p3 15e 24 y = 5x 4 ⇒___ dydx δy ≈ 20x 3 δx= 20x3= 20x 3 3____ x200 [ 0.5% of x =____ x200 ]24© Pearson Education Ltd 2010

i.e. δy =_____ 20x4200 = ___x4 10 % change in y5 y = 3x 2 ⇒ dy ___dx δy ≈ 6x δx % change in y= ___ δyy × 100 = _________ x 410 3 5x 4 × 100 = 2%= 6x= 6x 3____ x =____6x2100 100 = ___ δyy × 100 = __________ 6x 2× 1003x 2 3 100= 2%6 For a sphere: V = 4 __3pr 3 = dv __dr= 4pr 2δr ≈ 0.02 cmδv ≈ 4pr 2 drUse r = 1 ⇒δv ≈ 4p 3 0.02= 0.25 cm 3Exercise 9K1 20 m 3 40 m; 800 m 22 2000p cm 23 40 cm4 _____ 8004 + p cm25 27 216 mm 2Mixed Exercise 9L1 a x 5 4, y 5 20b ____ d2 ydx 2 5 ___15 . 0 minimum82 (1, 211) and (_ 7 3, 2125__27 )3 a 7 __ 3132 b___ x33 2 2x 2 __ 1x 2 22_ 3 c f9(x) 5_____( x 21x ) 2 . 0 for all values of x4 (1, 4)5 a y 5 1 2__ x 2___ px2 4 c______ 24 1 p m2 (0.280 m 2 )6 b __ 103 c ____ vd2 dx 2 , 0 maximumd_______ 2300p27 e 222_ 9 %7 a____ 250 2 2xx2 b (5, 125)8 b x = ±2√ __2 , or x = 0c OP = 3; f ″(x) > 0 so minimum when x = ±2√ __2 (maximum when x = 09 b A is (21, 0); B is ( 5_ 3, 913__27 )10 3x 2 cos 3x + 2x sin 3x12_____________x cos x 2 sin x x 213 b y = 2x + 114 a 2(x 3 2 2x)e x + (3x 2 2 2)e x15 ____ 56p5 Chapter 1010.1Exercise 10A1 a 9° b 12° c 75°d 90° e 140° f 210°g 225° h 270° i 540°2 a 26.4° b 57.3° c 65.0°d 99.2° e 143.2° f 179.9°g 200°3 a 0.479 b 0.156 c 1.74d 0.909 e 20.8974 a __ 2p45 b p__ 18 c p__ 8 d p__ 6 e p__ 4 f p__ 3 gjm 5p12 h 4p __ 9 i 5p __ 8 __3 k 3p __ 4 l 10p ___ 2p__ 4p3 n 3p9 __4 __ 2 o 7pp 11p6 5 a 0.873 b 1.31 c 1.75d 2.79 e 4.01 f 5.5910.2Exercise 10B1 a i 2.7 ii 2.025 iii 7.5p (23.6)b i 16 2_ 3 ii 1.8 iii 3.6c i 1 1_ 3 ii 0.8 iii 22 ___ 10p3 cm3 2p4 5√ __2 cm5 a 10.4 cm b 1 1_ 4 6 7.57 0.88 a__ p b 3( 6 +___ 4p3 )cm9 6.8 cm10 a (R 2 r) cm c 2.43Further Pure Mathematics© Pearson Education Ltd 201025

Further Pure Mathematics10.3Exercise 10C1 a 19.2 cm 2 b 6.75p cm 2c 1.296p cm 2 d 38.3 cm 2e 5 1_ 3 p cm2 f 5 cm 22 a 4.47 b 3.96c 1.983 12 cm 24 b 120 cm 25 40 2_ 3 cm6 a 12 c 1.48 cm 27 8.88 cm 28 a 1.75 cm 2 b 25.9 cm 2 c 25.9 cm 29 4.5 cm 210 b 28 cm11 78.4 cm12 b 28 cm10.4Exercise 10D√1 a _____ 2 2 b 2 √ _____ 3 2 c 2 1 __ 2√d _____ 3 2 e √ _____ 3 2 f 2 1_ 2 g __ 1 √ h 2 _____ 2 2 2 i 2 √ _______ 3 2 √j 2 _____ 2 k 21 l 21√m __2___ 3 3 n 2 √ __3 o √ __3 10.5Exercise 10E1 a x 5 84, y 5 6.32b x 5 13.5, y 5 16.6c x 5 85, y 5 13.9d x 5 80, y 5 6.22 (Isosceles )e x 5 6.27, y 5 7.16f x 5 4.49, y 5 7.49 (right-angled2 a 48.1 b 45.6 c 14.8d 48.7 e 86.5 f 77.43 a x 5 74.6, y 5 65.4x 5 105, y 5 34.6b x 5 59.8, y 5 48.4x 5 120, y 5 27.3c x 5 56.8, y 5 4.37x 5 23.2, y 5 2.064 a 3.19 cm b 1.73 cm (√ __3 cm)c 9.85 cm d 4.31 cme 6.84 cm (isosceles) f 9.80 cm5 a 108(.2)° b 90° c 60°d 52.6° e 137° f 72.2°6 a 23.7 cm 3 b 4.31 cm 3 c 20.2 cm 37 a 155° b 13.7 cm8 a x = 49.5, area = 1.37 cm 2b x = 55.2, area = 10.6 cm 2c x = 117, area = 6.66 cm 29 6.50 cm 210 a 36.1 cm 3 b 12.0 cm 310.6Exercise 10F1 a 11.7 cm b 14.2 cm c 34.4°d 63.4°2 a 18.6 cm b 28.1 cm c 48.6°3 a 14.1 cm b 17.3 cm c 35.4°4 a 28.3 cm b 34.6 cm c 35.1°c 19.5°5 a 4.47 m b 4.58 m c 29.2°d 12.6° e 26.6°6 a 407 m b 402 m c 8.57°c 13.3°7 a 43.3 cm b 68.7 cm c 81.2 cm8 a 28.9 cm b 75.7 cm c 22.4°9 a 16.2 cm b 67.9° c 55.3 cm 2d 71.6°10 a 26.5 cm b 61.8° c 1530 cm 211 a 30.3° b 31.6° c 68.9°12 a 36.9° b 828 cm 213 a 15 m b 47.7° c €91 30014 a 66.4° b 32.9°15 46.5 m16 a OW = 4290 m, OS = 2760 mb 36.0° c 197 km/h10.7Exercise 10G1 a _____ sin2 u2 b 5 c 2cos2 Ad cos u e tan x 0 f tan 3Ag 4 h sin 2 u i 12 a LHS 5 sin 2 θ + cos 2 θ + 2 sin θ cos θ= 1 + 2 sin θ cos θ= RHS26© Pearson Education Ltd 2010

5 sin__ p cos u 1 cos__ p sin u6 6b LHS 5_________ 1 2 cos2 θ=_____ sin2 θθ= sin θ 3_____sincos θ cos θ cos θ 5 1_ 2 cos u 2 √ _____ 3 2 sin uc LHS = sin θ tan θ = RHS5 sin( __ p 1 usin x°5 ______ x°+______coscos x° sin x° = x° + cos 2 x°6 ) 5 R.H.S.sin2 ______________sin x° cos x°1= ___________= RHSsin x° cos x° 10.8d LHS 5 cos 2 A 2 (1 2 cos 2 A) = 2 cos 2 A 2 1= 2 (1 2 sin 2 A) 2 1 = 1 2 2 sin 2 A = RHSExercise 10He LHS 5(4 sin 2 θ 2 4 sin θ cos θ + cos 2 θ)1 a 270° b 60°, 240°+ (sin 2 θ + 4 sin θ cos θ + 4 cos 2 θ)c 60°, 300° d 15°, 165°= 5 (sin 2 θ + cos 2 θ) = 5 = RHSe 140°, 220° f 135°, 315°f LHS 5 2 2 (sin2 θ 2 2 sin θ cos θ + cos 2 θ)g 90°, 270° h 230°, 310°= 2 (sin 2 θ + cos2 θ)i 45.6°, 134.4° j 135°, 225°2 (sin 2 θ 2 2 sin θ cos θ + cos 2 θ)2 a 2120, 260, 240, 300 b 2171, 28.63= sin 2 θ + 2 sin θ cos θ + cos 2 θc 2144, 144 d 2327, 232.9= (sin θ + cos θ) 2 = rhse 150, 330, 510, 690 f 251, 431g LHS 5 sin 2 x(1 2 sin 2 y) 2 (1 2 sin 2 x) sin 2 y3 a 2p, 0, p, 2p b 2___ 4p , 2___2p ,___2p= sin 2 x 2 sin 2 y = RHS3 3 3 3 a sin 35° b sin 35° c cos 210°c 2___ 7p , 2___5p4 4 , __ p ,___ 3p d 20.14, 3.00, 6.144 4d tan 31° e cos u f cos 7u4 a 0°, 45°, 90°, 135°, 180°, 225°, 270°, 315°, 360°g sin 3u h tan 5u i sin Ab 60°, 180°, 300°j cos 3x√4 a 1 b 0 c __c 22 1_ 2___ 3 2 °, 2021_ 2 °, 2921_ 2 °2 d 30°, 150°, 210°, 330°√d _____ 2 2 e √ _____ 2 2 f 2 1_ 2 e 300° f 225°, 315°g 90°, 270° h 50°, 170°g √ __√3 h __i 165°, 345°___ 3 3 i 1j √ __5 a 2___ 7p2 12 , 2 ___p b 1.48, 5.85125 a LHS 5 sin A cos 60° 1 cos A sin 60°1 sin A cos 60° 2 cos A sin 60°5 2 sin A cos 60°5 2 sin A ( 1_ 2) 5 sin A 5 R.H.S.10.9Exercise 10Ibcos A cos B 2 sin A sin BLHS 5_____________________ cos (A 1 B) 5__________sin B cos Bsin B cos B 1 a 30°, 210°b 135°, 315°5 R.H.S.2 a p, 2psin x cos y 1 cos x sin yb 0.59, 3.73c LHS 5____________________ cos x cos y3 a 60°, 120°, 240°, 300°sin x cos y5_________ x sin yb 0°, 180°, 199°, 341°, 360° 1_________coscos x cos y cos x cos y c 60°, 300°5 tan x 1 tan y 5 R.H.S.d 30°, 60°, 120°, 150°, 210°, 240°, 300°, 330°e 270°cos x cos y 2 sin x sin yd LHS 5 ____________________1 1f 0°, 18.4°, 180°, 198°, 360°sin x sin yg 194°, 270°, 346°5 cot x cot y 2 1 1 1 5 cot x cot y 5 R.H.S.e LHS 5 cos u cos__ p 2 sin u sin__ p 4 a ___ 5p1 √ __12 , ____11p , 17p____, 23p____12 12 12 3 sin u3 3 b 0.841,___ 2p ,___4p, 5.445 1_ 2 cos u 2 √ __3 3___ 3 2 sin u 1 √ __3 sin uc 4.01, 5.41© Pearson Education Ltd 2010Further Pure Mathematics27

Further Pure MathematicsMixed Exercise 10J1 a cos 2 u sin 2 ub sin 4 3uc 12 a 1 b tan y =__________ 4 + tan x2 tan x 2 3 3 a 2 sin 2u 5 cos 2u ⇒ 2 sin 2u \ cos 2u 5 1⇒ 2 than 2u 5 1 ⇒ than 2u 5 0.5b 13.3, 103.3, 193.3, 283.34 a 225, 345b 22.2, 67.8, 202.2, 247.85 a ____ 11p ,____23p12 12 b ___2p ,___5p ,___5p ,____11p3 6 3 6 6 0°, 131.8°, 228.2°7 0, p, 2p8 a Max 1, u = 100°; Min = 21, u = 280°b Max 1, u = 330°; Min = 21, u = 150°9 a i 1_ 2 ii 1_ 2 iii √ __3 \3b 23.8°, 203.8°Review exercise1 a A 5 5, B 5 2 5_ 2 , C 5 2281_ 4 2b f min 5 2 ___ 1134 , x 5 5_ 2 y432O2 3 43 a 3 b 2 1_ 2 c 1251_ 2 4 8145 a – b P 5 126.8°, Q 5 R 5 26.6°6 a i 1_ 4 b 2 a ii 1_ 3 a 1 1_ 6 b iii 1_ 3 a 2 5_ 6 bb 2_ 5 c 2_ 3 7 a – b 54.5°, 234.5°8 4 1_ 2 m9 21.8°, 38.2°,120°10 a 3.18, 6.69, 13.04 b –c 2.3 d 0.6 (0.5 acceptable)11 x 6 2 18x 4 1 135x 212 23 , p , 213 –x14 a (3.14), (5, 24) b 1 1_ 3 15 a – b 12 cm c 6√ __3 d 54.7° e 109.5°16 1_ 2 x%17 a–b (23, 2 5_ 3 ), (2, 5_ 2 ) cd –y 5 2 6_ 5(x 2 1)18 a 0.253, 2.89 b 1.11, 2.68c 1.91, 2.3019 a 625 b 2, 64c x 5 2, y 5 320 –21 82.8°22 a 8i 2 j b –23 449 2_ 5 p24 a cos 2u 5 2 cos 2 u 2 1 b sin 2u 5 2 sin u cos uc – d 0.767, 1.33, 2.86 e 425 a 26 b 50 c 1726 a (2, 4), (5, 16) b x < 2, x > 527 a 2i 2 11j b 135 i 2 2_ 5 j28 73.9°29 a – b 12 79130 15 20031 a r 5 1_ 2, r 5 23 b 1032 46.5°, 133.5°33 1_ 2 34 a 2y 1 x 5 25 b (25, 0) c (10, 0)35 a 4 m/s 2 b 25 1_ 3 m36 a 4y 5 x 1 23 b y 5 24x 1 26c (23, 38) d 6 1617 e 12 2851 37 a 22p 1 q 5 28, 3p 1 q 5 18b 22, 24c (x 1 2)(x 2 3)(x 2 4)d –e 2 125 , 2f 7__ 712 38 –39 a 2y 5 3x 2 18 b 3y 5 22x 1 51c 156 d 216p40 a 1 1___ x 2____x22p 8p2 1_____ x316p 3 b p 5 6 1_ 2 41 a – b – c 4.76d – e 40842 a – b 400043 a 1 2 2x 2 4x 2 b 2.76132c 0.087% d a 5 1, b 5 25, c 5 828© Pearson Education Ltd 2010

e|x| , 1_ 6 44 2y 5 x 2 245 p[ 1_ 4 e8 1 4e 4 1 27 3_ 4 ]46 a – b –c 15, 75, 105, 165 d 3_ 8 47 a i y 5 2 ii x 5 21b (0, 3), (2 3_ 2, 0)cy1 1 2148 a 4 m/s 2 b 90 m49 a 1_ 2 b 2 1_ 3 a b 2b 2 4_ 3 a c –50 a ___ dydx 5 10x cos 3x 2 15x2 sin 3xb ___ dydx 5 ___________________(x 2 1 3) 2 2x e 3x3e3x (x 2 1 3) 2 51 0.212 m/s52 a 1.39 b 28.7°53 p , 25, p . 254 a 1, 3.75, 5.89, 6.92 b –c 0.79 d 2.155 a A 5 2_ 7 2 , B 5 2 9_ 4 b 2 9_ 4 , x 5 _7 2 c (1, 4), (7,10) d (2, 0), (5, 0)e – f 2456 (22, 1), (21, 3)57 a i __ p21 6 4ii 9 b p 5 64c x 2 2 10x 1 9 5 058 a 2 911 , 5 b 2c 4 d 16 38059 a ___ dydx 5 10x e2x 1 2(5x 2 2 2)e 2xb ___ dydx 5 _________________2 x 4 1 4x 2 22x3 (x 2 x 2 ) 2 60 ln 461 91.1°62 23 2__5 63 a 1 1___ x 2____ x212 144 b 1 1 ___ x 2___x212 72 c |x| , 4 d 1 1 __ x 2___ x26 72 e 0.3083Ox64 a cos 2A 5 2 cos 2 A 2 1b sin 2A 5 2 sin A cos Ac –d 17.7°, 102.3°, 137.7°e 3 √ ______ 3 8 65 (6, 21), (1,4)66 a p 5 1 5p 4 qx 1 10p 3 q 2 x 1 10p 2 q 3 x 1 5pq 4 x 4 1 q 5 x 5b p 5 6_ 5, q 5 125 or p 5 22, q 5 467 a 3 b q 5 20c a 5 2, b 5 1 d 968 a i √ ___20 ii √ ___40 iii √ ___20 b A 5 90°, B 5 C 5 45°c (5, 5) d √ ___10 69 660°70 67.4°71 a 2 b log pc r 5 n 2 1, s 5 n d –72 a 2x 2 2 5x 1 2 5 0 b x 2 2___ 12 x 1___12p p 5 0c 8_ 3 d 3_ 2 73 24 , p , 3Practice examinationpapersPaper 11 80.4° or 99.6°2 a – b p 5 210, q 5 333 20 cm 2 /s4 x 5 2 y 5 3, x 5 3 y 5 25 a p 5 6, q 5 24 b 5i 1 4_ 3 j6 a ( 1_ 3, 2√__ 5_ 3) b __ 253 p7 a 5 b 7r ___2 1 3 __2 c 3 1_ 2 d 118 a a 6 1 6a 5 bx 1 15a 4 b 2 x 2 1 20a 3 b 3 x 3 1 15a 2 b 4 x 41 6ab 5 x 5 1 b 6 x 6b a 5 2 b 5 4_ 3 , a 5 22 b 5 2 4_ 3 9 a 5 b 28 c 9, 310 a (2, 4) b y 5 4x 2 4c y 5 4 b 8 units 211 a 11.0 cm b 11.9 cm c 40.1°d 101.4° e 61.9°Further Pure Mathematics© Pearson Education Ltd 201029

Paper 21 2e 2x sin 3x 1 3e 2x cos 3x2 a 37.0° b 17.2 cm 23 a i y 5 3 ii x 5 2 b i (2 2_ 3, 0) ii (0, 4)cy43O2223xFurther Pure Mathematics4 a x 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0y 1 0.649 21.28 24.52 28.61 212.8 215.9 215.9 29.40b graph drawn c i 1.9 ii 1.35 a 0, 3, 4 b – c 11.8 m6 a 6 1_ 2 b a 5 120x c 47 a (ln 3, 36), (0, 4) b 32.02 c 82.2 units 28 a – b 2.71 c – d 1389 a – b 2280 c 37d 46√ ___37 e 9x 2 1 280 1 3 5 0√10 a – b i _________ 3 1 1√ 3 2 1 ii √ _________ 3 2 1√ 3 1 1 cdtan 2u 5_________ 2 tan u1 2 tan 2 u 2 2 1√ ____29 e 2030© Pearson Education Ltd 2010

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