Cambridge Additional Mathematics

28 0.m

56 0.m

1

1.0

08.

06.

04.

02.

x

y

O

y = ~`x

498 Answers

15 av(t)=3¡^1

2

p

t+1

a(t)=^1

4(t+1)

(^32)
bx(0) =¡ 1 , v(0) = 2: 5 , a(0) = 0: 25
Particle is 1 cm to the left of the origin, is travelling to the
right at 2 : 5 cm s¡^1 , and accelerating at 0 : 25 cm s¡^2.
c Particle is 21 cm to the right of the origin, is travelling to the
right at 2 : 83 cm s¡^1 , and accelerating at 0 :009 26cm s¡^2.
dnever changes direction enever decreasing
16 bA= 200x¡ 2 x^2 ¡^12 ¼x^2 c
17 a v(0) = 0cm s¡^1 , v(^12 )=¡¼cm s¡^1 , v(1) = 0cm s¡^1 ,
v(^32 )=¼cm s¡^1 ,v(2) = 0cm s¡^1
b 06 t 61 , 26 t 63 , 46 t 65 , etc.
So, for 2 n 6 t 62 n+1,n2f 0 , 1 , 2 , 3 , ....g
18 x=
k
2
³
1 ¡p^13
́
19 3 : 60 ms¡^1
20 increasing at 0 : 128 radians per second
21 a
p 3
2 ¼cm s
¡ (^1) b 0 cm s¡ 1
22 a i y=¡
a^2
4 b

• a
  2 b
  x iiwhen y=0,x=
  a
  2
  biy=¡
  2 b
  a
  x+b iiwhen x=0,y=b
  iii Hint: Let P^00 be the point on the line y=¡bwhere
  the distance to P is shortest.
  Show that FP=P^00 P.
  ciHint: Show that 4 FPP^0 »= 4 P^00 PP^0.
  iiHint: Show that the tangents meet at
  ³a+c
  2
  ,
  ac
  4 b
  ́
  .
  EXERCISE 15A.1
  1a i 0 : 6 units^2 ii 0 : 4 units^2 b 0 : 5 units^2
  2a 0 : 737 units^2 b 0 : 653 units^2
  3 n AL AU
  10 2 : 1850 2 : 4850
  25 2 : 2736 2 : 3936
  50 2 : 3034 2 : 3634
  100 2 : 3184 2 : 3484
  500 2 : 3303 2 : 3363
  converges to^73
  4a i n AL AU
  5 0 :160 00 0 :360 00
  10 0 :202 50 0 :302 50
  50 0 :240 10 0 :260 10
  100 0 :245 03 0 :255 03
  500 0 :249 00 0 :251 00
  1000 0 :249 50 0 :250 50
  10 000 0 :249 95 0 :250 05
  ii n AL AU
  5 0 :400 00 0 :600 00
  10 0 :450 00 0 :550 00
  50 0 :490 00 0 :510 00
  100 0 :495 00 0 :505 00
  500 0 :499 00 0 :501 00
  1000 0 :499 50 0 :500 50
  10 000 0 :499 95 0 :500 05
  iii n AL AU
  5 0 :549 74 0 :749 74
  10 0 :610 51 0 :710 51
  50 0 :656 10 0 :676 10
  100 0 :661 46 0 :671 46
  500 0 :665 65 0 :667 65
  1000 0 :666 16 0 :667 16
  10 000 0 :666 62 0 :666 72
  iv n AL AU
  5 0 :618 67 0 :818 67
  10 0 :687 40 0 :787 40
  50 0 :738 51 0 :758 51
  100 0 :744 41 0 :754 41
  500 0 :748 93 0 :750 93
  1000 0 :749 47 0 :750 47
  10 000 0 :749 95 0 :750 05
  bi^14 ii^12 iii^23 iv^34 c area=^1
  a+1
  5a n Rational bounds for¼
  10 2 : 9045 <¼< 3 : 3045
  50 3 : 0983 <¼< 3 : 1783
  100 3 : 1204 <¼< 3 : 1604
  200 3 : 1312 <¼< 3 : 1512
  1000 3 : 1396 <¼< 3 : 1436
  10 000 3 : 1414 <¼< 3 : 1418
  b n= 10 000
  EXERCISE 15A.2
  1a
  b n AL AU
  5 0 : 5497 0 : 7497
  10 0 : 6105 0 : 7105
  50 0 : 6561 0 : 6761
  100 0 : 6615 0 : 6715
  500 0 : 6656 0 : 6676
  c
  R 1
  0
  p
  xdx¼ 0 : 67
  2aAL=^2
  n
  nP¡ 1
  i=0
  p
  1+xi^3 , AU=^2
  n
  Pn
  i=1
  p
  1+xi^3
  b n AL AU
  50 3 : 2016 3 : 2816
  100 3 : 2214 3 : 2614
  500 3 : 2373 3 : 2453
  c
  R 2
  0
  p
  1+x^3 dx¼ 3 : 24
  t


• 
  

  0
  a(t)
  t

  
  


• 
  

  0
  v(t)
  cyan magenta yellow black
  (^05255075950525507595)
  100 100
  (^05255075950525507595)
  100 100 IB HL OPT
  Sets Relations Groups
  Y:\HAESE\CAM4037\CamAdd_AN\498CamAdd_AN.cdr Tuesday, 8 April 2014 8:40:04 AM BRIAN