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389
Answers
389Exercise 12K
1 2t β 3 2 2Ο 3 β 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 + 2Οrh = 600Ο β h = 300 β r2 ________ r
V = Ο
r2h = Οr (300 β r2) = 300Οr β Οr3
b 2000Ο cm3
8 a Let ΞΈ = angle of sector .
Οr2 Γ ΞΈ ____ 360 = 100 β ΞΈ = 36 000 _______ Οr2
P = 2r
+ 2Οr Γ ΞΈ ____ 360 = 2r + 200Οr ______ Οr2
= 2r + 100 ____ r
ΞΈ < 2Ο
β Area < Οr2, so Οr2 > 100
β΄ r > β ____
100 ____ Ο
b 40 cm
9 a Let h = height of rectangle.
P
= Οr + 2r + 2h = 40 β 2h = 40 β 2r β Οr
A = Ο __ 2 r2 + 2rh = Ο __ 2 r2 + r (40 β 2r β Οr)
= 40r β 2r2 β Ο __ 2 r2
b 800 _____ 4 + Ο cm2
10 a 18x + 14y = 1512 β y = 1512 β 18x ___________ 14
A = 12xy
= 12x ( 1512 β 18x ___________ 14 )
= 1296x β 108x2 ______ 7
b 27 216 mm2
Mixed exercise
1 f9(x) = lim
hβ0 10(x + h)2 β 10x2 ________________ h = lim
hβ0 20xh + 10h2 ____________ h
= lim
hβ0 (20x + 10 h) = 20x
2 a y-coordinate of B
= (Ξ΄x)3 + 3(Ξ΄x)2 + 6Ξ΄x + 4
Gradient = ((Ξ΄x
)3 + 3(Ξ΄x)2 + 6Ξ΄x + 4) β 4 __________________________ (1 + Ξ΄x) β 1
= (Ξ΄x
)3 + 3(Ξ΄x)2 + 6Ξ΄x __________________ (Ξ΄x) = (Ξ΄x)2 + 3Ξ΄x + 6
b 6
3 4, 11 3 _ 4 , 17 25 __ 27
4 2, 2 2 _ 3
5 (2, β13) and (β2, 15)
6 a 1 β 9 __ x2 b x = Β±3
7 3 _ 2 x β 1 _ 2 + 2 x β 3 _ 2
8 a dy ___ dx = 6 x β 1 _ 2 β 3 _ 2 x 1 _ 2 = 3 _ 2 x β 1 _ 2 (4 β x ) b (4, 16)
9 a x + x 3 _ 2 β x β 1 _ 2 β 1 b 1 + 3 _ 2 x 1 _ 2 + 1 _ 2 x β 3 _ 2 c 4 1 __ 16
10 6x2 + 1 _ 2 x β 1 _ 2 β 2xβ211 a = 1, b = β4, c = 5
12 a 3x2 β 10x + 5
b i 1 _ 3 ii y = 2x β 7 iii 7 _ 2 β __
5
13 y = 9x
β 4 and 9y + x = 128
14 a ( 4 _ 5 , β 2 _ 5 ) b 1 _ 5
15 P is (0, β1),
dy ___ dx = 3x2 β 4x β 4
Gradient at P = β4,
so L is y = β4x β 1.
β4x
β1 = x3 β 2x2 β 4x β 1 β x2(x β 2) = 0
x = 2 β
y = β9, so Q is (2, β9)
Distance PQ = β ___________________ (2 β 0 ) 2 + (β9 β (β1) ) 2 = β ___
68 = 2 β ___
17
16 a x = 4,
y = 20
b d2y ____ dx2 = 3 __ 4 x β 1 __ 2 + 96x3
At x = 4, d2y ____ dx2 = 15 ___ 8 > 0
(4, 20) is a local minimum.
17 (1, β11) and ( 7 _ 3 , β 329 ___ 27 )
18 a 7 31 __ 32
b f9(
x) = (x β 1 __ x ) 2
> 0 for all values of x
19 (1, 4)
20 a (1, 33) maximum, (2,
28) and (β1, 1) minimum
b
20y
x O(2, 28)(1, 33)
(β1, 1)y = f(x)
21 a 250 ____ x 2 β 2x b (5, 125)
22 a P (x, 5 β 1 __ 2 x2)
OP2 = (x β 0)2 + (5 β 1 __ 2 x2 β 0 ) 2
= 1 __ 4 x4 β 4x2 + 25
b x = Β±2 β __
2 or x = 0
c OP = 3; f9
(x) > 0 so minimum when x = Β±2 β __
2 ,
maximum when x = 0
23 a 3 + 5(3) + 32 β 33 = 0 therefore C on curve
b A is (β1, 0);
B is ( 5 _ 3 , 9 13 __ 27 )
24
1.2 2.1 0, 5Velocity (cm/s)OTime (s)
25 10 ___ 3 , 2300Ο _______ 27
26 dA ___ dx = 4Οx β 2000 _____ x 2
dA ___ dx = 0: 4Οx = 2000 _____ x 2 β x 3 = 2000 _____ 4Ο = 500 ____ Ο
27 a y = 1 β x __ 2 β Οx ___ 4
|
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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 β x __ 2 β Οx ___ 4 ) + Οx2 ____ 8
= x β x2 __ 2 β Οx2 ____ 4 + Οx2 ____ 8
= x __ 8 (8 β 4x β Οx)
c 2 _____ 4 + Ο m2 (0.280 m2)
28 a Οx2 + 2Οx + Οx2 + 2Οxh = 80Ο
h = 40 β x
β x2 __________ x
V = Ο
x2h = Οx2 ( 40 β x β x2 __________ x )
= Ο(40
x β x2 β x3)
b 10 __ 3 c d2V ____ dx2 < 0 β΄ maximum
d 2300Ο _______ 27 e 22 2 _ 9 %
29 a Length of short sides = x ___
β __
2
Area = 1 __ 2 Γ base Γ height
= 1 __ 2 ( x2 __ 2 ) = 1 __ 4 x2 m2
b Let l be length of EF.
1 __ 4 x2 l = 4000 β l = 16 000 _______ x2
S = 2 ( 1 __ 4 x2
) + 2x l ____
β __
2
= 1 __ 2 x2 + 32 000x ________
β __
2 x2 = x2 __ 2 + 16 000 β __
2 _________ x
c x = 20 β __
2 , S = 1200 m2 d d2S ____ dx2 > 0
Challenge
a x7 + 7x6h + 21x5h2 + 35x4h3
b d ___ dx (x7) = lim
hβ0 (x + h)7 β x7 ____________ h = lim
hβ0 7x6h + 21x5h2 + 35x4h3 ______________________ h
= lim
hβ0 (7x6 + 21x5h + 35x4h2) = 7x6
CHAPTER 13
Prior knowledge check
1 a 5 x 5 _ 2 b 2 x 3 _ 2 c x 5 _ 2 β β __
x d x β 3 _ 2 + 4x
2 a 6x2 + 3 b x β 1 c 3x2 + 2x d β 1 ___ x 2 β 3x2
3 a
O xy
β1 3
β3 b
Oy
x β15
β5
Exercise 13A
1 a y = 1 _ 6 x6 + c b y = 2x5 + c
c y = xβ1 + c d y = 2xβ2 + c
e y = 3 _ 5 x 5 _ 3 + c f y = 8 _ 3 x 3 _ 2 + c
g y = β 2 _ 7 x7 + c h y = 2 x 1 _ 2 + c
i y = β10 x β 1 _ 2 + c j y = 9 _ 2 x 4 _ 3 + c
k y = 3x12 + c l y = 2xβ7 + c
m y = β9 x 1 _ 3 + c n y = β5x + c
o y = 3x2 + c p y = 10 __ 3 x 0.6 + c
2 a y = 1 _ 4 x4 β 3 x 1 _ 2 + 6xβ1 + c b y = x4 + 3 x 1 _ 3 + xβ1 + c
c y = 4x
+ 4xβ3 + 4 x 1 _ 2 + c d y = 3 x 5 _ 3 β 2x5 β 1 _ 2 xβ2 + c
e y = 4 x β 1 _ 3 β 3x + 4x2 + c f y = x5 + 2 x β 1 _ 2 + 3xβ4 + c
3 a f(x) = 6
x2 β 3 x β 1 _ 2 + 5x + c b f(x) = x6 β xβ6 + x β 1 _ 6 + c
c f(x) = x 1 _ 2 + x β 1 _ 2 + c d f(x) = 2 x5 β 4xβ2 + c
e f(x) = 3 x 2 _ 3 β 6 x β 2 _ 3 + c
f f(x) = 3
x3 β 2xβ2 + 1 _ 2 x 1 _ 2 + c
4 y = 4 x 3 ____ 3 + 6x2 + 9x + c
5 f(x
) = β3xβ1 + 4 x 3 _ 2 + x 2 ___ 2 β 4x + c
Challenge
y = β 12 ____
7 x 7 _ 2 β 4 ____
5 x 5 _ 2 + 3 ____ 2 x 2 + 1 __ x + c
Exercise 13B
1 a x 4 ___ 4 + c b x 8 ___ 8 + c
c βxβ3 + c d 5 x 3 ____ 3 + c
2 a 1 __ 5 x5 + 1 __ 2 x4 + c b x 4 ___ 2 β x 3 ___ 3 + 5 x 2 ____ 2 + c
c 2 x 5 _ 2 β x3 + c
3 a β4xβ1 + 6 x 1 _ 2 + c b β6xβ1 β 2 _ 5 x 5 _ 2 + c
c β4 x β 1 _ 2 + x 3 ___ 3 β 2 x 1 _ 2 + c
4 a x4 + xβ3 + rx + c b 1 _ 2 x 2 + 2 x 1 _ 2 β 2 x β 1 _ 2 + c
c px5 ____ 5 + 2tx β 3xβ1 + c
5 a t3 + t β1 + c b 2 _ 3 t3 + 6 t β 1 _ 2 + t + c
c p __ 4 t4 + q2t + px3t + c
6 a 2x β 3 __ x + c b 4 _ 3 x3 + 6x2 + 9x + c
c 4 _ 5 x 5 _ 2 + 2 x 3 _ 2 + c
|
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391
Answers
3917 a 1 _ 3 x3 + 2x β 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 β 2 ___ x 2 + c b β 1 __ x 2 β 1 _ x + 3x + c
c 1 __ x x4 β 1 _ 3 x3 + 3 _ 2 x2 β 3x + c d 8 _ 5 x 5 _ 2 + 8 _ 3 x 3 _ 2 + 2 x 1 _ 2 + c
e 3x + 2 x 1 _ 2 + 2x3 + c f 2 _ 5 x 5 _ 2 + 3x2 + 6 x 3 _ 2 + c
9 a β A __ x β 3x + c b 2 _ 3 β __
P x 3 _ 2 β 1 ___ x 2 + c
c β p __ x + 2 qx 3 _ 2 _____ 3 + rx + c
10 β 6 __ x + 8 x 3 _ 2 ____ 3 β 3 x 2 ____ 2 + 2x + c
11 2x4 + 3x2 β 6 x 1 _ 2 + c
12 a (2 + 5 β __
x )2 = 4 + 10 β __
x + 10 β __
x + 25x = 4 + 20 β __
x + 25x
b 4x
+ 40 x 3 _ 2 _____ 3 + 25 x 2 _____ 2 + c
13 x 6 ___ 2 β 8 x 1 _ 2 + c
14 p = β4,
q = β2.5
15 a 1024 β 5120x + 11 520x2
b 1024x β 2560x2 + 3840x3 + c
Exercise 13C
1 a y = x3 + x2 β 2 b y = x4 β 1 ___ x 2 + 3x + 1
c y = 2 _ 3 x 3 _ 2 + 1 __ 12 x 3 + 1 _ 3 d y = 6 β __
x β 1 _ 2 x 2 β 4
e y = 1 _ 3 x3 + 2x2 + 4x + 2 _ 3 f y = 2 _ 5 x 5 _ 2 + 6 x 1 _ 2 + 1
2 f(x
) = 1 _ 2 x 4 + 1 __ x + 1 _ 2
3 y = 1 β 2 ___ β __
x β 3 __ x
4 f(x
) = 3x3 + 2x2 β 3x β 2
5 y = 6 x 1 _ 2 β 4 x 5 _ 2 ____ 5 + 118 ___ 5
6 a p = 1 _ 2 , q = 1 b y = 4 x 3 _ 2 + 5 x 2 ____ 2 β 421 ___ 2
7 a f(t) = 10
t β 5 t 2 ___ 2 b 7 1 _ 2
8 a f(t) = β4.9
t2 + 35 b 23.975 m
c 35 m d 2.67 seconds
e e.g. the ground is flat
Challenge
1 f2(x) = x 3 ___ 3 ; f4(x) = x 4 ___ 12 b x n+1 _____________________ 3 Γ 4 Γ 5 Γ β¦ Γ (n + 1 )
2 f2(x) = x + 1; f3(x) = 1 _ 2 x2 + x + 1; f4(x) = 1 _ 6 x 3 + 1 _ 2 x 2 + x + 1
Exercise 13D
1 a 152 1 _ 4 b 48 2 _ 5 c 5 1 _ 3 d 2
2 a 5 1 _ 4 b 10 c 11 5 _ 6 d 60 1 _ 2
3 a 16 2 _ 3 b 46 1 _ 2 c 11 __ 14 d 2 1 _ 2
4 A = β7 or 4
5 28
6 β8 + 8 β __
3
7 k = 25 __ 4
8 450 mChallenge
k = 2
Exercise 13E
1 a 22 b 36 2 _ 3 c 48 8 __ 15 d 6
2 4 3 6 4 10 2 _ 3
5 21 1 _ 3 6 4 __ 81 7 k = 2
8 a (β1, 0) and (3, 0) b 10 2 _ 3
9 1 1 _ 3
Exercise 13F
1 a 1 1 _ 3 b 20 5 _ 6
y
xO β2 y
x O 4 β1
c 40 1 _ 2 d 1 1 _ 3
y
x Oβ3 3
2y
x O
e 21 1 __ 12
y
x O 5 2
2 a (β3, 0) and (2, 0) b 21 1 __ 12
3 a f(β3) = 0
b f(x
) = (x + 3)(βx2 + 7x β 10)
c f(x
) = (x + 3)(x β 5)(2 β x)
d (β3, 0), (2,
0) and (5, 0)
e 143 5 _ 6
Challenge
1 a 4 1 _ 2 b 9 c 9a ___ 2 d 4 1 _ 2 e 9 ___ 2a
2 a B has x
-coordinate 1
β«β1
0 (x3 + x2 β 2x)dx = [ 1 __ 4 x4 + 1 __ 3 x3 β x2] 1
0
= 1 __ 4 + 1 __ 3 β 1 = β 5 __ 12
So area under x-axis is 5 __ 12
Area above x-axis is
( 1 __ 4 04 + 1 __ 3 03 β 02) β ( 1 __ 4 x4 + 1 __ 3 x3 β x2) = 5 __ 12
So the x-coordinate of
a satisfies
3x4 + 4x3 β 12x2 + 5 = 0
Then use the factor theorem twice to get
(x β 1)2(3x2 + 10x + 5) = 0
|
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] |
392
Answers
392 Full worked solutions are available in SolutionBank.
Online
b A has coordinates ( β5 + β ___
10 _________ 3 , β80 + 37 β ___
10 _____________ 27 )
The roots at 1 correspond to point B.
The root β5 β β ___
10 _________ 3 gives a point on the curve to the
left of β2 below the x-axis,
so cannot be A.
Exercise 13G
1 a A(β2, 6), B(2, 6) b 10 2 _ 3
2 a A(1, 3),
B(3, 3) b 1 1 _ 3
3 6 2 _ 3
4 4.5
5 a (2, 12) b 13 1 _ 3
6 a 20 5 _ 6 b 17 1 _ 6
7 a, b Substitute into equation for y
c y = x
β 4 d 8 3 _ 5
8 3 3 _ 8
9 a Substitute x = 4 into both equations
b 7.2
10 a 21 1 _ 3 b 2 5 _ 9
11 a (β1, 11) and (3, 7) b 21 1 _ 3
Mixed exercise
1 a 2 _ 3 x 3 β 3 _ 2 x 2 β 5x + c b 3 _ 4 x 4 _ 3 + 3 _ 2 x 2 _ 3 + c
2 1 _ 3 x 3 β 3 _ 2 x 2 + 2 _ x + 1 _ 6
3 a 2x4 β 2x3 + 5x + c b 2 x 5 _ 2 + 4 _ 3 x 3 _ 2 + c
4 4 _ 5 x 5 _ 2 β 2 _ 3 x 3 _ 2 β 6 x 1 _ 2 + c
5 x = 1 _ 3 t3 + t2 + t β 8 2 _ 3 ; x = 12 1 _ 3
6 a A = 6,
B = 9 b 3 _ 5 x 5 _ 3 + 9 _ 2 x 4 _ 3 + 9x + c
7 a 9 _ 2 x β 1 _ 2 β 8 x β 3 _ 2 b 6 x 3 _ 2 + 32 x 1 _ 2 β 24x + c
8 a = 4,
b = β3.5
9 25.9 m
10 a f(t) = 5
t + t2 b 7.8 seconds
11 a β1,3 b 10 2 _ 3
12 a β 2 x 3 _ 2 ____ 3 + 5x β 8 β __
x + c b 7 __ 3
13 a (3, 0) b (1, 4) c 6 3 _ 4
14 a 3 _ 2 x β 1 _ 2 + 2 x β 3 _ 2 b 2 x 3 _ 2 β 8 x 1 _ 2 + c c A = 6, B = β2
15 a dy ___ dx = 6 x β 1 __ 2 β 3 __ 2 x 1 __ 2 = 3 __ 2 x β 1 __ 2 (4 β x)
b (4,16) c 133 (3 sf )
16 a (6,12) b 13 1 _ 3
17 a A(1, 0),
B(5, 0), C (6, 5) b 10 1 _ 6
18 a q = β 2 b C (6,17) c 1 1 _ 3
19 β 9 __ x β 16 x 3 _ 2 _____ 3 + 2 x 2 β 5x + c
20 A = β6 or 1
21 a f9(
x) = (2 β x2)(4 β 4x2 + x4) ___________________ x2 = 8xβ2 β 12 + 6x2 β x4
b f 0( x) = β16xβ3 + 12x β 4x3
c f(x) = β 8 __ x β 12x + 2 x 3 β x 5 ___ 5 β 47 ___ 5
22 a (β3, 0) and ( 1 _ 2 , 0) b 14 7 __ 24
23 a ( β 3 _ 2 , 0) and (4, 0) b 55 11 __ 24
24 a β2 and 3 b 21 1 __ 12 Challenge
10 5 __ 12
CHAPTER 14
Prior knowledge check
1 a 125 b 1 __ 3 c 32 d 49 e 1
2 a 66 b y21 c 26 d x4
3 gradient 1.5, intercept 4.1
Exercise 14A
1 a
2
O13y
4
x β1β2β3β4 2
2.634 1y = (1.7)x
y = 4
b x β 2.6
2 a
2
O13y
4
xβ1β2β3β4 2
β1.434 1y = (0.6)x
y = 2
b x β β1.4
3
2
O13y
4
xβ1β2β3β4 234 1y = 1x
4 a True, because a0 = 1 whenever a is positive
b False, for example when a
= 1 __ 2
c True, because when
a is positive, ax > 0 for all
values of x
5
2ba cd
O1y
β3x
|
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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 β 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 = 2.71828β¦
e3 = 20.08553β¦
3 a
Oy
x2y = ex + 1
y = 1
b y
xy = 4eβ2x
O4
c y
x
y = β3β1y = 2ex β 3
O d y
x4
y = 4 β ex3
e y
xy = 6 + 10e
16
61
2
Ox
f y
xy = 100eβx + 10110
10
O
4 a A = 1, C = 5, b is positive
b A = 4,
C = 0, b is negative
c A = 6,
C = 2, b is positive
5 A = e2, b = 3
Oy = f(x)
e2y
x
6 a 6e6x b β 1 _ 3 e β 1 _ 3 x c 14e2x
d 2e0.4x e 3e3x + 2ex f 2e2x + ex
7 a 3e6 b 3 c 3eβ1.5
8 f9(x) = 0.2e0.2x
The gradient of the tangent when x = 5 is
f9(5) = 0.2e1 = 0.2e.
The equation of the tangent is therefore e = 0.2e Γ 5 + c, so c = 0.
Exercise 14C
1 a Β£20 000 b Β£14 331
c
t20 000V (Β£)
OV = 20 000eβt
12
|
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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 for predictions that far into the future.
3
a 200
b Disease will infect up to 300 people.
c N
t200N = 300 β 100eβ0.5t300
O
4 a i 15 rabbits ii 132 rabbits
b The initial number of rabbits
c dR ____ dm = 2.4 e 0.2m
When m = 6,
dR ____ dm = 7.97 β 8
d The rabbits may begin to run out of food or space
5 a 0.565 bars
b dp ___ dh = β0.13 e β0.13h = β 0.13p , k = β0.13
c The atmospheric pressure decreases exponentially
as the altitude increases
d 12%
6 a Model 1: Β£15 733
Model 2: Β£15 723 Similar results
b Model 1: Β£1814
Model 2: Β£2484 Model 2 predicts a larger value
c
20k
OModel 1
Model 2T
t
d In Model 2 the tractor will always be worth at least Β£1000. This could be the value of the tractor as scrap metal.
Exercise 14D
1 a log4 256 = 4 b log3 1 _ 9 = β2
c log10 1 000 000 = 6 d log11 11 = 1
e log0.2 0.008 = 3
2 a 24 = 16 b 52 = 25
c 9 1 _ 2 = 3 d 5β1 = 0.2
e 105 = 100 0003 a 3 b 2 c 7 d 1
e 6 f 1 _ 2 g β1 h β2
i 10 j β2
4 a 625 b 9 c 7 d 9
e 20 f 2
5 a 2.475 b 2.173 c 3.009 d 1.099
6 a 5 = log2 32 < log2 50 < log2 64 = 6
b 5.644
7 a i 1 ii 1 iii 1 b a1 β‘ a
8 a i 0 ii 0 iii 0 b a0 β‘ 1
Exercise 14E
1 a log2 21 b log2 9 c log5 80
d log6 ( 64 __ 81 ) e log10 120
2 a log2 8 = 3 b log6 36 = 2 c log12 144 = 2
d log8 2 = 1 _ 3 e log10 10 = 1
3 a 3 loga x + 4 loga y + loga z
b 5loga x β 2loga y
c 2 + 2loga x
d logax β 1 _ 2 loga y β loga z
e 1 _ 2 + 1 _ 2 loga x
4 a 4 _ 3 b 1 __ 18 c β ___
30 d 2
5 a log3 (x + 1) β 2 log3 (x β 1) = 1
log3 ( x + 1 _______ (x β 1 ) 2 ) = 1
x + 1 _______ (x β 1 ) 2 = 3
x + 1 = 3(x
β1)2
x + 1 = 3(x2 β 2x + 1)
3x2 β 7x + 2 = 0
b x = 2
6 a = 9,
b = 4
Challenge
loga x = m and loga y = n
x
= am and y = an
x Γ· y = am Γ· an = am β n
loga ( x __ y ) = m β n = loga x β loga y
Exercise 14F
1 a 6.23 b 2.10 c 0.431
d 1.66 e β3.22 f 1.31
g 1.25 h β1.73
2 a 0, 2.32 b 1.26, 2.18 c 1.21
d 0.631 e 0.565, 0.712 f 0
g 2 h β1
3 a 5.92 b 3.2
4 a (0, 1)
1y
y = 4x
x O
b 1 _ 2 , 3 _ 2
5 a 0.7565 b 7.9248 c 0.2966
|
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] |
395
Answers
395Exercise 14G
1 a ln 6 b 1 _ 2 ln 11 c 3 β ln 20
d 1 _ 4 ln ( 1 _ 3 ) e 1 _ 2 ln 3 β 3 f 5 β ln 19
2 a e2 b e _ 4 c 1 _ 2 e4 β 3 _ 2
d 1 _ 6 ( e 5 _ 2 + 2) e 18 β e 1 __ 2 f 2, 5
3 a ln 2,
ln 6 b 1 _ 2 ln 2, 0 c e3, eβ5
d ln 4, 0 e ln 5, ln ( 1 _ 3 ) f e6, eβ2
4 ln 3, 2 ln 2
5 a 1 _ 8 (e2 + 3) b 1 _ 5 (ln 3 + 40) c 1 _ 5 ln 7, 0
d e3, eβ1
6 1 + ln 5 ________ 4 + ln 3
7 a The initial concentration of the drug in mg/l
b 4.91 mg/l
c 3 = 6 e β t __ 10
1 _ 2 = e β t __ 10
ln ( 1 _ 2 ) = β t ___ 10
t = β10 ln ( 1 _ 2 ) = 6.931β¦ = 6 hours 56 minutes
8 a (0, 3 + ln 4) b (4 β eβ3)
Challenge
As y = 2 is an asymptote, C = 2.Substituting (0, 5) gives 5 = A
e0 + 2, so A is 3.
Substituting (6, 10) gives 10 = 3e6B + 2.
Rearranging this gives B = 1 _ 6 ln ( 8 _ 3 ) .
Exercise 14H
1 a log S = log (4 Γ 7x)
log S = log 4 + log 7x
log S = log 4 + x log 7
b gradient log 7, intercept log 4
2 a log A = log (6x4)
log A = log 6 + log x4
log A = log 6 + 4 log x
b gradient 4, intercept log 6
3 a Missing values 1.52, 1.81, 1.94
b
log xlog y
0.20 0.40.60.811.21.40.5
011.522.5
c a = 3.5, n = 1.4
4 a Missing values 2.63, 3.61, 4.49,
5.82
b
x 2 0 4 6 8 101
0234567log y
c b = 3.4, a = 105 a Missing values β0.39, 0.62, 1.54,
2.81
b
O 1 β1 β2 2 31234log R
log m
c a = 60,
b = 0.75
d 1,600 kcal per day (2 s.f.)
6 a Missing values 2.94, 1.96, 0.95
b
log Rlog f
0.50 11.522.533.50.5
011.522.533.54
c A = 5800, b = β0.9
d 694 times
7 a Missing values 0.98, 1.08, 1.13,
1.26, 1.37
b P = abt
log P = log (abt)
log P = log a + log bt
log P = log a + t log b
c
tlog P
50 10152025303540450.2
00.40.60.811.21.4
d a = 7.6, b = 1.0
e The rate of growth is often proportional to the size
of the population
8 a log N = 0.095t
+ 1.6
b a = 40,
b = 1.2
c The initial number of sick people
d 9500 people. After 30 days people may start to recover
, or the disease may stop spreading as
quickly.
9 a log A = 2 log w β 0.1049
b q = 2,
p = 0.7854
c Circles: p is approximately one quarter
Ο, and the
width is twice the radius, so A = Ο __ 4 w 2 = Ο __ 4 (2r) 2 = Ο r 2 .
Challenge
y = 5.8 Γ 0.9x
|
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] |
396
Answers
396 Full worked solutions are available in SolutionBank.
Online
Mixed exercise
1 a
Oy
x1y = 2βx
y = 0
b
Oy
x4y = 5ex β 1
y = β1
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 β 2x + 1 β 15 = 0
22x β 2 Γ 2x β 15 = 0
(2x)2 β 2 Γ 2x β 15 = 0
u2 β 2u β 15 = 0
b 2.32
6 x = 6
7 a βeβx b 11e11x c 30e5x
8 a e 8 + 5 ______ 2 b ln 5 ____ 4 c β 1 _ 2 ln 14 d 3 + β ___
13 ________ 2
e 0 f e 4 ___ 2
9 a Β£950 b Β£290 c 4.28 years d Β£100
e
OP
t100950
P = 100 + 850eβt
2
f A good model. The computer will always be worth
something10 a y = ( 2 ____ ln 4 ) x
b (0, 0) satisfies the equation of
the line.
c 2.43
11 a We cannot go backwards in time
b 75Β°C
c 5 minutes
d The exponential term will always be positive, so the
overall temperature will be greater than 20Β°C.
12 a S = aV b
log S = log (aVb)
log S = log a + log (V b)
log S = log a + b log V
blog S 1.26 1.70 2.05 2.35 2.50
log V 0.86 1.53 2.05 2.49 2.72
c
log Slog V
0.50 0.00 1.00 1.50 2.00 2.50 3.000.50
0.001.001.502.002.503.00
d The gradient is approximately 1.5; a = 0.09
13 a The model concerns decay, not growth
b
OR
t140
R = 140ekt
c 70 = 140e30k
1 _ 2 = e30k
ln ( 1 _ 2 ) = 30k
k = 1 __ 30 ln ( 1 _ 2 )
k = β 1 __ 30 ln (2), so c = β 1 __ 30
14 a 6.3 million views
b dV ___ dx = 0.4e0.4x
c 9.42 Γ 1016 new views per day
d This is too big, so the model is not valid after
100 days
15 a 4.2
b i 1.12 Γ 1025 dyne cm
ii 3.55 Γ 1026 dyne cm
c divide b ii by
b i
16 a They exponentiated the two terms on RHS separately rather than combining them first.
b x = 2 Β±
β __
5
|
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] |
397
Answers
397Challenge
a y = 9x = 32x, log3(y) = 2x
b y2 = (9x)2 = 92x, log9(y2) = 2x
c x = β 1 __ 3 or x = β2
Review exercise 3
1 β4.5
2 β __
7
3 a All equal to β ____ 145
b (x
β 1)2 + (y + 3)2 = 145
4 a β2i β
8j
b | βΆ AB | = | βΆ AC | = β ___ 85
c | βΆ BC | = β ___ 68
cos β ABC
= 85 + 68 β 85 _____________
2 Γ β ___ 85 Γ β ___ 68 = 1 ___
β __
5
5 12
6 a 5N b 7
7 m = 50 β __
3 + 30, n = 50
8 a β _____________ (β75) 2 + 180 2 = 195 > 150 = β _________ 90 2 + 120 2
b Boat A: 6.5 m/s; Boat B: 5 m/s; Both boats arrive at
the same time β it is a tie.
9 lim
hβ0 5 (x + h) 2 β 5 x 2 _____________ h
= lim
hβ0 5x 2 + 10xh + 5 h 2 β 5 x 2 _____________________ h
= lim
hβ0 10xh + 5 h 2 ___________ h
= lim
hβ0 10x + 5h
= 10x
10 dy ___ dx = 12x2 + x β 1 __ 2
11 a dy ___ dx = 4 + 9 __ 2 x 1 __ 2 β 4x
b Substitute x = 4 into equation for
C
c Gradient of tangent = β3 so gradient of normal = 1 __ 3
Substitute (4, 8) into y
= 1 __ 3 x + c
Rearrange
y = 1 __ 3 x + 20 __ 3
d PQ = 8 β ___
10
12 a dy ___ dx = 8x β 5xβ2, at P this is 3
b y = 3x
+ 5
c k = β 5 __ 3
13 a P = 2,
Q = 9, R = 4
b 3 x 1 __ 2 + 9 __ 2 x β 1 __ 2 β 2 x β 3 __ 2
c When x = 1,
f9(x) = 5 1 __ 2 , gradient of 2y = 11x + 3 is
5 1 __ 2 , so it is parallel with tangent
14 f9
(x) = 3x2 β 24x + 48 = 3(x β 4)2 > 0
15 a A (1,0) and B (2,0)
b ( β __
2 , 2 β __
2 β 3)
16 a V = Οr2h = 128Ο, so h = 128 ____ r 2
S = 2Ο
rh + 2Οr2 = 256Ο _____ r + 2Οr2
b 96Ο cm217 a dy ___ dx = 6x + 2 x β 1 __ 2
b d2y ____ dx2 = 6 β x β 3 __ 2
c x3 + 8 __ 3 x 3 __ 2 + c
18 a 2x3 β 5x2 β 12x
b x(2x
+ 3)(x β 4)
c
xy
O 4 β3
2
19 6 3 _ 4
20 4
21 a βx4 + 3x2 + 4 = (βx2 + 4)(x2 + 1); x2 + 1 = 0 has no
real solutions; so solutions are A(β2, 0), B(2, 0)
b 19.2
22 4 1 _ 2 square units
23 a P(β1, 4),
Q(2, 1)
b 4.5
24 a k = β1,
A(0, 2)
b ln 3
25 a 425 Β°C b 7.49 minutes c 1.64 Β°C/minute
d The temperature can never go below 25 Β°C, so
cannot reach 20 Β°C.
26 a β0.179 b x = 15
27 a x = 1.55 b x = 4 or x
= 1 __ 2
28 a logp 2 b 0.125
29 a x = 2 b x = ln 3 or x = ln 1 = 0
30 a Missing values 0.88, 1.01, 1.14 and 1.29
b
tlog P
01 0 20 30 400.2
00.40.60.811.21.4
c P = abt
log P = log (abt) = log a + t log b
This is a linear relationship.
The gradient is log b
and the intercept is log a.
d a = 5.9,
b = 1.0
Challenge
1 a 0
b 1
2 a f9
(β3) = f9(2) = 0, so f9(x) = k(x + 3)(x β 2)
= k(x2 + x β 6); there are no other factors as f(x) is
cubic
b 2x3 + 3x2 β 36x β 5
3 51.2
4 a f(0) = 03 β k(0) + 1 = 1; g(0) = e2(0) = 1; P(0, 1)
b 1 _ 2
|
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] |
398
Answers
398 Full worked solutions are available in SolutionBank.
Online
Practice paper
1 a 1 __ 3 b 5 β __
2
2 y = 2 __ 3 x + 8 __ 3
3 a error 1: = β 3 ___
β __
x = β3 x β 1 _ 2 , not β3 x 1 _ 2
error 2: [ x5 __ 5 β 2 x 3 __ 2 + 2x] 2
1 = ( 32 ___ 5 β 2 β __
8 + 4) β ( 1 __ 5 β 2 + 2)
not ( 1 __ 5 β 2 + 2) β ( 32 ___ 5 β 2 β __
8 + 4)
b 5.71 (3 s.f.)
4 x = 30Β°, 90Β°,
150Β°
5 a 2x2(x + 3)
b 2x2(x + 3) = 980 β 2x3 + 6x2 β 980 = 0
β x3 + 3x2 β 490 = 0
c For f(
x) = x3 + 3x2 β 490 = 0, f(7) = 0, so x β 7 is a
factor of f(x) and x = 7 is a solution.
d Equation becomes (x β 7)(
x2 + 10x +70) = 0
Quadratic has discriminant 102 β 4 Γ 1 Γ 70 = β180
So the quadratic has no real roots, and the equation
has no more real solutions.
6 x + 15y
+ 106 = 0
7 a log10 P = 0.01t + 2
b 100, initial population
c 1.023
d Accept answers from 195 to 200
8 1 + cos4 x β sin4 x β‘ 1 + (cos2x + sin2x)(cos2 β sin2x)
β‘ (1 β sin2x) + cos2x β‘ 2cos2x
9 Magnitude = β ___
29 , angle = 112Β° (3 s.f.)
10 a 56.5Β° (3 s.f.) b Β£49.63
11 a
O
27β28β1y = g(x)
b β4, β1, 4
12 x = 3 or β 5 __ 3
13 a 1 β 15x + 90
x2
b 0.859
c Greater: The next term will be subtracting from this,
and future positive terms will be smaller
.
14 a f(x
) = β« ( x β 3 _ 2 β 1 β x β2 ) dx = β2 x β 1 _ 2 β x + x β1 + c
= β x 2 + 2 β __ x β 1 ____________ x + c
b c = 5 __ 3 + 2 __ 3 β __
3
15 a 52 + 12 β 4 Γ 5 + 6 Γ 1 = 12, so (5, 1) lies on C.
Centre = (2, β3), radius = 5
b y = β 3 __ 4 x + 19 __ 4
c 12 15 __ 32
|
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] |
399
Index
399elimination 39β40
equations
circle 117β119
normal 268β269
solving using logarithms
324β325
straight line 90, 93β95
tangent 268β269
equilater
al triangles 208
exhaustion, proof
by 150
expanding brack
ets 4β6,
163β164
exponential functions 312β318
gradient functions of
314β316
graphs 312β313, 326
inverses 326
modelling with 317
exponents 2β3, 312
expressions
expanding 4β6
factorising 6β8
simplifying 2β3
factor theorem 143β144
factorial notation 161
factorising
expressions 6β8
polynomials 143β144
quadratic equa
tions 19β21
force 248
for
mula
completing the square
22β23
distance between points
100β101
gradient of str
aight line
91β92
quadratic 21
fractions
, simplifying
algebraic 138β139
function notation
first order deriv
ative 260, 289
second order deriv
ative 271
functions 25β26
cubic 60β62
decreasing 270β271
domain 25
finding 90β91
gradient see
gradient
functions
increasing 270β271
quadra
tic see quadratic
functions
quartic 64β65
range 25
reciproca
l 66β67
roots 25, 30β31
transfor
ming 79β80, 194β197
fundamental theorem of
calculus 296
geometric prob
lems, solving
with vectors 244β246
gradient
curve 256β257
normal 268β269
paralle
l lines 97 perpendicular lines 98β99
straight line 91β92
tangent 256, 268β269
zero 264, 273
gradient functions 260β261
of exponentia
l functions
314β316
rate of
change of 271β272,
274
sketching 277β278
gra
phs 59β80
cosine 192β193, 195β197,
203
cubic 60β62, 68
exponential functions
312β313, 326
inequalities on 51β52
intersection points 42β43,
68β69
logarithms 326
periodic 192
quadratic functions 27β29
quartic 64β65
reciproca
l 66β67
reflections 77
regions 53β54
simultaneous equations
43β45
sine 192β194, 195β196, 203
stretching 75β77
tangent 193, 195β197
translating 71β73
identities 146, 209
identity symbol 146
images 79
βimplies thatβ
symbol 19
indefinite integrals 290β291
indices 2β3, 312
fractional 9β11
laws of 2β3, 9
negativ
e 9β11
inequalities 46β54
on graphs 51β52
linear 46β48
quadratic 49β50
regions satisfying 53β54
inflection, point of 273β274
initial va
lue 317
integration 287β304
areas between curv
es and
lines 302β304
areas under curves 297β298
areas under x
-axis 300β301
definite 295β296
finding functions 90β91
indefinite 290β291
polynomials 289, 290
symbol 290
xn 288β289
intersection points 42β43,
68β69
intervals 270
inv
erse trigonometric
functions 213
irra
tional numbers 12
isosceles right-angled triangles
208acute angles 206β207
algebr
a 1β17
alge
braic fractions,
simplifying 138β139
alge
braic methods 137β153
angles
acute 206β207
in all quadrants 203β207
negativ
e 204
obtuse 203
positive 203
reflex 211
appr
oximations 167
area
trapezium 303
triangle 100β101, 185β186,
303
under curve 297β298
assumptions 106
asymptotes 66
horizontal 277, 278
translation 73
vertical 277
bases 2, 319, 325
binomial estimation 167β168
binomial expansion 158β168
in ascending powers of
x
164
combinations 161
factorial notation 161
general ter
m 165
ignoring large pow
ers of x
167β168
Pascalβ
s triangle 159β160,
161
solving binomial prob
lems
165β166
brackets
expanding 4β6, 163β164
minus sign outside 2
CAST diagram 205
chords 123
perpendicular bisectors
123β125, 129β130
circles 113β130
equation 117β119
finding centre 129
intersections of straight lines
and 121
triangles and 128β130
unit 203, 209
circumcentre 117
circumcir
cle 128β129
column vectors 235β237
combinations 161
common factors 7
completed square for
m 22
completing the square 22β23
solving quadratic equa
tions
by 23β24
conjectures 146
constant of integr
ation 288,
293
coordinates 90
cos ΞΈ
of any angle 207 graph 192β193, 195β197, 203
quadrants wher
e positive or
negative 205
quadratic equa
tions in
219β221
ratio 174
value f
|
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-0.05548501014709473,
-0.032437991350889206,
-0.001173192635178566,
0.09727156907320023,
-0.018543338403105736,
0.031593214720487595,
-0.009377884678542614,
0.01926691085100174,
0.04316052421927452,
-0.0749821588397026,
-0.03320608660578728,
0.05628403276205063,
-0.03562207147479057,
-0.010617070831358433,
-0.022299909964203835,
-0.034063078463077545,
-0.02928769588470459,
-0.017095351591706276,
-0.0018517498392611742,
-0.024828191846609116,
0.015470292419195175,
-0.05752359703183174,
0.09977369755506516,
-0.008606942370533943,
0.057595349848270416,
-0.10790007561445236,
0.00543336383998394,
-0.022239085286855698,
-0.04112393409013748,
-0.06633035093545914,
-0.028508881106972694,
0.019128503277897835,
-0.026154842227697372,
-0.03388184681534767,
0.033820390701293945,
0.023471543565392494,
0.0255377646535635,
0.06012541055679321,
0.009340770542621613,
-0.029847467318177223,
0.06972253322601318,
-0.0003601949429139495,
-0.09488921612501144,
-0.06853747367858887,
-0.027658402919769287,
0.026331590488553047,
-0.037491921335458755,
-0.03000744618475437,
-0.034330882132053375,
-0.015291151590645313,
-0.012796543538570404,
-0.03405126929283142,
-0.025975460186600685,
0.0897430032491684,
-0.08812513947486877,
0.0019224691204726696,
-0.005449049174785614,
-0.0010943389497697353,
-0.019852159544825554,
0.010533488355576992,
0.08549441397190094,
-0.03035317361354828,
-0.06553031504154205,
0.03057974949479103,
0.07097375392913818,
-0.05070195719599724,
-0.01769619807600975,
-0.02863304316997528,
0.07544230669736862,
0.018915874883532524,
0.049057770520448685,
0.03139270469546318,
-0.02061668038368225,
-0.021261688321828842,
-0.005353532265871763,
-0.01663256622850895,
0.04299656301736832,
1.0386295452156522e-31,
-0.06735321134328842,
0.03286592662334442,
-0.0618763267993927,
0.015562422573566437,
0.054048795253038406,
-0.001445917529053986,
0.01881047897040844,
-0.012054764665663242,
-0.034649476408958435,
-0.03983209654688835,
0.08885978162288666,
0.04172034189105034,
-0.02170518785715103,
0.03316066786646843,
-0.056959182024002075,
-0.0082845538854599,
-0.09982765465974808,
0.04171699285507202,
-0.04410411790013313,
-0.05559539794921875,
-0.10788504779338837,
0.0694986879825592,
0.03662398084998131,
0.018492208793759346,
0.09265358746051788,
0.0037031969986855984,
0.02189776673913002,
-0.04008407145738602,
-0.07256889343261719,
-0.0700274407863617,
0.07874686270952225,
0.014695452526211739,
-0.059326354414224625,
-0.06651167571544647,
-0.07343479245901108,
0.06394044309854507,
-0.014769923873245716,
0.09507647901773453,
-0.07670058310031891,
0.05618244782090187,
0.006682863458991051,
-0.06258274614810944,
0.06664713472127914,
-0.050835736095905304,
-0.04376833513379097,
-0.03429349511861801,
-0.060645200312137604,
-0.03744681552052498,
-0.054313503205776215,
0.027687739580869675,
-0.0014630183577537537,
0.006245661061257124,
0.004608859308063984,
0.04834122955799103,
-0.01211458072066307,
-0.010531168431043625,
0.03196471557021141,
0.012930001132190228,
-0.014725669287145138,
0.003410382429137826,
0.04770699888467789,
0.09625041484832764,
0.036504097282886505,
-0.01964627392590046
] |
or 30Β°, 45Β° and 60Β°
208
value f
or multiples of 90Β°
192, 204
cos ΞΈ =
k 213β215
cos (ΞΈ
+ Ξ±) = k 218
cos nΞΈ
= k 217
cosβ1 ΞΈ 213
cosine rule 174β176,
187β189
counter-e
xamples 150β151
critical v
alues 49
cube roots 10
cubic functions 60β62
cubic graphs 60β62
compared with quadra
tic
graphs 68
curves
areas between lines
and 302β304
areas under 297β298
gradients 256β257
definite integra
ls 295β296
delta x
(Ξ΄x) 259
demonstra
tion 146
denominator
, rationalising
13β14
deriva
tive
finding 259β261
first order 271
second order 271β272
straight line 264
difference of tw
o squares 7
differentiation 255β280
decreasing functions
270β271
finding deriva
tive 259β261
from first principles 260
functions with two or more
ter
ms 266β267
increasing functions
270β271
modelling with 279β280
quadratic functions
264β265
second order
deriv
atives 271β272
sketching gradient
functions 277β278
stationary points 273β276
xn 262β263
directed line segments 231
direction of v
ector 231,
239β240
discriminant 30β31, 43β44
displacement 235, 248
distance 248
between points 100β101
division law of
logarithms
321β323
domain 25Answers
Index
|
[
0.022473951801657677,
0.015355157665908337,
0.00029580481350421906,
-0.0230964794754982,
-0.04679643735289574,
0.0022889147512614727,
-0.11842477321624756,
0.037406258285045624,
-0.08646530658006668,
-0.0455658994615078,
0.02838910184800625,
-0.06168826296925545,
0.00373494578525424,
0.01467196736484766,
0.04278171807527542,
0.025423871353268623,
-0.07261794060468674,
0.09458602219820023,
-0.024445151910185814,
-0.03148080036044121,
-0.024603458121418953,
-0.04083821922540665,
0.00966736488044262,
-0.07588065415620804,
-0.018021348863840103,
-0.010807107202708721,
0.034652192145586014,
-0.015026112087070942,
0.04333364963531494,
0.027969924733042717,
0.005982212256640196,
-0.019436178728938103,
0.05375728756189346,
-0.057980041950941086,
-0.05326513200998306,
-0.08805800974369049,
-0.00534099480137229,
-0.01390533335506916,
-0.012137848883867264,
0.04272584244608879,
-0.01636326126754284,
0.0542772151529789,
0.04790719226002693,
0.0877314880490303,
0.03365882486104965,
-0.012475760653614998,
-0.018621409311890602,
-0.008516157977283001,
-0.0011683928314596415,
0.028886957094073296,
0.06893599033355713,
0.059858329594135284,
-0.17199867963790894,
0.008566820062696934,
-0.015412759967148304,
0.0063189356587827206,
-0.03331592306494713,
-0.07703648507595062,
-0.024502627551555634,
0.023755822330713272,
0.059663377702236176,
0.028217332437634468,
-0.01762070134282112,
-0.00019740099378395826,
0.04177406057715416,
-0.012008057907223701,
0.037266530096530914,
-0.08483684062957764,
-0.046400777995586395,
0.08760194480419159,
-0.019557280465960503,
0.045324333012104034,
0.011737091466784477,
-0.08989414572715759,
0.005182343069463968,
-0.030554650351405144,
0.003747715847566724,
0.038459327071905136,
-0.08503259718418121,
-0.07222729176282883,
-0.03513695299625397,
0.00282475957646966,
0.039236459881067276,
0.027153918519616127,
-0.0014317702734842896,
0.04456483572721481,
0.02098962850868702,
0.1247813031077385,
-0.0000828938718768768,
-0.005412095692008734,
0.06512005627155304,
-0.02187894657254219,
-0.04784815013408661,
-0.09005332738161087,
-0.0188571996986866,
0.07993298768997192,
-0.010003916919231415,
-0.07435662299394608,
-0.014567503705620766,
0.051453687250614166,
-0.026334920898079872,
-0.07634887099266052,
-0.06171035394072533,
0.04518274962902069,
-0.020339837297797203,
-0.008418566547334194,
0.11939369887113571,
0.029580604285001755,
0.03591685742139816,
-0.020415056496858597,
-0.09178242832422256,
0.07052929699420929,
-0.009738505817949772,
-0.045162130147218704,
0.09408757090568542,
-0.04306695610284805,
0.019872838631272316,
0.022495275363326073,
0.049055855721235275,
0.006769524421542883,
-0.0286040510982275,
-0.042921099811792374,
-0.02069132588803768,
-0.01938595063984394,
-0.048646364361047745,
-0.03537728637456894,
-0.03593840450048447,
-0.05341131240129471,
-0.005169340874999762,
0.01122249010950327,
0.06895720958709717,
0.03202866017818451,
-0.02610625885426998,
-0.04515562951564789,
-0.03609558939933777,
0.004897012375295162,
0.09811808913946152,
0.09249450266361237,
-0.019625307992100716,
-0.014454788528382778,
0.03779870644211769,
0.061928875744342804,
0.034435249865055084,
0.061415817588567734,
0.03330954536795616,
-0.020954173058271408,
-0.13886772096157074,
0.030115924775600433,
-0.03349218890070915,
0.016220886260271072,
0.10022557526826859,
0.03205138444900513,
0.0491819828748703,
0.09949043393135071,
-0.037678707391023636,
0.0793270692229271,
-0.08368538320064545,
0.04591229185461998,
0.06444896012544632,
0.004512660671025515,
-0.0024350357707589865,
0.037552282214164734,
0.028100037947297096,
0.0512375645339489,
0.010837390087544918,
-0.014301074668765068,
0.04559267684817314,
-0.031909625977277756,
0.004550011828541756,
0.026385648176074028,
0.005825994070619345,
-0.020364638417959213,
0.0038109030574560165,
0.042716316878795624,
0.013725114054977894,
-0.005900203716009855,
0.04059525951743126,
-0.006432059220969677,
0.025657789781689644,
-0.03606471046805382,
0.04629344120621681,
-0.05809203162789345,
-0.0023382757790386677,
-0.008882332593202591,
0.03745398670434952,
0.024484360590577126,
-0.05516146123409271,
0.04182468727231026,
-0.03248074650764465,
0.0017178519628942013,
0.025461863726377487,
-0.06493998318910599,
-0.040689270943403244,
0.0801801010966301,
-0.1128232330083847,
0.030226385220885277,
-0.019928911700844765,
-0.06530509889125824,
-0.11582174897193909,
0.05590090900659561,
0.06550633162260056,
-0.047787971794605255,
-0.00937885046005249,
0.02919318526983261,
0.0472637414932251,
-0.03965556621551514,
-0.008578559383749962,
-0.04097616672515869,
-0.1280830204486847,
0.006514245644211769,
-0.04330257326364517,
-0.06855671852827072,
-0.03941259905695915,
-0.06725157052278519,
-0.008658979088068008,
0.004660331178456545,
0.049574293196201324,
-0.08918775618076324,
-0.11519763618707657,
-0.027224205434322357,
-0.0022995101753622293,
0.006877950392663479,
-0.03926076367497444,
-2.4075966524893377e-33,
-0.09714324027299881,
-0.03538701683282852,
-0.06743966788053513,
0.05772282928228378,
-0.04385410621762276,
0.020971236750483513,
0.11604348570108414,
0.052000779658555984,
0.08730611950159073,
-0.0006881935405544937,
0.06415411829948425,
0.0016829101368784904,
-0.09373854845762253,
-0.016795463860034943,
-0.06500975042581558,
-0.009391820058226585,
-0.018621446564793587,
-0.003594325389713049,
-0.0027166956569999456,
-0.021585887297987938,
0.044596295803785324,
0.06485588103532791,
-0.0016698932740837336,
0.06166413053870201,
-0.017946511507034302,
0.04021540284156799,
0.04094284027814865,
-0.12963174283504486,
-0.10752847790718079,
0.013014502823352814,
-0.03595355898141861,
0.13508330285549164,
-0.00648691039532423,
0.10478959232568741,
-0.1064273789525032,
-0.05698837339878082,
0.0473824217915535,
-0.031364262104034424,
-0.07865923643112183,
-0.019445018842816353,
0.05333629250526428,
0.0663227066397667,
0.12631292641162872,
0.007374232169240713,
0.010142186656594276,
-0.028799189254641533,
0.03038480505347252,
-0.022198516875505447,
-0.08605396747589111,
0.041277311742305756,
0.022483235225081444,
-0.04798687994480133,
0.0023717035073786974,
0.07019449025392532,
0.11108743399381638,
-0.01804414950311184,
-0.059633269906044006,
0.04456615820527077,
0.09523853659629822,
-0.04844560846686363,
-0.02351919375360012,
0.01740281470119953,
-0.07163678854703903,
0.05055193230509758,
-0.020191863179206848,
-0.031755343079566956,
-0.00006585331720998511,
-0.015378306619822979,
0.04569055885076523,
0.039681293070316315,
-0.003706823568791151,
0.04917503893375397,
-0.04928465932607651,
-0.004611415322870016,
-0.13772447407245636,
-0.0160331130027771,
0.09554673731327057,
0.04370597004890442,
0.001447472721338272,
-0.02548755705356598,
-0.09610569477081299,
0.05479644611477852,
-0.02643275074660778,
-0.05262349918484688,
-0.022934479638934135,
-0.06466419249773026,
-0.009162969887256622,
0.006665751338005066,
0.11535441130399704,
-0.018778430297970772,
-0.017741311341524124,
0.0365922786295414,
-0.05599566921591759,
0.00584834162145853,
0.048959847539663315,
9.99900180190805e-32,
-0.09879240393638611,
0.057362787425518036,
-0.04131266102194786,
0.052400797605514526,
0.0089588463306427,
-0.019262028858065605,
0.0023113403003662825,
-0.029491275548934937,
0.04427647218108177,
-0.04881870746612549,
0.09914984554052353,
0.04411754384636879,
-0.06967958062887192,
-0.015737826004624367,
-0.06955359131097794,
-0.06725338846445084,
0.0007641169358976185,
0.12214149534702301,
-0.0014826965052634478,
-0.024424895644187927,
0.010637423023581505,
0.012930133379995823,
0.01876635104417801,
-0.014330514706671238,
0.03767060115933418,
0.04978218302130699,
0.01826869323849678,
0.006209683138877153,
-0.004910064861178398,
-0.02170102298259735,
-0.05239417776465416,
-0.019468458369374275,
0.014703686349093914,
-0.017654040828347206,
0.01721230521798134,
0.00021458871196955442,
-0.06777430325746536,
0.026482749730348587,
-0.07887740433216095,
-0.0184714887291193,
0.01636243425309658,
-0.036495391279459,
-0.027207329869270325,
-0.0031628611031919718,
0.01361407246440649,
-0.0246158167719841,
-0.06600923091173172,
-0.020740779116749763,
0.00041385600343346596,
0.018007375299930573,
-0.0665498897433281,
0.010346248745918274,
-0.01337860431522131,
0.005408560391515493,
0.019521396607160568,
-0.02652636356651783,
0.014242999255657196,
-0.014894046820700169,
-0.015594897791743279,
0.003445212496444583,
-0.027833444997668266,
0.05484737828373909,
-0.03456367179751396,
-0.024999327957630157
] |
400
Index
400tan ΞΈ = p 213β215
tan (ΞΈ
+ Ξ±) = p 218
tan nΞΈ
= p 217
tanβ1 ΞΈ 213
tangents 43, 123β125, 256
equations 268β269
gradients 256, 268β269
theorems 146
transf
ormations 71β80
to functions 79β80, 194β197
translations 71β73
tra
peziums, areas 303
triangle law 231
triangles
areas 100β101, 185β186, 303
circles and 128β130
equilatera
l 208
isosceles right-angled 208
lengths of
sides 174 β176,
179β180
right-angled 187β188
sizes of
angles 174 β175,
180β181, 183β184
solving problems 187β188
trigonometric equations
213β221
trigonometric functions,
inv
erse 213
trigonometric gra
phs 192β194
transfor
ming 194 β197
trigonometric identities
209β211, 221
trigonometric ratios 173β197
signs in four quadrants
203β207
turning points 27β29, 264
unit circle 203, 209
unit vectors 236, 239
value
initial 317
produced by definite
integr
al 295
vectors 231β250
addition 231β233
column 235β237
direction 231, 239β240
magnitude 231, 239β240, 248
modelling with 248β250
multiplication b
y scalars 232
paralle
l 232, 233
position 242β243
representing 235β237
solving geometric problems
with 244 β246
subtraction 232
two-dimensional 236
unit 236, 239
zero 232
ve
locity 248
vertices 127
y
= ex 314 β316
y
= mx + c 90β92
y-inter
cept 91
zer
o gradient 264, 273
zero v
ector 232reciproca
l functions 66β67
reciproca
l graphs 66β67
reflections 77
refle
x angles 211
regions 53β54
resultant 231
right-angled triangles
187β188
isosceles 208
roots 19
of function 25, 30β31
number of 30β31
repeated 19
scalars 232, 248
scale factors 75β77
semicircle, angle in 129
set notation 46
simplifying expressions 2β3
simultaneous equations 39β45
on graphs 43β45
linear 39β40
quadratic 41β42
sin ΞΈ
of any angle 207
graph 192β194, 195β196, 203
quadrants wher
e positive or
negative 205
quadratic equa
tions in
219β221
ratio 174
value f
or 30Β°, 45Β° and 60Β°
208
value f
or multiples of 90Β°
192, 204
sin ΞΈ =
k 213β215
sin (ΞΈ
+ Ξ±) = k 218
sin nΞΈ
= k 217
sinβ1 ΞΈ 213
sin2 ΞΈ + cos2 ΞΈ β‘ 1 209, 221
sine rule 179β181, 187β188
small change 259
speed 248
square r
oots 10
statements 146
stationary points 273β276
straight lines 89β106
equation 90, 93β95
gradient 91β92
intersections of circles and
121
modelling with 103β106
stretches 75β77
substitution 39β40
surds 12β13
symmetry, lines of 27β29
tan ΞΈ
of any angle 207
graph 193, 195β197
quadrants wher
e positive or
negative 205
quadratic equa
tions in 219
ratio 174
value f
or 30Β°, 45Β° and 60Β°
208
value f
or multiples of 90Β°
193, 204
tan ΞΈ =
sin ΞΈ _____ cos ΞΈ 29non-linear data, lo
garithms
and 328β330
norma
ls 268
equations 268β269
number lines 46β48
obtuse angles 203
parabola 27
para
llel lines 97β98
paralle
logram law 233
Pascalβ
s triangle 159β160,
161
periodic gra
phs 192
perpendicular bisectors
chords 123β125, 129β130
line segments 116
perpendicular lines 98β99
product of gr
adients 98
βplus or minusβ symbol 20
point of inflection 273β274
points of intersection 42β43,
68β69
polynomials 139β144
dividing 139β141
factorising 143β144
integrating 289, 290
position vectors 242β243
positive integers 163
power la
w of logarithms
321β323
powers 2β3, 312
principal va
lue 213β214
products 4β6
proof 146β148
methods of 150β152
Pythagorasβ
theorem 117, 187
quadratic equa
tions
form 19
signs of solutions 19
solving by completing the
square 23β24
solving by factorisation
19β21
in trigonometric ratios
219β221
quadratic e
xpressions 7
quadratic f
ormula 21
quadratic functions 25β26
differentiating 264β265
graphs 27β29
modelling with 32β33
quadratic gr
aphs, compared
with cubic graphs 68
quadratic inequa
lities 49β50
quadratic sim
ultaneous
equations 41β42
quartic functions 64β65
quartic graphs 64β65
quotient 140
range 25
ra
te of change 105, 279
ratio theor
em 244
rationa
l numbers 9
rationa
|
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] |
lising denominators
13β14
real n
umbers 7, 232key points summaries
algebr
aic expressions 17
algebr
aic methods 156
binomial expansion 172
circles 135β136
differentiation 286
equations 58
exponentials and lo
garithms
336β337
graphs 84
inequalities 58
integration 310
quadratics 37
straight line gra
phs 111β112
trigonometric identities and
equations 224β225
trigonometric ratios 201
vectors 254
known facts
, starting proof
from 151β152
laws of
indices 2β3, 9
length 100β101
limits 260
integral betw
een 295β296
line segments 114
directed 231
midpoints 114
perpendicular bisectors 116
linear inequalities 46β48
local maximum 273β276
local minimum 273β276
logarithms 319β330
base 319, 325
division law 321β323
graphs of 326
multiplication la
w 321β323
natura
l 320, 326β327
and non-linear data
328β330
power la
w 321β323
solving equations using
324 β325
to base 10 320
magnitude-direction f
orm 240
magnitude of v
ector 231,
239β240, 248
mathematica
l models 32, 104
mathematica
l proof 146β148
methods of 150β152
maximum point 27β29
midpoints, line segments 114
minimum point 27β29
modelling
with differentiation
279β280
exponential 317
linear 103β106
with quadratic
functions 32β33
with vectors 248β250
modulus of vector 239
multiplication la
w of
logarithms 321β323
natur
al logarithms 320,
326β327
natura
l numbers 163
|
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