pure-mathematics-1-pdf

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( )6 a 121, −2 b k = −4 or k = −20 3 ay

7 a Proof b (6, 29) 6
c k = 1, C = (2, 5)
4
8 a Proof b (2, 1), (5, 7),
2
c 2,x,5

9 a 25 − (x − 5)2 b (5,  25)

c x < 1 or x > 9 O 2 4x

10 i 3 5 , − 1 , 5 ii k = 3 or 11 b each input does not have a unique output
2
Review
( )11 i2 1 , 2 1 ii m = −8, (−2, 16) 4 a domain: x ∈ R, −1 < x < 5
2 2 range: f(x) ∈ R, −8 < f(x) < 8

12 i 2(x − 1)2 − 1, (1, −1) ii  − 1 , 3 1 b domain: x ∈ R, −3 < x < 2
 2 2
range: f(x) ∈ R, −7 < f(x) < 20

iii y − 3 = − 1 (x − 2) 5 a f(x) . 12 b −13 < f(x) < −3
5
c −1 < f(x) < 9 d 2 < f(x) < 32

2 Functions e 1 < f(x) < 16 f 3 < f(x) < 12
32 2
Prerequisite knowledge
6 a f(x) > −2 b 3 < f(x) < 28
1 10
c f(x) < 3 d −5 < f(x) < 7
2 3 − 2x
3 f −1(x) = x − 4 7 a f(x) > 5 b f(x) > −7

5 c −17 < f(x) < 8 d f(x) > 1 289
4 2(x − 3)2 − 13
Review 8 a f(x) > −20 b f(x) > −6 1
3

9 a f(x) < 23 b f(x) < 5

Exercise 2A 10 a y

6

1 a function, one-one b function, many-one

c function, one-one d function, one-one 4

e function, one-one f function, one-one

g function, one-one h not a function 2

2a y
10

O 2 4x

8

–2

6 b −1 < f(x) < 5

Review 4 11 f(x) > k − 9
12 g(x) < a2 + 5
2
8
13 a = 2

14 a = 1 or a = −5

–4 –2 O 2 4x 15 a 2( x − 2 )2 − 3 b k=4

b Many-one c x ∈ R, −3 < x < 5

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16 a domain: x ∈ R c RR(x), domain is x ∈ R, x ≠ 0,
range: f(x) ∈ R range is f(x) ∈ R, f(x) ≠ 0

b domain: x ∈ R d QPR(x), domain is x ∈ R, x ≠ 0,
range: f(x) ∈ R, f(x) > 2 range is f(x) ∈ R, f(x) . 1

c domain: x ∈ R e RQQ(x), domain is x ∈ R, x ≠ −4,
range: f(x) ∈ R, f(x) . 0 range is f(x) ∈ R, f(x) ≠ 0

d domain: x ∈ R, x ≠ 0, f PS(x), domain is x ∈ R, x > −1,
range: f(x) ∈ R, f(x) ≠ 0 range is f(x) ∈ R, f(x) > −1

e domain: x ∈ R, x ≠ 2, g SP(x), domain is x ∈ R, x > −1,
range: f(x) ∈ R, f(x) ≠ 0 range is f(x) ∈ R, f(x) > −1

Review f domain: x ∈ R, x > 3, Exercise 2C
range: f(x) ∈ R, f(x) > −2

Exercise 2B 1 a f −1(x) = x + 8 b f −1(x) = x − 3
5

1 a7 b 3 c 231 c f −1(x) = 5 + x − 3 d f −1(x) = 3x + 8
x
2 a hk b kh c hh
7 − 2x
3 a a = 3, b = −12 b 5 7 e f −1(x) = x−1 f f −1(x) = 2 + 3 x + 1
9

4 a − x 6 1 b −4 2 a Domain is x > −4, range is f −1(x) > −2
+
b f −1(x) = −2 + x + 4
5 a (2x + 5)2 − 2 b −4 1 or − 1
2 2 3 a f −1(x) = 5 − x
2x
1 1 b x<1
2 2
290 6 or 3 4 a f −1(x) = −1 + 3 x + 4 b x > −3

7 − 4 or 0 5 a g is one-one for x > 3, since vertex = ( 2, 2 )
3
Review b g−1(x) = 2 + x − 2
8 −9 2

9 x+2 6 a −3 b f −1(x) = −3 + x + 32
4x + 9 7 a f(x) > −9 2

10 a fg b gf c gg

d ff e gfg f fgf b No inverse since it is not one-one

11 Proof 8 a k=3

12 ±4 b i f −1(x) = 3 + 9 − x

13 k > − 19 ii Domain is x < 9, range is 3 < f −1(x) < 7
2
1
14 Proof 9 a f(x) = 5−x b Domain is x < 4 2
3

15 a 2(x + 1)2 − 10 b −1 10 a = 5, b = 12
16 a x < −1 or x > 3 b (x − 1)2 + 3
11 a f −1(x) = x + 1, g−1(x) = 4x + 3
c f(x) > 3 3 2x

Review17 a 4x2 + 2x − 6 b fg(x) > −6 1 b Proof
4
12 a f −1(x) = 1 (1 + 3 x + 3 )
18 a ff (x) = 2(x + 1) for x ∈ R, x ≠ −3
x+3 2
b Domain is −2 < x < 122
b Proof c −2 or 1
13 a f(x) = (x − 5)2 − 25
19 a PQ(x), domain is x ∈ R, b f −1(x) = 5 + x + 25 , domain is x > −25

range is f(x) ∈ R, f(x) > −1

b QP(x), domain is x ∈ R,
range is f(x) ∈ R, f(x) > 1

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14 a f −1(x) = x + 1 b Proof cy y=x
x
6
c 1± 5
2 5

15 b and c ii 7 − x 4 f–1 f
6 3
16 a 7 − x
6 2

b i 14 − x 1
6
O 1 2 3 4 5 6x
c (fg)−1(x) = g−1 f −1(x)
d f −1 does not exist since f is not one-one
Review Exercise 2D y 2 a f −1(x) = x + 1
6 f –1 y = x
1a 4 2
b Domain is −3 < x < 5, range is −1 < x < 3
2f
cy

6 y=x

4f

2 f –1

–6 –4 –2 O 2 4 6x
–2
–4 –4 –2 O 2 4 6x
–6 –2
291

Review –4

3 a 0 , f(x) < 2 b f −1(x) = 4 − 2x
x

by c Domain is 0 , x < 2, range is f −1(x) > 0

6 d y f –1 y=x
y=x
f
f4

2

–6 –4 –2 O 6x Ox
–2
–4 24 4 a Symmetrical about y = x
–6 f –1
b Not symmetrical about y = x

Review c Symmetrical about y = x

d Symmetrical about y = x

5 a Proof b d = −a

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Exercise 2E 4a y
4
1 a y = 2x2 + 4 b y=5 x−2 3
2
c y = 7x2 − 2x + 1 d y = x2 + 1 1

e y = 2 f y = x − 3
x+5 x − 2

g y = (x + 1)2 + x + 1 h y = 3(x − 2)2 + 1 –4 –2 O 2 4x
–1

 0   0 –2
 4   −5 
2 a Translation b Translation –3

–4

Review  −1 d Translation  2  b a=2 c b = −1
c Translation  0   0 
5 y = (x + 1)(x − 4)(x − 7)

e Translation  1  f Translation  2  6 y = x2 − 6x + 8
 0   4 

3a y 7 a = 2,  b = −3,  c = 1

4

3 Exercise 2F

2 1a

1 y
4
–4 –3 –2 –1–O1 1 2 3 4 x 3
2
–2 1
292 –3
–4 –3 –2 –1–O1 1 2 3 4x
–4 –2
–3
Review by –4

4 by
3
2 4
1 3

–4 –3 –2 –1–O1 1 2 3 4x 2
–2
–3 1
–4
–4 –3 –2 –1–O1 1 2 3 4x
–2
Review cy –3
–4
4
3 2 a y = −5x2 b y = 2x4
2
1

–4 –3 –2 –1–O1 1 2 3 4x c y = 2x2 + 3x + 1 d y = 3x2 − 2x − 5
–2
–3 3 a Reflection in the x-axis
–4
b Reflection in the y-axis

c Reflection in the x-axis

d Reflection in the x-axis

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Exercise 2G y Exercise 2H y
6 4
1a 1a 3
4 2
1
2

–4 –3 –2 –1–1O 1 2 3 4x
–2
–6 –4 –2 O 2 4 6x –3
–2 –4
Review –4
–6 by

by 4
3
6
2
4
1
2
–4 –3 –2 –1–1O 1 2 3 4x
–2
–3
–4

293

–6 –4 –2 O 2 4 6x cy
–2
Review –4 4
–6 3
2
1

–4 –3 –2 –1–1O 1 2 3 4x
–2
2 a y = 6x2 b y = 3x3 − 3 –3
–4
d y = 1 x2 − 4x + 10
c y = 2x −1 + 2 2

e y = 162x3 − 108x dy

Review 3 a Stretch parallel to the x-axis with stretch 4
factor 1 3
2
2
b Stretch parallel to the y-axis with stretch
factor 3 1

c Stretch parallel to the y-axis with stretch –4 –3 –2 –1–1O 1 2 3 4x
factor 2 –2
–3
d Stretch parallel to the x-axis with stretch –4
factor 1
3

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ey 4 a y = 1 (x − 5)2 b y =  1 x − 5  2
4 2
4
3 cy ab

2 y = x2

1

–4 –3 –2 –1–1O 1 2 3 4x
–2
–3 O 5 10 x
–4
5 a y = 2x2 − 8 b y = −x2 + 4x − 5
Review fy 6 a y = 2g(−x) b y = 3 − f(x − 2)

4 7 a Stretch parallel to the y-axis with stretch
3
factor 1 followed by a translation  0
2 2  3 

1 b Reflection in the x-axis followed by a

–4 –3 –2 –1–1O 1 2 3 4x translation  0 
–2  2 
–3
–4 c Translation  6 followed by a stretch parallel
 0 
to the x-axis with stretch factor 1
g y 2
4
294 3 d Stretch parallel to the y-axis with stretch
2
1 factor 2 followed by a translation  0
 −8 
Review
8 a Translation  −5  followed by a stretch
 0 
–4 –3 –2 –1–1O 1 2 3 4x
–2
–3 parallel to the y-axis with stretch factor 1
–4 2

b Translation  −1 , stretch parallel to the
 0 

hy y-axis with stretch factor 1 , reflection in the
2
4
3 x -axis, translation  0 
2  −2 
1
c Translation  3  , stretch parallel to the
 0 
–4 –3 –2 –1–1O 1 2 3 4x
–2 y-axis with stretch factor 2, reflection in the
Review –3
–4 x -axis, translation  0 
 4 

2 a y = 2f(−x) b y = 2 − f(x) 9 a y = − 1 x − 1 + 3 b y = − 1 (x − 1) − 3
22
c y = 2f(x − 1) + 1
10 a y = 3[(−x + 4)2 + 2] = 3(4 − x)2 + 6
3 a y = 3(x − 1)2 b y = 3(x − 1)2 b y = 3[(−(x + 4))2 ] + 2 = 3(x + 4)2 + 2

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11 Translation  2 followed by reflection in the 4 a f −1 : x ֏ x + 2 for x > −2
 0 
b y=x
y-axis or reflection in the y-axis followed by
y
translation  −2  f
 0 
4
y = g(x) y y = f(x) 3

f–1
2

1

–2 O 2 4x
–1

–2

Review –2 O x –3

12 Translation  −10  followed by a stretch parallel 5 i −(x − 3)2 + 4 ii 3
 0 
iii f −1(x) = 3 + 4 − x, domain is x < 0

to the x-axis with stretch factor 1 or stretch 6 i (x − 2)2 − 4 + k
2 ii f(x) > k − 4
iii p = 2
parallel to the x-axis with stretch factor 1 iv f −1(x) = 2 + x + 4 − k , domain is x > k − 4
2
 −5  7 i −5 < f(x) < 4
followed by translation  0 

End-of-chapter review exercise 2 ii y y = x

7 2 f–1
6
1 25 − 9 x − 295
4
f

Review 2a y

Ox

O 6x
–2

b Translation  −3  followed by a reflection in  1 (x + 2 ) for −5 < x < 1
 0   3 for 1 , x < 4
iii f −1(x) =  4
 5 −x
the y-axis or reflection in the y-axis followed

by translation  3  8 i 4(x − 3)2 − 25, vertex is (3, −25)
 0 
ii g(x) > −9
3a y
iii g−1(x) = 3 − 1 x + 25 , domain is x > −9
8 2

Review 9 i 2(x − 3)2 − 5 ii 3

–4 –2 O x iii f(x) > 27

b y = 3x2 + 6x iv f −1(x) = 3 + x + 5 , domain is x > 27
2

10 i (x − 1)2 − 16 ii −16

iii p = 6, q = 10 iv f −1(x) = 1 + x + 16

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Cambridge International AS & A Level Mathematics: Pure Mathematics 1e-ssC-aRmebrviidegweCUonipvyersitCyoPprye-ssC-aRmebrviidegweCUonipvyersitCyoPprye-ssC-aRmebrviidegweCUonipvyersitCyoPprye-ssC-aRmebrviidegweCUonipvyersity

11 i 2(x − 3)2 − 11 ii f > −11 11 a (5, 2) b 82
iii −1 , x , 7 iv k = 22
12 A(−5, 5), B(7, 3), C(−3, −3)

12 i fg(x) = 2x2 − 3, gf(x) = 4x2 + 4x − 1

ii a = −1 iii b = 2 Exercise 3B

iv 1 (x2 − 3) v h−1(x) = − x + 2 1 a 1, 1 b Not collinear
2 56

13 i 2(x − 2)2 + 2 ii 2 < f(x) < 10 2 Proof

iii 2 < x < 10 3 − 2 , 5
5 2
iv f(x): half parabola from (0, 10) to (2, 2);
4 (7, −1)
Review g(x): line through O at 45°;
f −1(x): reflection of f(x) in g(x) 5 k=5
7
v f −1(x) = 2 − 1 (x − 2)
2 6 k = 2 or k = 3

3 Coordinate geometry 7 (0, −26)

Prerequisite knowledge 8 a1 b5
9 a = 10, b = 4
( )1 −4 10 a 1 b −2
1 , −2 , 13
2 2 b a = −4, b = 16, c = 11
c a = 6 or a = −4 d 100
2 a −1 b6 11 a (6, 6)
6 b −5 b y = −3x − 1
c 4 145
296 3 a 2 b 4 − 21, 4 + 21 b 9x + 5y = 2
3
b x + 2y = −8
c 7 1 d 3x + 2y = 18
2 b 5x + 3y = 9

Review4 a (x − 4)2 − 21 Exercise 3C

Exercise 3A 1 a y = 2x + 1
c 2x + 3y = 1
1 a PQ = 5 5, QR = 4 5, PR = 3 5 ,
right-angled triangle 2 a 2y = 3x − 3
c 2x − 3y = 9
b PQ = 197, QR = 146, PR = 3 5 ,
not right angled 3 a y = 3x + 4
c x + 2y = 8
2 17 units2
4 a y = 2x + 2
3 a = 3 or a = −9 c 7x + 3y = −6

4 b = 3 or b = −5 4 5 (8, 2)
5
6 a y = 3x + 8
5 a = 2, b = −1 2 b (0, 8)

Review6 a (−2, −1) b (−1, 9) c 39

c 2 41, 2 101 7 a (6, 3) b y = −2x + 7
3
7 k=4
8 a y = 4 x + 10 −7( )b1 ,0 , (0, 10)
3 2
8 38 1 units2
2
c 12 1
2
9 k=2
9 a 2y = 5x + 33 b 33

10 (−2, 6)

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10 E(4, 6), F (10, 3) 10 (x − 3)2 + ( y + 1)2 = 16, (3, −1), 4

11 10 11 y = 3 x − 21
42
12 (14, −2)
12 (x − 5)2 + ( y − 2)2 = 29

13 a y = −3x + 2 b (−1, 5) 13 a Proof b (x + 1)2 + ( y − 4)2 = 20

c 5 10, 4 10 d 100 14 (x − 5)2 + ( y + 3)2 = 40

14 ai y = 4 1 ii x + y = 7 15 (x − 9)2 + ( y − 2)2 = 85
2

( )b 2 1 , 4 1
2 2
16 (x + 3)2 + ( y + 10)2 = 100,
15 a y = 2x − 7 ( )b (x − 13)2 + ( y + 10)2 = 100
4 2 , 154
5

Review 16 x + y = 8, 3x + y = 3. Other solutions possible.

Exercise 3D 17 a i 1 + 2 ii Student’s own answer
b i 3+2 2 ii Student’s own answer
1 a (0, 0), 4
c (0, 2), 5 b (0, 0), 3 2 Exercise 3E
2
1 (−1, −4), (5, 2)
d (5, −3), 2

e (−7, 0), 3 2 f (3, −4), 3 10 2 25
2
g (4, −10), 6
2 a x2 + y2 = 64 ( )h3 1 , 2 1 , 10 3 Proof
2 2

b (x − 5)2 + ( y + 2)2 = 16 4 − 2 , m , 2
29
c (x + 1)2 + ( y − 3)2 = 7

x 1 2  3 2 25 5 a (0, 6), (8, 10) b y = −2x + 16 297
2  2 4
d − + y + = c ( 5 − 5, 6 + 2 5 ), ( 5 + 5, 6 − 2 5 )

Review 3 (x − 2)2 + ( y − 5)2 = 25 d 20 5

4 (x + 2)2 + ( y − 2)2 = 52 6 (4, 3)

5y 7 a (x − 12)2 + ( y − 5)2 = 25 and
(x − 2)2 + ( y − 10)2 = 100
2
b Proof

O End-of-chapter review exercise 3
2 4 6x
1 2 , a , 26
–2

–4 2 i 4 and 1 ii 49
94 24

3 a = −4, b = −1 or a = 12, b = 7

–6 4 10

Review 6 (x − 6)2 + ( y + 5)2 = 25 5 a a = 5, b = −2 b (4, −5)
7 Proof
8 (x − 5)2 + y2 = 8 and (x − 5)2 + ( y − 4)2 = 8 c y = −2 x − 3 2
9 (x − 4)2 + ( y − 2)2 = 20 5 5

6 i 16t2 ii Proof

7 (13, −7)

8 a (−2, 2), (4, 5) b y = −2x + 5 1
2

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9 a (−2, −3) b y = −1 x + 4 3 3 a = 5, b = −2
2 4

c 19 − 113, 19 + 113 4 Translation  5  , vertical stretch with stretch
22  
0

10 i 2, m = 1 ii (−1, 6) factor 2

iii (5, 12) 5 y = −x2 + 6x − 8

11 i y = 2x − 2 ii (0, −2),  8, 6  6y
 5 5 

12 i y = −2x + 6, (3, 0) ii Proof 5

iii (−1, 8), 2 10

Review13 a y = − 2 x + 3 b p = −1 4
3

c (x − 6)2 + ( y + 1)2 = 26 3

14 a (19, 13) b 104
b k , −12, k . 12
15 a  10 , 10  2
3

16 a y = − 4 x + 2 b Proof 1
3

c (x − 15)2 + ( y − 7)2 = 325 –3 –2 –1 O 1 2 3 x

17 a 4, (4, −2) b 4 − 2 3, 4 + 2 3

c Proof d Proof 7 a 1< x <5 b −13, 3
8 (k2, −2k)
298

Cross-topic review exercise 1 9 65

Review1 x = ± 2, x = ± 2 10 a k = 14 b y = 1x + 4
32 3

2a y
30
(–2, 26) 11 a (−1, −11), (6, 3) b k , − 25
12

12 a k = −1 b x.5
2

13 i fg(x) = 5x, range is fg(x) > 0

ii g−1(x) = 4 − 2x , domain is 0,x<2
5x

O 4x 14 a b = −5, c = −14
(2, –6)
–4 y (2, 22) b i (2.5, −20.25) ii −3 , x , 8

b 15 a 2y = 3x + 25 b (−3, 8)

16 a 36 − (x − 6)2 b 36

Review c x < 36, g−1(x) > 6 d g−1(x) = 6 + 36 − x

17 a 3(x + 2)2 − 13 b (−2, −13)

c 6 , x , 18

–4 O 4x 18 a a = 12, b = 2 b −3

c g−1(x) = −3 + 26 − x
2

19 a (x − 8)2 + ( y − 3)2 = 29 b 5x + 2y = 75

(–2, –42)

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20 a x = 1 3 a 0.489 b 0.559
2 d 3.49
c 0.820
b f −1(x) = x + 7 , g−1(x) = 5 − 18 b 45.8°
3x e 5.59 d 87.1°
4 a 68.8°
c Proof
c 76.8°
21 a 17 −  x + 3 2 b  − 3 , 17  e 45.3°
4 2  2 4  5 a.

c −5 and −1 d (1, −2), (−1, 4)

22 a (8, 0) b 10 Degrees 0 45 90 135 180 225 270 315 360
c (−2, 0), (18, 0) d y = −3 x + 6
Review Radians 0 π π 3π π 5π 3π 7π 2π
23 a k = −2 4 4 2 4 4 2 4

b i fg(x) > 28 b. Degrees 0 30 60 90 120 150 180 210
ii (fg)−1(x) = − x + 26 , domain is x > 28,
6 Radians 0 π π π 2π 5π π 7π
range is (fg)−1(x) < −3 632 3 6 6

24 a i (4, 5), (10, 2) ii 4x − 2y = 21 Degrees 240 270 300 330 360
b k = ±4 10
Radians 4π 3π 5π 11π 2π
4 Circular measure 3236

Prerequisite knowledge 6 a 0.644 b 14.1

1 (12 + π) cm, 3π cm2 c 0.622 d0 299
e3 f 0.727
2 13, 67.4
Review 2
3 5.14, 15.4 cm2 7 7.79 cm

8 12.79°

Review Exercise 4A b 2π Exercise 4B b 3π cm
9 d 28π cm
1 aπ 1 a 2π cm b 2.275 cm
9 d 5π c 6π cm b 0.8 rad
18
c 5π 2 a 13 cm b 20.5 cm
36 f 5π 3 a 0.5 rad
6 4 15.6 m b 4 cm
eπ
36 h 7π 5 a 19.2 cm b 11.8 cm
6 c 50.4 cm
g 3π b 2.35 rad
4 j 5π 6 a 0.927 rad
3 c 17.6 cm b 43.4 cm
i 5π
4 l 3π 7 a 14 cm
c 25.8 cm
k 13π n 7π
36 36 8 a 13 cm
c 56.6 cm
mπ b 60° c 30° d 15°
20 h 105° 9 a Proof
f 80° g 54° l 48° 10 Proof
o 10π j 810° k 252°
3 n 420° o 202.5°

2 a 90°

e 240°

i 81°

m 225°

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e-ssC-aRmebrviidegweCUonipvyersitCyoPprye-ssC-aRmebrviidegweCUonipvyersitCyoPprye-ssC-aRmebrviidegweCUonipvyersitCyoPprye-ssC-aRmebrviidegweCUonipvyersityCambridge International AS & A Level Mathematics: Pure Mathematics 1

Exercise 4C b 20π cm2 5 Trigonometry
d 54π cm2
1 a 12π cm2 b 3.042 cm2 Prerequisite knowledge
c 9π cm2 b 1.5 rad
4 b 40 cm2 1 a 1+ r2 b r
b 4.79 cm 1+ r2
2 a 867 cm2 c1 r
b 25 ( 2 )3 − π cm2 1+ r2
3 a 1.125 rad 6 π d −5, 3
ii 4π 2
4 a 1.25 rad 2 a i4 ii 630° iii 5π
b i 30° 6
5 a 1.75 rad
iii 195°
c 5.16 cm2

Review6  32 3 − 32π  cm2 3 a 0, 5 b
 3 

7 a 5 3 cm Exercise 5A b3 c 24
3 4 25
1 a3
8 1.86 cm2 5 e4 f 12
5 19
9 a 29.1cm2 b 36.5 cm2 d 20
c 51.7 cm2 d 13.9 cm2 3 b5
3
10 a Proof b Proof 2 a2
3 d5
11  2πr2 − 3r2  cm2 2
 3 2  c1
f 15(3 − 5 )
300 12 100  1 + π − 3  cm2 e6 4
 3  5
b 15
Review13 a  tan x + 1 x  cm b 0.219 rad 3 a 15 15
 tan  4
d 15
14 a Proof b Proof c 15 16
16
End-of-chapter review exercise 4 f 75 − 4 15
e 4 + 15 15
1 a 15 − 5 3 + 5 3π b 25 3 − 25π
6 48 4 a1 b1
4 2
2 i π ii 8 + 5π
α= ii 4 + 4 tan α + 2α c 3+ 2 d3
2
8 cos α
3 i 8 tan α − 2α e 2− 3
2 f1
4 i r(1 + θ + cos θ + sin θ ) ii 55.2 c1
5 a1 b1 2
5 i AC = r − r cos θ ii 7π + 2 3 − 2 2 4
3 f 2+ 3
d 2 −2 6 e −1 3
2
6 i Proof ii 36 − (r − 6)2

Review iii A = 36, θ = 2 6 θ=π θ=π θ=π
436
7 i Proof ii r2

8 i r  cos θ + 1 − sin θ + π − θ  ii 6.31 tan θ 1 1
2 33

9 i 4α cos α + 4α + 8 − 8 cos α ii π 1 1 3
3
cos θ
ii 3r2α + πr2 22 2
10 i 2πr + rα + 2r 2

12 32
iii α = 2 π sinθ 2
5

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Exercise 5B 9 a −12 b −5
13 12
1 a 70° b 40°
d −3
c 20° d 40° c3 4
5
2 a 2nd quadrant, 80° b 3rd quadrant, 80° b3
2 13
c 4th quadrant, 50° d 3rd quadrant, 30° 10 a − 13
d−7
e 1st quadrant, 40° f 2nd quadrant, π c −7 3
3 4

g 3rd quadrant, π h 1st quadrant, π 11 θ = 120° θ = 135° θ = 210°
6 3
Review tan θ − 3 −1 1
i 3rd quadrant, 4π j 4th quadrant, π 3
9 8 sin θ

3 a 125° b −160° 1 3 1 −1
c 688° d 5π cos θ 2 22

e 8π 4 Exercise 5D
3
f − 13π 1 a 360°
6 c 360°
e 180° −2
Exercise 5C −2 −2 3
2 a1
1 a −sin 10° b cos 55° c7 b 180° 301
c −tan 55° d −cos 65° e4 d 120°
Review e −cos π f −sin π f 180°
5 3 ay b5
g −cos 3π 8
10 h tan 2π 2 d3

2 a −1 9 f2
2 b−3
1
c−2 3
2 O
d3 90 180 270 360 x
e−3
2 f1 –1
2
g−3
3 h −1
2
3 4th quadrant
b−2
4 a − 21 21
5
b2 –2
5 a−2
3 b 12 by
13
6 a −5 2
13
Review ba 1
7 aa 1+ a2

ca d − a O
1+ a2 1 + a2 90 180 270 360 x

8 a 1 − b2 bb –1
1 − b2
c − 1 − b2
d 1 − b2

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Cambridge International AS & A Level Mathematics: Pure Mathematics 1e-ssC-aRmebrviidegweCUonipvyersitCyoPprye-ssC-aRmebrviidegweCUonipvyersitCyoPprye-ssC-aRmebrviidegweCUonipvyersitCyoPprye-ssC-aRmebrviidegweCUonipvyersity

cy gy

2

1

O O
60 120 180 240 300 360 x 90 180 270 360 x

–1

Review dy –2

4 hy 60 120 180 240 300 360 x
3
2 90 180 270 360 x 3
1 2
1
O O
–1 –1
–2 –2
–3 –3
–4
iy

Review302 90 180 270 360 x

ey

5
4
3
2
1

O 90 180 270 360 x
–1
4 ai y

–2 3

–3 2

1

fy O π π 3π 2π x
–1 22
2 –2
1 90 180 270 360 x –3

Review O ii y
–1
–2 2
–3
–4 1

O π π 3π 2π x
22

–1

–2

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iii y 8 a = 4, b = 2, c = 5 y
9 a = 3, b = 2, c = 3 3
2 10 a

1

O π π 3π 2π x 2
22

–1

1

–2

Review  π , 1  5π −1 ,  9π , 1  13π −1 –π – π O π πx
8 8 8 8 2 2
b , , , ,

–1

5 ay –2

3

2 y = 1 + cos 2x –3
1
4
b k = π

O π π 3π 2π x c (0, 0),  − π , −2 
22  2 
–1
y = sin 2x 11 a = 3, b = 1, c = 5 303
–2
Review y = 2 + cos 3x 12 a a = 3, b = 2 b 1 < f(x) < 5
b4 13 a a = 3, b = 5
6 ay π π 3π 2π x f 360 x
22 by 180 270
4
3 y = 2 sin x 10 b5
2 8
1 6
O 4
–1 2
–2
–3 O
–2 90
b2
7 ay –4

4 14 a = 5, b = 4, c = 3
15 a A = 2, B = 6

Review 2 y = 3 sinx
O π π 3π 2π x

22
–2 y = cos 2x

–4

b2
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Cambridge International AS & A Level Mathematics: Pure Mathematics 1e-ssC-aRmebrviidegweCUonipvyersitCyoPprye-ssC-aRmebrviidegweCUonipvyersitCyoPprye-ssC-aRmebrviidegweCUonipvyersitCyoPprye-ssC-aRmebrviidegweCUonipvyersity

cy 7 a −9 < f(x) < −1

8 b f −1(x) = 2 cos−1 x + 5  , 0 < f −1(x) < 2π
4
6

Exercise 5F

4 1 a 56.3°, 236.3° b 23.6°, 156.4°

f c 45.6°, 314.4° d 197.5°, 342.5°

2 e 126.9°, 233.1° f 116.6°, 296.6°

g 60°, 300° h 216.9°, 323.1°
O 30 60 90 120 x 2 a 0.305, 2.84 b π , 5π
Review 33
–2
c 1.25, 4.39 d 3.92, 5.51

–4 e 1.89, 5.03 f 2π, 4π
33

16 y = 2 + sin x g 0.848, 2.29 h 2.19, 5.33
17 y = 6 + cos x
3 a 26.6°, 153.4°

b 17.7°, 42.3°, 137.7°, 162.3°

Exercise 5E c 38.0°, 128.0° d 105°, 165°

e 24.1°, 155.9° f 116.6°, 153.4°

1 a 0° b 30° g 58.3°, 148.3° h 5.77°, 84.2°
4 a 90°, 210° b π , 7π
c 60° d −90°
26
304 e −60° f 135°
Review bπ c 139.1°, 175.9° d 0.0643, 2.36, 3.21, 5.51
2 a0 e 278.2° f 0, 3π
4
cπ d −π 2
4
6 5 a 26.6°, 206.6° b 56.3°, 236.3°
e 2π f −π
3 c 119.7°, 299.7°
3
d 18.4°, 108.4°, 198.4°, 288.4°
3 a 16 b 16
25 9 6 0.298, 1.87

4 a −7 < f(x) < −1 b f −1 ( x ) = sin−1 x + 4  7 a 0°, 150°, 180°, 330°, 360°
3 
b 0°, 36.9°, 143.1°, 180°, 360°

5 a 2 < f(x) < 6 y c 0°, 78.7°, 180°, 258.7°, 360°
6 d 0°, 116.6°, 180°, 296.6°, 360°

4f e 0°, 60°, 180°, 300°, 360°
f 0°, 76.0°, 180°, 256.0°, 360°

Review 8 a 60°, 120°, 240°, 300°
2 f–1 b 56.3°, 123.7°, 236.3°, 303.7°

O π π 3π 2π x 9 a 30°, 150°, 270°
22 b 45°, 108.4°, 225°, 288.4°
c 0°, 109.5°, 250.5°, 360°
b f is one-one, f −1(x) = cos−1 4 − x  d 60°, 180°, 300°
2  e 0°, 180°, 199.5°, 340.5°, 360°
f 70.5°, 120°, 240°, 289.5°
6 a 3π g 19.5°, 160.5°, 270°
2 h 30°, 150°, 270°

b f −1(x) = sin−1 5 − x  , 3 < x < 7
2 

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10 a 0.565, 2.58 b π , 5π End-of-chapter review exercise 5
11 2.03, 3π , 5.18, 7π 66
1 a = 1, b = 2
44
2 1.95
Exercise 5G b 1− k2
3 a 1− k2 k
1 9 sin2 x − 3
c −k
2 a Proof b Proof 4 x=± 3

c Proof d Proof 2
5 39.3° or 129.3°
e Proof f Proof

Review 3 a Proof b Proof 6 30° or 150°

c Proof d Proof 7 30° or 150°

4 a Proof b Proof

c Proof d Proof 8iy

5 a Proof b Proof 1 y = cos 2θ y= 1
O π 2
c Proof d Proof
2π θ
e Proof f Proof

g Proof h Proof –1

6 a Proof b Proof ii 4 iii 20
ii 45°, 135°, 225°, 315°
c Proof d Proof 9 i Proof ii 120°
10 i 60° or 300° ii 109.5° or 250.5°
e Proof f Proof b −2.21, 0.927 305
ii 3 − 2 3
Review 74 b 4 < f(x) < 7 11 i Proof
8 a 4 + 3 sin2 x b 4, −4 12 a π, 5π πx
9 a (sin θ + 2)2 − 5
66

10 a Proof 13 i f(x) < 3

b sin θ = 1− 4a2 , cos θ = 4a iii y f
1+ 4a2 1 + 4a2
3

Exercise 5H Oπ
2
1 a Proof
b 76.0°, 256.0°

2 a Proof b 18.4°, 116.6°  3 − x 
2
3 a Proof iv f −1( x ) = 2 tan−1

b 60°, 131.8°, 228.2°, 300° 14 i 30° or 150° ii n = 3, θ = 290°

4 a Proof b 30°, 150°, 210°, 330° 15 i Proof

5 a Proof b 72.4°, 287.6° ii 54.7°, 125.3°, 234.7°, 305.3°

Review 6 a Proof b 65.2°, 245.2° 16 i Proof ii 194.5° or 345.5°

7 a Proof b 41.8°, 138.2°, 270° 17 i 1.68
8 a Proof b 30°, 150°
9 a Proof b 66.4°, 293.6°

10 a Proof b 70.5°, 289.5°

11 a Proof b 30°, 150°, 210°, 330°

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Cambridge International AS & A Level Mathematics: Pure Mathematics 1e-ssC-aRmebrviidegweCUonipvyersitCyoPprye-ssC-aRmebrviidegweCUonipvyersitCyoPprye-ssC-aRmebrviidegweCUonipvyersitCyoPprye-ssC-aRmebrviidegweCUonipvyersity

ii y 7 16 + 112x + 312x2 + 432x3 + 297x4 + 81x5

8 f 8 a x8 − 4x6 + 6x4 − 4x2 + 1 b −16
9 −216
6
10 54
4
11 a 1 + 4y + 6y2 b 142
2
12 5

13 x11

Review O π π 3π 2π x 14 a x5 + 5x4y + 10x3y2 + 10x2 y3 + 5xy4 + y5
22

iii f is one-one b 113 100 b 36 2
15 a p = 8,  q = 8 b y5 − 5y3 + 5y
iv f −1( x ) = 2 cos−1  x − 5  16 a y3 − 3y
3

6 Series Exercise 6B

Prerequisite knowledge 1 a 35 b 84

1 a 4x2 + 12x + 9 b 9x3 − 9x2 − x + 1 c 495 d 5005
2 a n(n − 1) bn
2 a 125x6 b −32x15
2
3 a 2n + 3 b 11 − 3n c n(n − 1)(n − 2)

306 Exercise 6A 6

1 a x3 + 6x2 + 12x + 8 3 a 45 b 56

Review b 1 − 4x + 6x2 − 4x3 + x4 c 364 d 792
4 a 1 + 16x + 112x2 b 1 − 30x + 405x2
c x3 + 3x2 y + 3xy2 + y3
c 1 + 7 x + 21 x2
d 8 − 12x + 6x2 − x3 24 d 1 + 12x2 + 66x4

e x4 − 4x3y + 6x2 y2 − 4xy3 + y4 e 2187 + 5103 x + 5103 x2
24
f 8x3 + 36x2 y + 54xy2 + 27 y3

g 16x4 − 96x3 + 216x2 − 216x + 81 f 8192 − 53248x + 159 744x2

h x6 + 9x + 27 + 27 g 256 + 1024x2 + 1792x4
2 4x4 8x9
h 512 + 1152x2 + 1152x4

2 a 12 b 10 5 a −84 b 5940

c −90 d 16 c 35 d −9720
4
e 40 f −32

g 768 h −5 6 7920
2

3 A = 486, B = 540, C = 30 7 −224 000

Review4 ±2 8 −41184

5 a 16 + 32x + 24x2 + 8x3 + x4 9 40 095

b 97 + 56 3 10 a 128 + 320x + 224x2 b 1 − 28x + 345x2
c 1 − 3x + 3x2
6 a 1 + 3x + 3x2 + x3
11 a 1024 + 5120x + 11520x2
b i 16 + 8 5 ii 16 − 8 5 b 1024 + 10 240y + 30 720y2

c 32

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12 a 1 − 4x + 7x2 b1 7 −8, 2

13 16 + 224x + 1176x2 8 a 765 b 255
c −85
14 a = −2, b = 1, p = −364 d 700 5
9

15 n = 8, p = 256, q = −144 9 21

 3  n
 4 
Exercise 6C 10 a 8 b 48.8125 m
b 2, 1
1 a + 6d, a + 18d 11 a 48
x +1 3
2 a 22, 1210 b 35, 3535
b −1957 12 40, −20
3 a 1037 d −3160x
b 2059
Review c 38 1 13 a $17 715.61 b $94 871.71
3 b 20 14 Proof
15 Proof
47 b 9a 16 Proof
b Proof
5 a 7, 29 b 900
6 1442

7 1817

8 31 Exercise 6E
9 5586
1 a3 b 191

10 25 c 26 2 d −36 4
3 7

11 $360 24
12 a 17, −4 3
13 7, 8
3 32 307

Review 14 10, −4 4 2 , 810
3
15 1 (5n − 11)
5 a 0.5ɺ7ɺ = 57 + 57 + 57 + …
2 100 10 000 1000 000
16 9°
17 a a = 8d b Proof
18 Proof
19 a 4 − 3 sin2 x 6 0.25, 199.21875
20 a Proof
7 0.5, 9 b 40.5
Exercise 6D b 3, 15 309
d No 8 52 b 204.8
1 a No f −1, −1 165 b 405
c −1, − 1 b 192
3 27 9 a 2 , 13.5
Review e No 3

2 ar5, ar14 10 a −0.25, 256
11 a 90
32 12 a 36
3 13 93.75

4 −10.8 14 a = 2, r = 3
5 3, 8 5

2 15 π , x , π
6 64 3 2

16 a 5π b 11π
8
17 a Proof
c Proof b Proof

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Cambridge International AS & A Level Mathematics: Pure Mathematics 1e-ssC-aRmebrviidegweCUonipvyersitCyoPprye-ssC-aRmebrviidegweCUonipvyersitCyoPprye-ssC-aRmebrviidegweCUonipvyersitCyoPprye-ssC-aRmebrviidegweCUonipvyersity

Exercise 6F 21 a 2 1 b 115.2°
4 b a = 12 , r = 5

1 a 352 b 788.125 22 a d = 6, a = 13 77

2 a 100 b 16

3 a2 b 384, 32 Cross-topic review exercise 2

4 −2.5, 22.5 1 x180

5 a3 b 12.96, 68 2 3840
5

6 a4 b 3, n = 6 3 a 729x6 − 2916x3 + 4860 b −5832
2
4 a 1 − 10x + 40x2 b 12
7 a x = −3 or 5, 3rd term = 24 or 40
Review 5 a −25.6 b 27 7
b −4 9
5
6 a 14 b 112

End-of-chapter review exercise 6 7 625
8
1 240
8 i Proof

25 ii 35.3°, 144.7°, 215.3°, 324.7°

32 9 a Proof b 225 − (r − 15)2
4 − 864
c 15 d 225, maximum
25
5 16800 10 a Proof b 625 − (r − 25)2

c 25 d 625, maximum
11 a Proof
6 40 b 40 000 − π  r − 200 2
π  π 
308 7 135
2 c Proof d 40 000 , maximum
π
Review 8 a 6561x16 − 17 496x15 + 20 412x14
b −37 908 12 i Proof ii 0.9273

9 a 1 + 8 px + 28 p2x2 b −17 , 3 iii 5.90 cm2 ii 12 + 12 3 + 4π
7 13 i r2(tan θ − θ )

10 a i 1 + 10x + 40x2 ii 243 − 405x + 270x2 14 i 2 − 5 cos2 x ii −3 and 2

b 5940 iii 0.685, 2.46

11 23 15 i Proof ii 26.6°, 153.4°

12 a −2 b 2187 16 i −5 < f(x) < 3 ii (0.253, 0), (0, −1)
3

c 1312.2 iii y

13 a d = 2a b 99a 4

14 a a = 44, d = −3 b n = 22 2 y = f(x)

15 a a = 60, d = −10.5 b 42 2
3

Review16 a 17 b r = −5,S = 7 – π – π O π πx
74 2 4 42

17 a 1 b 16 –2
5

18 i 41000 ii 22100 –4

19 a a = 10, b = 45

bi 0 , θ , π ii 1.125 –6
3
20 i x = −2 or 6, 3rd term = 16 or 48 ii 16
27

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iv f −1(x) = sin−1  x + 1  , domain is −5 < x < 3, g 14x + 3 − 4 h 3 − 5 + 1
4 x2 x3 x2 x3
4

range is −1 π < f −1(x) < 1 π i 6 x+ 3 + 1
2 2 2 x x3

17 a 250 6 a3 b 0.5

b i Proof ii 70 c −5
4

7 Differentiation 7 15

Prerequisite knowledge 8 −3

3 2 9 −8

Review 1 a 3x 2 b 5x 3 10 (−2, −10), (2, −6)

c 1 1 d 1 x−1 11 5
2 4
2 x2
12 a (−2, 7), (3, −8)
e 3x−2 f −2 5 b −8, 2
5
x3

2 a 4(x − 2)−3 b 2(3x + 1)−5 13 a = 2, b = −7

−3 14 a = −5, b = 2
2
3 15 a = 4, b = −6

4 y = 2x + 1 16 a = −10.5, b = 18

Exercise 7A 17 −2 , x , 3

1a AF BF CF DF EF 18 x < −1 and x > 1 309
Chord 2

Gradient 2 2.5 2.8 2.95 2.99 19 Proof

Review b3 Exercise 7B

2 a4 b −2 1 a 6(x + 4)5 b 16(2x + 3)7
d −3
c1 c −20(3 − 4x)4 d 9  1 x + 1 8
2 2  2

3 a 5x4 b 9x8 e 10(5x − 2)7 f 50(2x − 1)4
g −56(4 − 7x)3
c − 4 d − 1 h 21 (3x − 1)6
x5 x2 5

e0 f 2 i 10x(x2 + 3)4 j −16x(2 − x2 )7
33 x
 5(x3 − 5)4(2x3 + 5)
g 5x4 h 3x2 k 6x2(x + 2)(x + 4)2 l x6

4 a 8x3 b 15x4 2 a − (x 1 2)2 b − (x 3 5)2
c 3x5 + −
3
d − x2 c 16 d − 32x
(3 − 2x)2 x2 + 2
10 ( )2
3x3
Review e − f0 e − 72 f − 45
(3x + 3x +
g2 h−1 1)7 2 ( 1 )6
x x5
g − 16(x + 1) h − 49(4x − 5)
x2(x + 2)2 (2x − 5)8 x8
5 a 10x − 1 b 6x2 + 8
c 10x − 3 d 2x + 1 3 a 1 b 1
2 x−5 2x + 3

e 16x3 − 24x f 10 − 2 c 2x d 3x2 − 5
x3 x2 2x2 − 1 2 x3 − 5x

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Cambridge International AS & A Level Mathematics: Pure Mathematics 1e-ssC-aRmebrviidegweCUonipvyersitCyoPprye-ssC-aRmebrviidegweCUonipvyersitCyoPprye-ssC-aRmebrviidegweCUonipvyersitCyoPprye-ssC-aRmebrviidegweCUonipvyersity

e −2 f 3 2 a 30 − 45 b − 3
33 (5 − 2x)2 3x + 1 x4 x7 x3

g − 1 h6 c 4 − 45 d− 9
(2x − 5)3 3 (2 − 3x)4 x3 4 x7 4 (1 − 3x)3

4 10 e 15 x − 6 f 80
4 33 (2x + 1)7
5 12
6 4, 4 3 4 − 8(2x − 1)3, −48(2x − 1)2

3 4 a −1 b4
7 (5, 1)
c 10
8 a = 5, b = 3
48
ReviewExercise 7C 5 − (2x − 1)9

1 a y = 3x − 7 b 8x + y = 17 62
c y = 3x + 9 d 2y = x −1 81
b y = x −1
2 a 4y = x + 4 d 5x − 6y = 3 7 7
c x − 6y = 9 b y = 4x − 7.5 x 0123456 +
b (0.6, 2.48) +
3 a x + 4y = 4 dy ++0−−0+
b (17, 0) dx
4 a Proof b Proof
d2y − − − − + + +
5 (0, 7.5) dx2

6 (−2.5, 8.5)

310 7 a y = 4x − 68 8 x.5
9 Proof
+ 20
8 a 2 x3 10 Proof

Review9 (−7.5, 2.5) 11 a Proof b −8, 8
12 a = −4, b = 4
10 (−6.2, 6.6)

11 a (−32.6, 28.4) b 317.2 units2 End-of-chapter review exercise 7

12 a (2 3, 8 3 ), (−2 3, −8 3 ) 1 3x3 + 7
4x2
b x + 4y = 0
2 −3 5
13 (4, −1) 9

14 y = 0.6x + 1.6, 30.96° 3 Proof

4 −15(3 − 5x)2 − 2, 150(3 − 5x)

15 73 −8
15
16 a −7 b (0.4, −5.48) 5

6 i 4x + 5y = 66 ii (16.5, 0)

ReviewExercise 7D 7 i 5 − 24 ii Proof
x3
1 a2 b 30x − 14
d 48(2x − 3)2 8 37.4
− 36
c x4 f 27 9 (4.25, −7.5)
2 (3x + 1)5
e − 4 10 a y = −4x + 18 b y = 1 x+1
(4x − 9)3 h 24x2 − 36x + 20 4

g 4x − 30 11 a Proof
x4
b i (−0.8, 15.5) ii (2.1, 8.25)
i − 5x + 12
4 x5 12 y = − 1 x
2

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13 i Proof ii 10 3 2 a (9, 6) minimum b (1, 12) minimum
14 i Proof 4 c (−3, −12) maximum; (3, 0) minimum

ii − 9 , 9 d (−2, −28) maximum; (2, 36) minimum
2 4

iii  1 , 4 3  e (4, 4) maximum f (2, 6) minimum
 2 4 
18 18
15 i y − 2 = 1 (x − 1), y − 2 = −2(x − 1) 3 x3 , x3 ≠ 0
2
4 a −2, 3 b −44, 81
1
ii 2 2 5 a a=3 b −3 , x , 1

iii  161,  12  , E not midpoint of OA 6 a a = −15, b = 36 b (2, −2), maximum
 11 
7 Proof
Review 8 Further differentiation 8 x = 3 + k , minimum; x = 3 − k , maximum

Prerequisite knowledge 22
9 (0, 1), minimum; (1, 2), maximum;

1 a x , −1 and x . 3 b −2 , x , 3 (2, 1), minimum

2 a 6x − 1, 6 b − 3 , 9 y
x3 x4

c 9 x, 9 b 18 4
2 4x (1 − 3x)3 3
2
3 a 10(2x − 1)4 1

Exercise 8A –1 O 1 2 3 x

1 a x.4 b x.1 10 a a = −6, b = 5 b minimum 311

c x , −1.75 d x , 0 and x . 8 c (0, 5), maximum d (2, − 11), −12

Review e x , 1 and x . 4 f −2 2 , x , 2 11 a a = 4,  b = 16 b minimum
3

2 a x , 113 b x . 4.5 c x , 0 and x . 2
12 a a = 54, b = −22 b minimum
c 2,x,5 d −1 , x , 3

e x , 2 and x . 6 2 f x , −4 and x . 2 c x , 0 and 0 , x , 3
3 13 a a = −3, b = −12 b (−1, 14)

3 1.5 , x , 3.5

4 (1 − 8 )2, increasing c (2, −13) = minimum, (−1, 14) = maximum
2x d (0.5, 0.5), −13.5

5 (2xx+−26)3, neither Exercise 8C

6 Proof 1 a y=9−x

7 8x + 20, 8x + 20 ù 0, if x ù 0 b i P = 9x2 − x3 ii 108

8 Proof c i Q = 5x2 − 36x + 162 ii 97.2

9 7 , x , 20 2 a θ = 40 − 2r b Proof
r
Review
Exercise 8B c r = 10 d A = 100, maximum

1 a (2, 4) minimum 3 a y = 50 − x b Proof
b (−0.5, 6.25) maximum 2
c (−2, 22) maximum; (2, −10) minimum
d (−3, −17) minimum; (1, 15) maximum c A = 312.5, x = 25
e (−1, −4) minimum
f (1, −7) maximum; (2, −11) minimum 4 a 3x2 − 10x + 160 b x = 123 , 15123 cm2
5 a Proof b A = 37.5, maximum

6 a QR = 9 − p2 b Proof

c p= 3 d A = 12 3, maximum

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7 a Proof b V = 486, x = 3 Exercise 8E

c Maximum 1 4 π cm2 s−1
5
8 a y = 288 b Proof
x2 2 18 cm3 s−1

c 432, 12 cm by 6 cm by 8 cm 3 π cm3 s−1

9 a y = 1 − 1 x − 1 πx b Proof 4 0.003 cm s−1
24
5 0.125 cm s−1
c dA = 1− x − 1 πx, d2 A = −1 − 1 π 6 1 cm s−1
dx 4 dx2 4
320
d 4 e A = 2 π, maximum 7 9π cm2 s−1
4+π 4+
Review
10 a h = 5 − 2r − πr b Proof b 3 cm s−1
2 120
8 a Proof
dA d2A
c dr = 5 − 4r − πr, dr2 = −4 − π 9 a 1 cm s−1 b 1 cm s−1
3 7
d 5 e 25 , maximum
4+π 8 + 2π 9
10 a Proof b − 100π cm s−1
50 − 2x
11 a r = π b Proof
11 32π cm2 s−1
100 2500
c (π + 4) = 14.0, (4 + π) = 350, minimum 10 5 π cm s−1
π 4π 10
12 a 2 cm b

12 a h = 432 b Proof 13 a 40π cm3 s−1
r2 d A = 216π, minimum

312 c r = 6 b i 1.024 cm s−1 ii 8π cm2 s−1

13 a y = 500 − 24x2 b Proof End-of-chapter review exercise 8
10x
Review
c 5 10 d Proof 1 2 cm s−1
6 45π

14 a h = 160 − 3 r b Proof 2 300π m2 hr−1
r2
3 Maximum
c r=8
4 k = 0.0032 kg cm3, 0.096 kg day-1
15 a r = 20h − h2 b Proof
5 i A = 2 p2 + p3
c 13 1 d 1241, maximum ii 0.4
3
6 i Proof ii 20 m by 24 m

Exercise 8D 7 i Proof ii 120, minimum

1 −0.315 units per second, decreasing 8 i y = 4(6 − x) ii Proof
2 1 units per second 3

300 iii A = 72
3 −0.04 units per second
Review4 0.08 units per second 9 i dy = − 8 + 2, d2 y = 16
dx x2 dx2 x3
5 1.25 units per second
ii (2, 8), minimum since d2 y . 0 when x = 2
6 0.09 units per second, increasing dx2

7 1, 6 (−2, −8), maximum since d2 y , 0 when x = −2
dx2
8 0.016 units per second
9 1, 3 10 i y = 30 − x − πx ii Proof
4
3
iii x = 15 iv Maximum

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11 a −2 , x , 4 e 4 x +c f − 10 + c
3 3 x

b Maximum at  − 5 , 364  , minimum at (1, 4) 5 a x3 + 5x2 + 4x + c b x3 − 3x2 + 9x + c
 3 27  32 3

12 i Proof ii Maximum 3 10 4

13 i  − 2p , 4 p3  c − 8x2 + 2x2 + x + c d 3x 3 + 3x 3 + c
 3 27  3 10 4

 2p 4 p3  e x + 1 + c f x − 3 +c
 3 27  2 2x 2 2x2
ii (0, 0) minimum, − , maximum
7
Review iii 0 , p , 3 g x2 + 4 x + c
63 h 2x 2 + 20 + c
7x

9 Integration i 2x2 + 12 − 9 +c
x 4x4

Prerequisite knowledge Exercise 9B

1 a −19 b −1 1 a y = x3 + x + 2 b y = 2x3 − x2 + 5

2 a  − 2 , 5  b (0, 9) c y = 10 − 4 d y = x2 + 6 − 4
 3  b 10x − 4 + 5 x x

3 a 24x7 − 13 x e y =4 x −x+2

Exercise 9A f y=2 x − 2x + 2 3 −1

1 a y = 5x3 + c b y = 2x7 + c 3 x2

c y = 3x4 + c d y = −3 +c 2 y = 3 +2 313
x x
e y = − 1 +c
4x2 f y =8 x +c 3 y = 2x3 − 6x2 + 5x − 4

Review 2 a f(x) = x5 − x4 + 2x + c 4 y = 5x3 + 3 − 2 b y = 42x − 97
2 x2

b f(x) = x6 + x3 − x2 + c 5 a y = 2x2 x + 2x − 1
23
6 y = 2x2 + 3x − 7

c f( x ) = 3x2 − 1 − 8 +c 7 f(x) = 4 + 8x − x2
x2 x
y (4, 20)

d f( x ) = − 2 3 6 + 3 − 4x + c
x x

3 a y = x3 + 5x2 + c 4
32 Ox

b y = 3x4 + 2x3 + c
23

c y = x4 − 2x3 − 8x2 + c 8 a y = x3 + 1 x2 − 10x + 3
4 2

Review d y = x2 − 5 + 1 +c b x , −2 and x . 1 2
4 4x2 x 3

75 9 a y = 2x3 + 6x2 + 10x + 4 b Proof

e y = 2x 2 − 12x 2 3 +c 10 y = 2 + 4x − 2x2 − x3

75 + 6x 2

53 11 a −3.5

f y = 2x2 + 2x2 + 2 x + c b P = minimum, Q = maximum

4 a 2x6 + c b 5x4 + c 12 a −6 b y = 1 x2 − 6x + 2
c −3 +c 2
x 2
d − x2 +c 13 f ′(x) = 2x − 2 , f(x) = x2 + 2 − 4
x2 x

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Cambridge International AS & A Level Mathematics: Pure Mathematics 1e-ssC-aRmebrviidegweCUonipvyersitCyoPprye-ssC-aRmebrviidegweCUonipvyersitCyoPprye-ssC-aRmebrviidegweCUonipvyersitCyoPprye-ssC-aRmebrviidegweCUonipvyersity

( )14 1 Exercise 9E
−11, 408 3

15 a y = 9 + 3x − x2 b 5y = x + 21 1 a7 b 16
9
16 a y = 2x x − 6x + 10 b (4, 2), minimum c −6
d 21
17 (1, 7), maximum e9 f5

18 a y = x2 − 5x + 2 b x + y = −1 2 a 11 2
c (1, −2) 2
b3
c 107
Exercise 9C 6 d4
15
Review1 a 1 (2x − 7)9 + c b 1 (3x + 1)6 + c e 37
18 18 8 f 18

c 2 (5x − 2)9 + c d − 1 (1 − 2x)6 + c 3 a 10 b 26
45 4 3

e − 3 4 +c f 1 (2x 5 + c c2 d4
5
16 (5 − 4x)3 5 + 1)2

g 4 3x − 2 + c h −2 +c e2 f8
3 (2x + 1)2 b4
4 a − 4x
5 + (x2 + 5)2 45
i − c
32(7 2x)4
5 a 15x2 (x3 − 2)4 b 2 1
y = 1 (2x − 1)4 + 2 = 1 + 3 − 15
2 a b y (2x 7
83 5)2 6 a ( x + 1)4
4x
c y =2 x−2 +5 = 2 +6 b 84 2
3 − 2x 5
d y
314

3 y = 3(x − 5)4 − 1 Exercise 9F

Review4 a x + 5y = 7 b y = 5 2x − 3 − 4 1 a 2113 b8

5 a Maximum c 20 5 d 5 1
b y = 8 3x + 1 − 2x2 − 2x + 5 6 3

2 Proof

6 y = 4 2x − 5 − 2 3 a 1156 b 40 1
2

c 4 3 d 21112
32

Exercise 9D 4 a 7 1 b 5 1
5 2
1 a 8x(x2 + 2)3
b 1 (x2 + 2)4 + c c 15.84 d 14
8
e 16 f9

2 a 20x(2x2 − 1)4 b 1 (2x2 − 1)5 + c 5 a 48 3 b6
20 4

−2 6 3 1
x2 − 3
3 a k = −2 +
b 5 c 7 10 2
3
6x 1
4 a (4 − 3x2 )2 b 8 − 6x2 + c 8 a9 b 1.6
b 3− 5
Review5 a 6(2x − 3)(x2 − 3x + 5)5 9 a Proof
b 90
10 18 2
3
b (x2 − 3x + 5)6 + c
3 11 a (−1, 0)

6 a 4( x + 3)7 b 1 ( x + 3)8 + c 12 10 1
x 4 2

7 a 15 x (2x x − 1)4 b 1 (2x x − 1)5 + c 13 34
5
14 26

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Exercise 9G 7 a (25, 0) b 3125π
6
1 26 2 8 a (0, 3)
3 b 16π
b 250π
2 10 2 9 a  1 , 7  , (2, 7)
3 3 9

3 57 1 10 Proof b 128π
6 3
11 a 52π
4 a 36 b 10 2 3 b 32π
3 5
12 a 8π
c 36 3 b 171π cm3

51 13 a 1888π cm3
3 3

Review 6 1 1
3

7 a y = 1 x+2 b 1 (2 3 − 3) 14 Proof
3 2
End-of-chapter review exercise 9
8 a y = −3x + 46 b 64
b 108 1 f(x) = 3x4 + 5x2 − 7
9 a y = −8x + 16 b 8.83

10 a 2y = x − 1 b1 2 25 x3 − 20x − 4 + c
256 3x

Exercise 9H d4 3 y = 30 − 6 − 5 x2
f 16 x2
1 a2 h1
4
b Proof 4 f(x) = 6 x+2 + x2 − 10
d Proof
c −5 f Proof 5 194π
4 9
315
e 50
b f(x) = 3x2 − 6x + 8
g 63 6 a1
i 20
Review 7 10 2
9 3

8 a ∪-shaped curve, y-intercept (0, 11),

2 Proof vertex at (3, 2)
b 483π
3 a Proof
c Proof 5
e Proof
9 i Proof ii 9
4

Exercise 9I 10 i B(0, 1), C(4, 3) ii y = −3x + 15

1 a 71π b 16π iii 2π ii 1 iii 5π
5 3 15 3

c 15π d 25π 11 i Proof
8 4 12 i 1 or 9

2 a 81π b 124π 9
2 15
ii f ′′(x) = 3 −1 − 3 −3 at x = 1 max, at

Review x2 x2
22 9
36
x = 9 min
4 39π
4 31

5 a 24π iii f(x) = 2x 2 + 6x 2 − 10x + 5

b 24π 13 i  4,  20  ii P and Q are both 32
3 3
6 a ∪-shaped curve, y-intercept = 4,

vertex = (2, 0) 14 i y = −24x + 20 ii 9
b 32π 15 i 29.7° 8

5 ii 1

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Cross-topic review exercise 3 14 i B  5 , 0  , C  0, 3  ii 17
4 4 4
1 y = 2 x3 − 3x + 7
3 iii 3
40
2 i −0.6
15 i Proof
ii y = 2x − 16 (3x + 1 + 12 ii 284π
3
3 4)2

3 Practice exam-style paper

3 i y = 2x 2 − 6x + 2 ii x = 4, minimum

4 i y = − 3 +1 ii x . 1, x , −2 1 f ′(x) = 2 + 15 0 for all x
2(1 + 2x) x4
.
 0 
Review5 i 1 , ii 1 2 y = −x3 + 6x2 − 12x + 11
2 6

6 i4 ii − 2 3 Proof
9 27
4 a 2187 − 10 206x + 20 412x2 b −30 618
7 a Proof b Proof
5 a 3 (4π − 3 3 ) cm2
c 1250 = 175 (3 s.f.), maximum 2
(π + 4)
b (2π + 3 + 3 3 ) cm
8 i 12 ii x = −1 or x = −3, 113
6 a (x − 3)2 + ( y + 2)2 = 20 b x − 2y = 17
dy 9 dy
9 i dx = −x )2 , no turning points since dx ≠ 0 7 a 7, −8

(2 bi 1 ii 27
3 8
ii 81π iii k , −8, k . 4
2 8 a 21 − 2(x − 3)2 b (3, 21)

316 f ′(x) = − 8 8 <
+ (2x + 1)2
10 a (2x 1)2 , 0 c x < 3 − 13, x > 3 + 13

Review b f −1(x) = 4 − x , 0 , x < 4 9 a 1 < f(x) < 11
2x
by
cy
12
f–1
y=x 10

8

4 6

4

2

f

O4 x O π π 3π 2π x
22

11 i y = 2 3 1 − 2 ii 1 −1 + 1 −3 c 0.927 rad, 5.36 rad

x2 − 2x2 x2 x2
33 22
 6 − x 
iii (1, −2), minimum d g−1( x ) = cos−1 5

Review12 i y − 6 = − 2 (x − 2) ii y = x2 + 2 + 1 10 a 3x + 4y = 17 b 1 units per second
7x 240

iii x = 1, minimum since f ′′(1) . 0 11 a − 16 − 2x, 32 − 2 b (−2, −12), maximum
x2 x3
13 i 13
c 431π
ii x = −1 (max), x = 2 (min) 5

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Review A F

Amplitude: the distance between a maximum (or minimum) Factorial: 6 × 5 × 4 × 3 × 2 × 1 = 6 ! (read as ‘6 factorial’)
point and the principal axis of a sinusoidal function First derivative: see Derivative

Arithmetic progression: each term in the progression differs Function: a rule that maps each x value to just one y value
from the term before by a constant for a defined set of input (x) values

Asymptote: a straight line such that the distance between a G
curve and the line approaches zero as they tend to infinity
General form of a circle: the equation
B x2 + y2 + 2gx + 2 fy + c = 0, where (−g, − f ) is the centre

Basic angle: the acute angle made with the x-axis and g2 + f 2 − c is the radius of the circle
Binomial: a polynomial with two terms Geometric progression: each term in the progression is a
Binomial coefficients: the coefficients in a binomial expansion constant multiple of the preceding term
Binomial theorem: the rule for expanding (1 + x)n or (a + b)n Gradient function: the derivative f ′(x) is also known as the
gradient function of the curve y = f(x)
C 317
I
Review Chain rule: the rule for computing the derivative of the
composition of two functions Identity: a mathematical relationship equating one
quantity to another
Common difference: the difference between successive
terms in an arithmetic progression Improper integrals: a definite integral that has either one
limit or both limits are infinite, or a definite integral where
Common ratio: the constant ratio of successive terms in a the function to be integrated approaches an infinite value
geometric progression at either or both endpoints in the interval (of integration)

Completed square form: the equation Increasing function: a function whose value increases as x
(x − a)2 + ( y − b)2 = r2, where (a, b) is the centre and r is increases
the radius of the circle
Completing the square: writing the expression ax2 + bx + c Indefinite integral: an integral without limits whose result
in the form d(x + e)2 + f contains a constant of integration

Composite function: a function obtained from two given Integration: the reverse process of differentiation
functions by applying first one function and then applying Inverse function: the inverse of a function, f −1(x) , is the
the second function to the result function that undoes what f(x) has done

Convergent series: a sequence that tends to a finite number M

D Many-one: a function which has one output value for
each input value but each output value can have more
Decreasing function: a function whose value decreases as than one input value
x increases
Mapping: a diagram to show how the numbers in the
Definite integral: an integral between limits whose result domain and range are paired

does not contain a constant of integration Maximum point: a point, P, on a curve where the value
of y at this point is greater than the value of y at other
Review Derivative: denoted by dy of f (x); gives the gradient of a points close to P
dx
curve Minimum point: a point, Q, on a curve where the value of
y at this point is less than the value of y at other points
Differentiation: the process of finding the gradient of a close to Q
curve
N
Differentiation from first principles: the process of finding
the gradient of a curve using small increments Normal: the line perpendicular to the tangent at a point on
a curve
Discriminant: the part of the quadratic formula
underneath the square root sign

Domain: the set of input values for a function

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Cambridge International AS & A Level Mathematics: Pure Mathematics 1e-ssC-aRmebrviidegweCUonipvyersitCyoPprye-ssC-aRmebrviidegweCUonipvyersitCyoPprye-ssC-aRmebrviidegweCUonipvyersitCyoPprye-ssC-aRmebrviidegweCUonipvyersity

O Range: the set of output values for a function

One-one: a function where exactly one input value gives Reference angle: the acute angle made with the x-axis
rise to each value in the range
Roots: if f(x) is a function, then the solutions to the
P equation f(x) = 0 are called the roots of the equation

Parabola: the graph of a quadratic function S d2 y
dx2
Pascal’s triangle: a triangular array of the binomial Second derivative: denoted by or f ″(x) and is used to
coefficients, where each number is the sum of the two
numbers above determine the nature of stationary points on a curve

Period: the length of one repetition or cycle of a periodic Series: the sum of the terms in a sequence
function Self-inverse function: a function f  where f −1(x) = f(x) for

ReviewPeriodic functions: a function that repeats its values in all x

regular intervals or periods Solid of revolution: the solid formed when an area is
Point of inflexion: a point on a curve at which the direction rotated through 360° about an axis

of curvature changes Stationary point: a point on a curve where the gradient is

Principal angle: the angle that the calculator gives is called zero, also known as a turning point

the principal angle

Q T

Quadrant: the Cartesian plane is divided into four Tangent: a straight line that touches a curve at a point
quadrants Term: a number in a sequence
Quadratic formula: the formula x = −b ± b2 − 4ac , Turning point: a point on a curve where the gradient
is zero, also known as a stationary point
2a
which is used to solve the equation ax2 + bx + c = 0 V

318 R Vertex: the vertex of a parabola is the maximum or

Radian: one radian is the angle subtended at the centre of minimum point

Reviewa circle by an arc that is equal in length to the radius of a Volume of revolution: the volume of the solid formed when
circle an area is rotated through 360° about an axis

Review

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Index

Review addition/subtraction rule, completing the square 6–8 scalar multiple rule 195
differentiation 195 graph sketching 19 second derivatives 205–6
proof of the quadratic formula 10 stationary points 216–19
amplitude of a periodic function 129 tangents and normals 201–3
angles composite functions 39–41 differentiation from first principles 193
conic sections 71 discriminant 24
degrees 100 connected rates of change 228–9 domain of a function 35–7
general definition of 121–2 inverse functions 43, 45
radians 101–2 practical applications 230–2
arcs, length of 104–5 constant of integration 244–5 equation of a circle 82–3, 86–7
area bounded by a curve and a line convergent series 176 completed square form 83–4
260–1 cosine expanded general form 84–5
area bounded by two curves 260–1
area enclosed by a curve and the y-axis angles between 0° and 90° 118–20 equation of a straight line 78–80
255–6 general angles 123–6 tangents and normals 202
area under a curve 253–5 graph of 128–9
arithmetic progressions 166–7, 180 inverse of 137, 138 factorial notation 163–4
sum of 167–9 transformations of 132 factorisation, quadratic equations 3–5
asymptotes 129 trigonometric equations 140–4 first derivative test 217
trigonometric identities 145–7, 149 first derivatives 205
functions
basic angle (reference angle) 121–2 decreasing functions 213–15 319
binomial coefficients 160–4 definite integration 250–2 composite 39–41
Review binomial expansions 156–8 definitions 34
binomial theorem 161 area bounded by a curve and a line domain and range 35–7
Review 260–1 graph of a function and its inverse
calculus 191
see also differentiation; integration area bounded by two curves 260–1 48–9
area enclosed by a curve and the increasing and decreasing 213–15
Cantor, Georg 181 inverse of 43–5, 48–9
Cartesian coordinate system 83 y-axis 255–6 many-one 35
area under a curve 253–5 one-one 34–5
quadrants of 122 improper integrals 264–7 quadratic see quadratic functions
chain rule, differentiation 198–200 volumes of revolution 268–70 self-inverse 44
degrees 100 transformations of 51
connected rates of change 228–9 converting to and from radians
circles combined transformations 59–63
101–2 reflections 55–6
area of a sector 107–8 derivatives 193 stretches 57–8
equation of 82–7 translations 52–3
intersection with a line 88–90 first and second 205–6 trigonometric see cosine; sine;
length of an arc 104–5 see also differentiation tangent; trigonometry
right-angle facts 85 Descartes, René 83 use in modelling 34
codomain (range) of a function 35–7 differentiation 191–6
inverse functions 43, 45 addition/subtraction rule 195 Galileo Galilei 2
combined transformations of functions chain rule 198–200 Gauss, Carl 167
59–60 constants 195 geometric progressions 171–2, 180
trigonometric functions 132 increasing and decreasing functions
two horizontal transformations 61–3 infinite geometric series 175–8
two vertical transformations 60–1 213–15 sum of 173–4
common difference, arithmetic notation 193 gradients 75, 192–3
progressions 166 power rule 194–5 and equation of a straight line 78–80
common ratio, geometric progressions 171 practical applications of connected of tangents and normals 201–3
communication vi see also differentiation
completed square form, equation of a rates of change 230–2
circle 83 practical maximum and minimum

problems 221–3
rates of change 227–9
real life uses of 212

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gradients of parallel and perpendicular line segments quadratic formula 10

lines 75–7 gradient 75 quadratic functions 2

graph of a function and its inverse 48–9 length and mid-point 72–4 graph sketching 17–19

graph sketching 17–19 parallel and perpendicular 75–7 lines of symmetry 17–19

quadratic inequalities 22 see also straight lines maximum and minimum values

graphs of trigonometric functions lines of symmetry, quadratic functions 17–19

transformations 130–3 17–19 quadratic inequalities 21–3

sin x and cos x 128–9 longitude 104

tan x 129 radians 101

many-one functions 35 area of a sector 107–8

horizontal transformations, mappings see functions converting to and from degrees

combination of 61–3 maximum points 17–19, 216–17 101–2

Review practical problems 221–3 length of an arc 104–5

identities, trigonometric 145–7, 149 and second derivatives 218–19 simple multiples of π 102

image set of a function see range mid-point of a line segment 72–4 range (codomain) of a function 35–7

(codomain) of a function minimum points 17–19, 216–17 inverse functions 43, 45

improper integrals practical problems 221–3 rates of change 227–8
type 1 264–6 and second derivatives 218–19 connected 228–9
type 2 266–7 modelling vi–vii practical applications 230–2

increasing functions 213–15 Newton, Isaac 193, 239 recurring decimals 176
indefinite integrals 241 normals, gradient 201–3 reference angle (basic angle) 121–2
inequalities, quadratic 21–3 nth term of a geometric progression reflections 55–6
infinite geometric series 175–8 revolution, volumes of 268–70
infinity 181 171–2 right-angle facts for circles 85
nth term of an arithmetic progression roots of a quadratic equation 24–5
improper integrals 264–7
320 inflexion, points of 216 166–7

Reviewintegration 239, 249 one-one functions 34–5 scalar multiple rule, differentiation
area bounded by a curve and a line oscillations 117 195
260–1
area bounded by two curves 260–1 parabolas 17–19 second derivatives 205–6
area enclosed by a curve and the intersection with a line 27–8 and stationary points 218–19
y-axis 255–6 lines of symmetry 17–19
area under a curve 253–5 maximum and minimum values sectors, area of 107–8
definite 250–2 17–19 self-inverse functions 44
of expressions of the form (ax + b)n
247–8 paraboloids 18 graphs of 48
finding the constant of 244–5 parallel lines 75 sequences
formulae for 241–3 Pascal’s triangle 157–8
arithmetic progressions 166–9
geometric progressions 171–4
infinite geometric series 175–8

improper integrals 264–7 periodic functions 129 series 156, 167

notation 240 perpendicular lines 75 binomial coefficients 160–4

as the reverse of differentiation points of inflexion 216 binomial expansion of (a + b)n 156–8

239–40 power rule, differentiation 194–5 simultaneous equations 11–13

volumes of revolution 268–70 principal angle 137, 138 sine

intersection of a line and a parabola problem solving vi angles between 0° and 90° 118–20

Review 27–8 general angles 123–6

inverse functions 43–5 quadrants of the Cartesian plane 121 graph of 128–9

graphs of 48–9 quadratic curves, intersection with a inverse of 136, 138–9

trigonometric functions 136–9 line 27–8 transformations of 130–1

quadratic equations trigonometric equations 140–4

Lagrange’s notation 193 completing the square 6–8 trigonometric identities 145–7, 149

latitude 104 functions of x 15–16 sketching a graph 17–19

Leibnitz, Gottfried 193, 239 number of roots 24–5 quadratic inequalities 22

length of a line segment 72–4 simultaneous equations 11–13 solids of revolution 268

limits of integration 250 solution by factorisation 3–5 sphere, surface area of 14

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Review stationary points 17–19, 216–17 tangent (trigonometric ratio) trigonometric identities 145–7, 149
practical maximum and minimum angles between 0° and 90° 118–20 trigonometry 117
problems 221–3 general angles 123–6
and second derivatives 218–19 graph of 129 angles between 0° and 90° 118–20
inverse of 137 general definition of an angle 121–2
Stevin, Simon 176 trigonometric equations 141–3 graphs of functions 127–33
straight lines trigonometric identities 145–7, 149 inverse functions 136–9
ratios of general angles 123–6
equation of 78–80 tangents to a curve 27 transformations of functions 130–3
intersection with a circle 88–90 gradient 201–3 turning points 17–19, 216
intersection with a quadratic curve see also stationary points
terms of a sequence 166
27–8 trajectories 2 Underground Mathematics vii
see also line segments transformations of functions 51
stretch factors 57–8 vertex of a parabola 17–19
stretches 57–8 combined transformations 59–63 vertical transformations, combination
trigonometric functions 130 reflections 55–6
sum of an arithmetic progression 167–9 stretches 57–8 of 60–1
sum of a geometric progression 173–4 translations 52–3 volumes of revolution 268–9
sum of an infinite geometric series trigonometric functions 130–3
175–8 translations 52–3 rotation around the x-axis 269–70
sum to infinity of a series 176–8 trigonometric functions 131 rotation around the y-axis 268–9
trigonometric equations 140–4, 149
waves 117

321

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Review

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