Cambridge International A and AS Level Mathematics Pure ...

Cambridge International A and AS Level

Mathematics

Pure Mathematics 1 Practice Book

University of Cambridge International Examinations bears no responsibility for the example answers to

questions taken from its past question papers which are contained in this publication.

Answers

1 Algebra

x = ? 1 or 3

2

(vii) x =

3

2

or ?1

(viii) x = ? 1 or 2

3

Exercise 1.1

1 (i)

(vi)

11a ? 9b

(ii)

?17cd 3 ? 14cd ? 15c 4 d 2

(iii)

5a + 3ab

3

2

5

(ix)

x =

(x)

x =1

(xi)

x = ¡À1

or ? 3

3

(iv)

2f g

(xii) x =

(v)

6x

y

(xiii) x = 0 or 2

(vi)

8

x

(xiv) x = ? 4 or

2 (i)

60 x

(viii) x + 15

20

3mn2 ( 4 + 3n)

(ii)

( p ? 4 )( p + 3)

(iii)

(3q ? 1)(q + 2)

(iv)

(t ? 2u )( s + p)

3 (i)

(ii)

(ii)

(iii)

(iv)

x =

2

?12

= ?1

e =

6

bd

v ? bc

p?n

m2 + w

fu

v =

u?f

k =

p

d

e =

?

3 12¦Ð 2w

Exercise 1.2

1 (i)

2

1

2

x = ¡À1

1

2

(ii)

x =

(iii)

x = 4

(iv)

x = 1 or x = 2

(v)

x = ¡À1

(vi)

x = 4

or 2

2

Exercise 1.3

x = ?17

4 x = 168km

5 (i)

or ? 3

3

(vii) 149

2 (i)

1

3

x = 0 or ? 5

(ii)

x = 5 or ? 5

(iii)

x = 4 or ? 2

(iv)

x = ?7 or 2

(v)

x = 8 or ? 5

1 (i)

( x ? 3)2 ? 8

(ii)

( x + 2)2 ? 4

(iii)

(x ? )

(iv)

( x + 1)2 + 4

2 (i)

3

2

2

1

4

?

x = 3¡À

8

(ii)

x = 0 or ? 4

(iii)

x = 2 or 1

3 (i)

2( x ? 1)2 + 5

(ii)

2 ( x + 3)2 ? 7

(iii)

3( x + 2)2 ? 16

(iv)

5( x ? 4)2 ? 8

(v)

4( x + 3)2 ? 52

(vi)

9 x ?

(

1

3

)

2

?1

1

P1

4 (i)

(ii)

a = 19, b = 4

(v)

a = 2, b = 1

5 a = 15, b = 2, c = 2

6 (i)

y

5

?7 ?6 ?5 ?4 ?3 ?2 ?1 0

?1

?2

?3

?4

?5

4

3

2

Answers

1

?7 ?6 ?5 ?4 ?3 ?2 ?1 0

?1

1

2 3 4 5 6 7x

1 2 3 4 5 6 7x

Vertex(¨C

1

2

, 1)

?2

?3

(vi)

?4

?5

Vertex (¨C1, 0)

(ii)

4

3

2

4

3

¨C7 ¨C6 ¨C5 ¨C4 ¨C3 ¨C2 ¨C1 0

¨C1

2

¨C2

1

¨C3

1

¨C5

Vertex (¨C 2, ¨C1)

?3

(vii)

?4

?5

Vertex (¨C4, ¨C2)

3

2

1

?7 ?6 ?5 ?4 ?3 ?2 ?1 0

?1

1 2 3 4 5 6 7x

?2

?3

?4

?5

Vertex (0.5, 2.25)

(viii)

y

5

4

3

y

2

5

4

3

2

1

?7 ?6 ?5 ?4 ?3 ?2 ?1 0

?1

?2

?3

?4

?5

y

5

4

y

5

4

3

2

1

?7 ?6 ?5 ?4 ?3 ?2 ?1 0 1 2 3 4 5 6 7 x

?1

?2

?3

?4

?5

Vertex (2, 1)

(iv)

1 2 3 4 5 6 7x

¨C4

1 2 3 4 5 6 7x

?2

(iii)

y

5

y

5

?7 ?6 ?5 ?4 ?3 ?2 ?1 0

?1

1

?7 ?6 ?5 ?4 ?3 ?2 ?1 0

?1

1 2 3 4 5 6 7x

?2

1 2 3 4 5 6 7x

Vertex ( 12 , ¨C4 )

2

y

5

4

3

2

1

?3

?4

?5

Vertex (1, 1)

7 (i)

y = ( x ? 4)2 ? 2

(ii)

y = ?( x ? 1) + 5

(iii)

y = x ( x + 2) = ( x + 1)2 ? 1

(iv)

y = ?( x ? 3)2 + 4

2

(iii)

k >

(iv)

k < ?1

8 (i)

(ii)

2 or k < ? 2

P1

m = ?4, n = ?12

p = ?16

Exercise 1.5

Exercise 1.4

1 (i)

1 (i)

x = 3, y = ?1

(ii)

x = ?2, y = 5

(ii)

x = 6.14 or ?1.14 (3 s.f.)

(iii)

x = 3, y = 12

(iii)

x = 0.180 or ?1.85 (3 s.f.)

(iv)

x = 6, y = ?3

(iv)

x = ?0.354 or 2.35

2 (i)

3 2

(ii)

5 3

(iii)

3 5

Exercise 1.6

1 (i)

3 a = 3.

Answers

x = 1.79 or ?2.79 (3 s.f.)

x ¬Ý ?9

(ii)

x < ?8

(iii)

x > ?2

2 (i)

or

?2 < x

y

5

x < 0 or x > 1

4 Using ¦¤ to be the value of the discriminant:

(i)

No solutions

(ii)

No solutions

4

3

2

(iii) Two solutions

1

(iv)

One solution

(v)

No solutions

(vi)

Two solutions

¨C3

k = ?4

¨C4

5 (i)

¨C5

¨C4

¨C3

¨C2

¨C1

0

¨C1

2

3

4

5x

1

2

3

4

5x

¨C2

¨C5

(ii)

k = 4 or ? 4

(iii)

k = 2

(iv)

k = 0 or ? 4

4

1> k

k 6

1

(iii)

k < 0 or k > 4

(iv)

k >

6 (i)

1

(ii)

y

5

0<

¨Cx<

¨C1

2

?36

24

¨C5

¨C4

¨C3

¨C2

¨C1 0

¨C1

¨C2

k > ?3

¨C3

2

¨C4

7 (i)

(ii)

? 2 1

(vi)

y

5

4

3

3

2

2

1

1

¨C4

¨C3

¨C2

Answers

¨C5

(iv)

¨C3 < x < 3

¨C4

x

4

0

2

3

4

5x

?2

¨C2

?3

¨C3

?4

¨C4

?5

¨C5

?6

1

3x

2

y

10

9

8

7

6

5

4

3

2

1

(vii) ¨C 2 x 4

1 2 3 4 5x

y

16

15

14

13

12

11

10

9

8

7

6

5

4

3

2

1

?5?4?3?2?1¨C10 1 2 3 4 5 x

¨C2

¨C3

¨C4

¨C5

4

1

?7 ?6 ?5 ?4 ?3 ?2 ?1 0

?1

¨C1

y

5

4

3

2

1

?5 ?4 ?3 ?2 ?1 0

?1

?2

?3

?4

?5

?6

?7

?8

?9

?10

(v)

¨C1

y

x < ¨C5 or x > 0

?4 ?3 ?2 ?1 0

?1

?2

?3

?4

?5

?6

?7

?8

?9

?10

(viii)

1 2 3 4 5 6 x

x < ¨C7 or x > 2

¨C10 ¨C8

¨C6

¨C4

y

14

12

10

8

6

4

2

¨C2 ¨C20

¨C4

¨C6

¨C8

¨C10

¨C12

¨C14

¨C16

¨C18

¨C20

¨C22

¨C24

2

4

6

8

x

(ix)

x < ¨C5 or x >

3

2

(xii)

y

x < 0 or x > 2

y

4

P1

3

20

2

1

10

?2

?1

0

2

1

3x

?1

¨C4

0

¨C2

2

x

Answers

¨C6

?2

?3

¨C10

(x)

¨C

1

2

?4

Stretch and challenge

y

30

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download