Cambridge International AS and A Level Mathematics Pure Mathematics 1

Answers

292

P1^

9 (i) f(x)  2

(ii) k = 13

10 (i) k = 4 or −8; x = 1 or − 5

(ii)  7

(iii)  92 – xx, x ≠ 0

11 (i) 2(x − 2)^2 + 3

(ii) f(x)  3

(iii) f is not one-to-one

(iv)  2

(v)  2 − x 2 –3, g−^1 (x)  2

12 (i)

(ii) − 9 x^2 + 30 x − 16

(iii)  9 − (x − 3)^2

(iv) 3 + 9–x

Chapter  5

Activity 5.1 (Page 124)

See text that follows.

Activity 5.2 (Page 126)

6.1; 6.01; 6.001

Activity 5.3 (Page 127)

(i) 2

(ii) −4

(iii) 8

Gradient is twice the x co-ordinate.

Exercise 5A (Page 129)

  2 4 x^3

(^3)
f(x) f ́(x)
x^22 x
x^33 x^2
x^44 x^3
x^55 x^4
x^66 x^5

xn nxn−^1
●?^ (Page^ 129)
When f(x) = xn, then
f(x + h)
= (x + h)n
= xn + nhxn−^1 + terms of order h^2
and higher powers of h.
The gradient of the chord
= ff()xh+h–(x)
= nxn−^1 + terms of order h and
higher powers of h.
As h tends to zero, the gradient
tends to nxn−^1.
Hence the gradient of the tangent is
nxn−^1.
Activity 5.4 (Page 130)
When x = 0, all gradients = 0
When x = 1, all gradients are equal.
i.e. for any x value they all have the
same gradient.
Activity 5.5 (Page 130)
y = x^3 + c ⇒ ddyx
= 3 x^2 , i.e. gradient
depends only on the x co-ordinate.
Exercise 5B (Page 133)
  1 5 x^4
  2 8 x
  3 6 x^2
  4 11 x^10
  5 40 x^9
  6 15 x^4
  7 0
  8 7
  9 6 x^2 + 15 x^4
10 7 x^6 − 4 x^3
11 2 x
12 3 x^2 + 6 x + 3
13 3 x^2
14 x + 1
15 6 x + 6
16 8 πr
17 4 πr^2
18 12 t
19 2 π
20 3 l^2
(^21 32)
x^12
22 − x^12
23 1
2 x
(^24 12)
(^32)
x
25 −^23
x
26 −^154
x
27 −x−
(^32)
28 2 4 32
x

• x−
  (^29 3232)
  (^1252)
  xx− −
  (^30 5323)
  (^2353)
  xx+ −
  x
  y
  –2 O
  –2
  y = f(x)
  y = x
  y = f–1(x)
  (^23)
  (^23)
  y
  2 x
  y = x^3 + 2
  y = x^3 – 1
  y = x^3 + 1
  y = x^3
  1
  2
  –1