45

1 y = px 2 + qx

yx

= px + q

For the point (2, 6), x = 2 and yx

= 6.

∴ 6 = p(2) + q … 1

For the point (10, 2), x =10 and yx

= 2.

∴ 2 = p(10) + q … 2

2 – 1 : –4 = 8p ⇒ p = – 12

For 1 : 6 = �– 12�(2) + q ⇒ q = 7

2 y = 7x 2 – x 3

yx 2 = 7 – x

The straight line passes through the point (2, h).

Thus, x = 2 and yx 2

= h.

yx 2 = 7 – x

h = 7 – 2 h = 5

The straight line passes through the point (k, 3).

Thus, x = k and yx2

= 3.

yx 2 = 7 – x

3 = 7 – k k = 4

3 (a) y = mx 2

log10 y = log10 mx 2

log10 y = log10 m + log10 x2

log10 y = log10 m + 2 log10 x

log10 y = 2 log10 x + log10 m

Gradient

Y-intercept

(b) (i) Y-intercept = –1 log10 m = –1

m =10–1

m = 110

(ii) Gradient = 2

k – (–1)

2 – 0 = 2

k + 1 = 4 k = 3

4 y = –3x 3 + 4yx3 = –3 +

4x3

Divide throughout by x 3.

yx3 = 4� 1

x3� – 3 Rearrange

(Y = 4X + c)

By comparison, Y = yx3

and X =

1x3.

Form 5: Chapter 13 (Linear Law)SPM Practice

Fully-Worked Solutions

Paper 1

46

5 y 2 = 12

x(20 – x)

y 2 = 10x – 12

x 2

y 2

x = 10 –

12

x

Point (6, k) � �x, y 2

x �: k = 10 –

12

(6)

= 7

Point (h, 0) � �x, y 2

x � 0 = 10 –

12

h

12

h = 10

h = 20

6 (a) y = q2x

log10 y = log10 q2x

log10 y = log10 q – log10 2x

log10 y = log10 q – x log10 2

log10 y = –x log10 2 + log10 q

(b) Y-intercept = log10 q

–3 = log10 q

q = 10–3

q = 1

1000

47

2 (a)

→ x 2 4 6 8 10 12

y 5.18 11.64 26.20 58.95 132.63 298.42

↑ log10 y 0.71 1.07 1.42 1.77 2.12 2.47

y = pkx log10 y = log10 p + x log10 k log10 y = (log10 k) x + log10 p

The graph of log10 y against x is a straight-line graph, as shown below:

O

log10 y

2.2

2.0

1.8

1.6

1.4 4

0.7

1.2

1.0

0.8

0.6

0.4

0.2

2

0.36

4 6 8 10 12

2.4

x

Graph of log10 y against x

(b) (i) log10 p = Y-intercept

log10 p = 0.36 p = 2.29

(ii) log10 k = gradient

log10 k = 2.12 – 1.4210 – 6

log10 k = 0.74

log10 k = 0.175

k = 1.5

1 (a)

x 1 2 3 4 5

y 1.32 1.76 2.83 5.51 13.00

→ x2 1 4 9 16 25

↑ log10 y 0.121 0.246 0.452 0.741 1.114

The graph of log10 y against x 2 is as shown below.

x2O

log10 y Graph of log10 y against x2

0.1

5

0.08

15

0.62

10 15 20 25

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

(b) y = abx2 Non-linear

log10 y = log10 a + x 2 log10 b log10 y = (log10 b) x 2 + log10 a

(i) Y-intercept = 0.08 log10 a = 0.08 a = antilog 0.08 a = 1.2

(ii) Gradient = 0.74 – 0.1216 – 1

log10 b = 0.6215

= 0.04133 b = antilog 0.04133 = 1.1

Paper 2

Linear

48

3 (a)

x 2.5 3.0 3.5 4.0 4.5 5.0

y 7.0 7.7 8.4 9.9 10.1 11.0

→ xy 17.5 23.1 29.4 39.6 45.5 55.0

↑ x 2 6.3 9.0 12.3 16.0 20.3 25.0

The graph of xy against x 2 is as shown below.

x2O

xy

5

5 10 15 20 25

25

10

15

20

25

30

35

40

45

50

50

Graph of xy against x2

55

Incorrect

Correct

(b) (i) From the graph, the value of y which is incorrectly recorded is 9.9.

The actual value of y is given by: xyactual = 37 4(yactual) = 37 yactual = 9.25

(ii) y = qx + pqx

xy = qx 2 + pq

q = Gradient

q = 55 – 525 – 0

q = 2

4 (a)

x 1 3 5 7 9 11

y 5 20 80 318 1270 5050

→ x + 1 2 4 6 8 10 12

↑ log10 y 0.70 1.30 1.90 2.50 3.10 3.70

The graph of log10 y against (x + 1) is as shown below.

128 10

10

3

642O

(x + 1)

1.0

0.5

0.1

1.5

2.0

2.5

3.0

3.5

4.0

log10

y Graph of log10

y against (x + 1)

(b) y = hqx +1 log10 y = log10 h + (x + 1) log10 q log10 y = (x + 1) log10 q + log10 h Gradient Y-intercept

Y-intercept = 0.1 log10 h = 0.1 h = 1.26

Gradient = 3.7 – 0.712 – 2

log10 q = 0.3 q = 2

pq

= Y-intercept

pq

= 5

p2

= 5

p = 10

49

5 (a)

→ x 2 3 4 5 6 7

y 7.0 11.3 16.0 21.2 27.0 33.2

↑yx 3.50 3.77 4.00 4.24 4.50 4.74

O 1

0.5

1.0

1.5

2.0

2.5

3.0

3.53.4

4.0 4.5 – 3.5= 1.0

6 – 2 = 4

4.5

5.0

21.6 3 4 5 6 7x

yx

Graph of against xyx

(b) y = 5hx 2 + kh

x

yx

= 5hx + kh

(i) 5h = Gradient

5h = 4.5 – 3.56 – 2

5h = 0.25 h = 0.05

(ii) kh

= Y-intercept

k0.05

= 3

k = 0.15

(iii) When x = 1.6, from the graph,

yx

= 3.4

y

1.6 = 3.4

y = 5.44

6 (a)

→ x 1.5 3.0 4.5 6.0 7.5 8.0

y 2.66 3.54 4.72 6.28 8.35 9.19

↑ log10 y 0.42 0.55 0.67 0.80 0.92 0.96

O 1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.80.85

8 – 3 = 5

0.96 – 0.55= 0.41

0.9

1.0

2 3 4 5 6 6.7 7x

8

log10

y Graph of log10

y against x

(b) (i) When y = 7.1, log10 y = 0.85. From the graph, x = 6.7

(ii) y = bd 2x

log10 y = log10 (bd 2x) log10 y = log10 b + log10 d

2x

log10 y = log10 b + 2x log10 d log10 y = (2 log10 d ) x + log10 b

log10 b = Y-intercept log10 b = 0.3 b = 2.0

(iii) 2 log10 d = Gradient

= 0.96 – 0.55

8 – 3

= 0.41

5

= 0.082

log10 d = 0.041

d = 1.1

Divided throughout by x.