43

Paper 1 1

y = px2 + qxyx

= px + q

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

= 6.

∴ 6 = p(2) + q …

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

= 2 .

∴ 2 = p(10) + q …

− : −4 = 8 p ⇒ p = −

12

From : 6 = −

12

⎛

⎝ ⎜

⎞

⎠ ⎟ (2) + q ⇒ q = 7

2

y = 7x2 − x3

yx2

= 7 − x

The straight line passes through the

point (2, h). Thus, x = 2 and y

x2= h .

yx2

= 7 − x

h = 7 − 2h = 5

The straight line passes through the

point (k, 3). Thus, x = k and y

x2= 3.

yx2

= 7 − x

3 = 7 − kk = 4

3 (a) 2mxy =

log10 y = log10 mx2

log10 y = log10 m + log10 x2

log10 y = log10 m + 2 log10 xlog10 y = 2 log10 x + log10 m

(b) (i) Y-intercept = −1

log10 m = −1m = 10−1

m =1

10

(ii) Gradient = 2

k − (−1)2 − 0

= 2

k +1 = 4k = 3

4 y = −3x3 + 4

yx3

= −3 +4x3

yx3

= 4 1x3

⎛

⎝ ⎜

⎞

⎠ ⎟ − 3

(Y = 4X + c)

By comparison, Y =

yx 3

and X =

1x 3

.

Form 5: Chapter 13 (Linear Law) SPM Flashback

Fully-Worked Solutions

Gradient Y-intercept

Divide throughout by x3.

Rearrange

44

Paper 2 1 (a)

x 1 2 3 4 5 y 1.32 1.76 2.83 5.51 13.00x2 1 4 9 16 25

log10 y 0.121 0.246 0.452 0.741 1.114 The graph of log10 y against x2 is as shown below.

(b)

y = abx2

log10 y = log10 a + x2 log10 b

(i) Y-intercept = 0.08

log10 a = 0.08a = antilog 0.08a = 1.2

(ii) Gradient =

0.74 − 0.1216 − 1

log10 b =0.6215

= 0.04133

b = antilog 0.04133b = 1.1

→↑

Non-linear

Linear

45

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 = pk x

log10 y = log10 p + x log10 k

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

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

log10 p = 0.36p = 2.29

(ii) log10 k = gradient

log10 k =2.12 − 1.42

10 − 6

log10 k =0.74

log10 k = 0.175k = 1.5

→

↑

46

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 55.0x2 6.3 9.0 12.3 16.0 20.3 25.0

The graph of xy against x2 is as shown below.

(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 = 374(yactual) = 37

yactual = 9.25

(ii)

y = qx +p

qx

xy = qx2 +pq

q = Gradient

q =55 − 525 − 0

q = 2

pq

= Y-intercept

pq

= 5

p2

= 5

p = 10

→↑

47

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.

(b) y = hq x +1

log10 y = log10 h + (x +1) log10 qlog10 y = (x +1) log10 q + log10 h

Gradient Y-intercept

Y-intercept = 0.1log10 h = 0.1

h = 1.26

Gradient =3.7 − 0.712 − 2

log10 q = 0.3q = 2

→↑