293

ANSWERS TO PROGRESS TEST 1Award marks as shown. If your method is correct but theanswer wrong, award I mark where 2 have been allowed.Add up the marks, and then look at the comments given atthe end.

10. (i) I I I I Iii I I I I I I-1.5 .-1 -0.5

I Io

I mark

1.5 10

Since a = 18'shade any 10 squares. 11.

(ii)

2.5.

I mark

2 marks

I mark

12. 5n - 3. I mark

Z. (i)(ii)

all ringed correct, rn(2)M '2Y M 49 ~@all crossed correct.A' 1\ \!:2J 7'\ ~

I markI mark

13. i litre = 500 cm 3

500760 = 8.38 glasses. 2 marks

3- (4 + 3) x (6 - 2) = 28. I markOnly one possibility.

f. (i) e.g. 12,20,28. I markMany possibilities.

(ii) 5 and 8 for example. I mark

5. (i) 20% of £8 is 0.2 x 8 = £1.60Sale price = £8 - £ 1.60 = £6.40. 2 marks

(ii) 20% = £16.80. I mark100% = £ 16.80 x 5 = £84.00Original price = £84.00. I mark

I. 2 x 1f X 4.5 = 28.3 cm. 2 marks

If. (6,5,2).

15. (£141 7 117.5) X 100= £120.

II. (4 X 4.5 + 7 X 14.5 + 6 X 24.5+ 7 X 34.5 + 1 X 44.5) 7 25= 22.1

2 marks

2 marks

2 marks

7. 21.Easy to get 210, but remember the calculator does12 7 10 = 1.2 first, i.e. 21 X 6.4 + 1.2 = 135.6.

& 87 should be 870. 1.07 should be 1.1.0.11 should be 0.10.Subtract 1 mark for any extra found wrong.

•• Difference = 3.25 km3.25

Hence % = 2.25 X 100 = 144.4%.

I mark

2 marks

2 marks

How to interpretyour scores22-30 Grade C/D: (a good start).15-21 Grade E: check mistakes carefully.8-14 Grade F: a lot of work ahead.0-7 Grade G: only just registering on the GCSE scale.

294

ANSWERS TO PROGRESS TEST 2

12 x 95p = £11.40. Profit = £11.40 - £7.00 = £4.40.1 mark

'I.2.

V9500 = 97m (to the nearest m).

£4.40% Profit = £7.00 x 100 = 62.9%.

1 mark

2 marks

'12. (i)

(ii)

(iii)

All probabilities are 1.I I I) x) =9.

Easier to do (I - probability both the same)I I I 2

=1- 9 - 9 - 9=).

2 marks

2 marks

2 marks

3. (i) IS - 3 = 7x - 4x, so 12 = 3x,x = 4.(ii) 4 - 5 = 3x, -I = 3x, x = -1.

4. 8 x £13.20 = £105.60.Saving = £105.60 - £98.95 = £6.65.

1 mark2 marks

1 mark1 mark

'13.5

Find LACE first, tan ACE = TI = 0.4167, hence ACB = 22.6°.

So L ACD = 180° - 22.6° = 157.4°. 2 marks6. ADCis isosceles, so x = !( 180 - 157.4°) = 11.3°. 1 mark(There are other methods.)

•• Taxable earnings = £135 - £52 = £83.Tax = 83 x £0.25 = £20.75.Take home pay = £135 - £20.75 = £114.25.

1 mark1 mark1 mark

'14. (i)(ii)(iii)(iv)

(4,3). 1 mark!. 1 marky = ix + 1. 1 markputx = -3 in equation (iii),y =! x -3 + I = -i.The point is (- 3,-!). 1 mark

.. Work out the cost of l lb in each case.SSp + 3 = 18.3p, 86p + 5 = 17.2p, £3.60 + 20 = 18p.

1 markThe SIb pack is the best buy. 1 mark

7.800

$800 = £T6 = £500. 1 mark

The total cost = £228 + £500= £728. 1 mark

B. (i) 4(x + 2y) 1 mark(ii) 2xy2(2xy - 3) 1 mark

fl.35

Scale factor = 22 1 mark35

new height = 29.5 x 22 = 46.9 em 1 mark

'I'. (IO x 8 x 4) + (2 x I x 0.5) = 320 1 mark

2 marks

2 marks

80m

csw80

In triangle ECS, tan 26° = SC'80

so SC= tan 260 = 164m.

80Similarly in triangle ECW, WC = tan 180 = 246 m.

The distance between the swimmer and windsurfer= 246 - 164 = 82m. 3 marks

E

'I'. (i) 4x + 4 = 3 - x,4x+x=3-4=-1,5x = -I,x= -1.

(ii) 5 = 3(x + 6) = 3x + 18,so 5 - 18 = 3x,

-13-13 = 3x,x=-3-.

'I" Note: it is easier to put the angles of depression in as angles ofelevation.

How to interpretyour scores1140time

112011001040

See graph.

1020

'1'1. (a)

(km)12

10

8

6

4

2

01000

(b) Arrival time = 11.25 am.

Notes:(i) 18 km/h for 20 mins covers 6 km.(ii) 5 mins on the scale is !cm.(iii) Remaining 6 km must take I hour.

1 mark}1 mark each1 mark" line1 mark

36-45 Grade B: well done.28-35 Grade C/D: check carefully topics not correctly

answered.20-27 Grade E: thorough revision needed.10-19 Grade F: do not do the next paper before redoing

this one.0-9 Grade G: only just registering on the GCSE scale.

295

ANSWERS TO PROGRESS TEST 3

I mark

2 marks& (9.45 7 117.5) x 17.5

= £1.41

.. Sector - triangle60 I

= 1T X 52 X 360 - '2 X 5 2s in 60°

= 2.26 crrr'

ff. This means two correct with probability i and one incorrectwith probability t. This can happen in 3 ways.

I I 3 9Probability = 3 x 4' x 4' x 4'= 64 2 marks

fZ. The missing values are -5, 1,21, 3, -5, -15. 3 marksSubtract 1 for each error

f.. (i) Length of walls = 2 x 8'6" + 2 x 15'= 17' + 30' = 47'.

Look at the column headed 48' and height 8'7" to 9'0".This gives 7 rolls. 1 mark

(ii) Length of walls = 2 x 4.6 + 2 x 2.8 m= 9.2 + 5.6 = 14.8m.

Look at the column headed 15.85 m and height 2.75 to2.90m.This gives 8 rolls. 1 mark

B

T2 1 mark

A

f. (a) 0.000682. 1 mark(b) 6.98 x 107

• 1 mark 20

Z. 4 x 9 = 36 days for I bricklayer,36 7 3 = 12 days for 3 bricklayers. 1 mark

15

3. (i) A = (2 x 1TX 9) + (2 x 1TX 3 x 5)= 181T + 301T = 481T

= 150.8cm2 (I decimal place). 2 marks(ii) This requires rearrangement of the formula 10

40 = (2 x 1Tx 4) + (2 x 1Tx 2 x h)40 = 81T + 41Th, 40 - 25.13 = 12.57h

14.872 marks14.87 = 12.57h, h = 12.57 = 1.18cm.

c:)5

4- (iv) translation = 1 mark

09.00 10.00 11.00 noon 1.00pm

.. v'2l, v'TI etc.

7. 10.25 a.m. (It takes 40 mins + 15 min stop).

.. (25 x 2 + 8 x 2 + 15 x 4)+ 20 = 146cm

x

-2-~----f-ft----+------''--+;----r-

2

y

3I mark

1 mark

2 marks

2 marks

1 mark

204= 5m.p.h.

approximately 10.15 a.m.

(i)

(ii)

How to interpret your scores28-34 Grade A: very good.23-27 Grade B: a good standard for GCSE.16-22 Grade C/D: not good enough for the higher tier.8-15 Grade E: more work needed to reach Grade C

standard.0-7 below Grade E standard.

(a) (i) -1.7 (±O.I).(ii) 2.5 (±O.I).

(b) x = 2 - 2.x2add one to each sidex+I=3-2.rthe required line isy=x+1The solutions arex= 0.80r- 1.3

I markI mark

I mark

2 marks

296

ANSWERS TO PROGRESS TEST 4k = 4 (the length of the side of the square). 1 mark(2,7,4) (6,7,4) (6,7,0) in the order youhave labelled it. 2 marks

,I. (i) 0.308151812.(ii) 0.308.

n. (i) 1(or 0.25).

I(ii) ~x 16~.

16 4= 36 ="9'

,. (i)(ii)

2. To make E as large as possible,take H = 6, andx - y = 7 - 2 = 5.

6:. E =5 =, 1.2.

To make E as small as possible,take H = 4, andx - y = 12 - I = 11.

4:. E =rr =0.4.

3. (b) (i) 11.2 miles,±O.I±0.2(ii) 133°,±Io±2°

4. 14 x 5 = 3(x + 2),70 = 3x+ 6.64 = 3x.x = 211.

1 mark

1 mark

2 marksonly 1 mark

2 marksonly 1 mark

1 mark

1 mark

,2. (x+I)+2(x+2)(x+2)(x+I) .

x+I+2x+4(x + 2)(x + I)'

3x + 5(x + 2)(x + I)'

5x - 13 < 3x + 4.2x < 17.x< 8.5.3x + 4 < 6x - II.15 < 3x.5 <x.So 5 < x < 8.5.But x is an integer, sox = 6,7,8.

1 mark1 mark

1 mark

1 mark

1 mark

1 mark

1 mark

1 mark

1 mark

1 mark

1 mark

.. (a)

(b)

I. (i)

14 720m~ lm~

Black followed by red or red followed by black is:7 6 3 14 42

10 x 19 + 10 x 19 = 95' 2 marks

5.9 X 109 -7 5.791 X 107 is approximately 100.1 mark

,4. Ratio of volumes is 120:200 = 3:5 = I: Ij.So ratio of lengths is I :t1T = 1 : 1.186.Height of larger bottle = 20 x 1.186 = 23.7cm. 2 marksRatio of areas = I : (1.186)2 = 1 : 1.407.The area of the larger label = lOX 1.407 = 14.07 crrr',

2 marks

I mark1 mark1 mark

c

Bcm

B

:~X2 (~x):!SD = 'J

n n

starting at Astarting at Bstarting at C

Find BC first using the cosine rule:BC2 = 52 + 82 - 2 x 5 x 8 cos 80 = 75.11.Hence BC = 8.67 cm. 2 marksSo BM = 8.67 + 2 = 4.335 em.Now find angle B using the sine rule:sin B sin 80°-8- = 8.67 . Hence B = 65.32°. 2 marksNow use the cosine rule in triangle BAM:AM2 = 52 + 4.335 2 - 2 x 5 x 4.335 cos 65.32 = 25.69.So AM = 5.07 em. 2 marks

,I. The question involves more work than at first sight.

,I. Sean has to make up 400 m to catch Pete up again.400

This takes -2- = 200 seconds.

200In this time, Pete has run 90 = 2~ laps.

Hence Sean has run. 3~ laps. 2 marks

'1& Using

'7.

(i) There are two rectangles = 2 x I x b = 21b.1 mark

The overlap b2 is counted twice, so subtract b2•

The area of the ~T' shape = 21b - b2 = b(2/- b).1 mark

(i) Angle OTP = 90° (angle between tangent andradius). .Sox = 180° - 90° - 38° = 52°. 2 marks

(ii) OC= OT (radii), so ~OCTis isosceles.L TOC = 180° - 52° = 128°.Hencey = 1(180° - 128°) = 26°. 2 marks

BA must be parallel to OT, so LBAC = LBCA = x = 52°.Hence z = 180° - 104° = 76°. 2 marks

6cm

(ii) A = 21b - b", soA + b2 = 21b.A + b2

Hence/=~. 1 markPerimeter I + 2b + (I - b) + (l- b) + (l- b) + b = 4/.

2 marks

Aangle AMB is required.Find BM by Pythagoras.BM2 = 62 + 32 = 45. So BM = V45cm.

6Hence tan AMB = ~ ~ = 0.8944, so AMB = 41.8°.

v45 2 marksAngle MBC is required.

3Tan MBC = 6 = 0.5, so MBC = 26.6°. 2 marks

(ii)

(i)

(ii) 4.498 x 109- 1.496 X 108 = 4.35 X 109 km.

2 marks

(b)

(b)

& (a)

.. (a)

7. This is a difficult question.

B v'45cm M

1.96 - 1.77

.43

How to interpretyour scores50-60 Grade A*: an excellent result.40-49 Grade A: a high standard.30-39 Grade B: perhaps still more work to do.20-29 Grade C/D: only just on the highter tier.0-1 9 Below Grade D certainly not good enough for the

higher tier.

297

298

AISWERS AID SOME HIlTS TOQUEmOIS TO TRY

... (a) (1.8 -:- 9) x 2 = 0.4 tonnes. (b) 8.

• (i) 105. (ii) 131.(iii) (£10.25 -:- 25) x 60 = £24.60.(iv) £24.60 -:- 131 = £1.87.

•• (a) 36. (b) No, the answer should be an exact wholenumber. (c) 19.

I. (i) 8532. (ii) 5832 for example. (iii) 35 and 82.

3. 60

.. (a) 31. (b) 17. (c) O. (d)R,U.

Dr (i) 7, 12. (ii) 18, 12. (iii) 24, 7.

(ii) for example I + 4 + 13 + 16,I + 3 + 14 + 16.

.. (i)

I 12 14 7

4 15 9 6

13 2 8 II

16 5 3 10

• (a) Aberdeen. (b) Aberdeen. (c) not really,temperature differences are roughly equal, but perhapsSwansea is on average slightly warmer.

27. £1.

• (i) 28kg.

.. (i) 3km is 300 000 em : -:- 20 000 to give 15em.(ii) On the map 1em = 20 000 em = 0.2 km.

:. 1crrr'on the map = 0.2 x 0.2 = 0.04km2 •

:. 10cm2 is 10 x 0.04 = 0.4km2•

.. (i) 17. (ii) 175. (iii) 57 and 81. (iv) 175 (anynumber ending in 75 and using these digits).

• (a) 4. (b) 30 and 120 for example. (c) Number onlarge wheel = 4 x Number on small wheel.

7. (i) 847 em. (ii) 8856 gm.(v) 8.8. (vi) 880 000.

(iii) 9000. (iv) 6.". (a) 60 km/h.

tV. (i) 0.556.

(b) 3 hours 18 minutes.

(iii) 5.73.

& (i) 8812. (ii) 8800.1& 8 x 4 - 7 = 25.

(ii) 38000. (iii) 40000.

(ii) 29.77. (iii) 4.39 (2 dec. pl.).

(iii) 2 x 10-3 •

(vi) 1.6 x 10-2 •

(ii) 9.57 x 102 •

(v) 1.8 X 104•

II. (i) 8.64 x 104•

(iv) 4.06 x 10-2 •

K (i) 9856.(iv) 8.0.

(viii) 85.(iv) 19.0.

(viii) 4.9.

(iii) 1000~

(vii) 9.0.(iii)' 8.1.

(vii) 10.0.

a (a) (i) 19. (ii) 8500.(v) 92. (vi) 85000.(b) (i) 8.7. (ii) 9.0.(v) 9.1. (vi) 8.9.

,a (i) 37 809.

,I. Change each fraction to a decimal; (i) Ir.(iii) ti and H.

n. (i) 4.22 x 106•

(iv) 1.2 x 106 •

(iii) 4.6 x 10-2 •

(vi) 1.25 x 10- 1•

". (i) •. (ii) 0.049. (iii) 0.52.

(ii) I.

• (i) 6000. (ii) 6. (iii) I.

(ii) 3.36 x 106•

(v) 3.6 X 10 13 •

(iv) 2000. (v) 6000.

,,. Start with 3.5 which is between v'9 and Vi6 : 3.4.• (i) 0.167. (ii) 3.833 (iii) 1.688.

,.. 414. • (i)t\. (ii) £76.50.

,.. (a) i. (b) i.(b) 19. (c) 114700.

,.. (i) 42!. (ii) £3.80.••• £19350.

,a nothing about rr and n; -h = O.OS,if = 0.0476190.

17. (aUiv), (yi), ~i), (ix), (x).VI -:- vB = 0.25 = 0.5

,& (a) i.

Z2. (i) 255.

(d) h. (e) I.

(ii) still 7°C.

(ii) 120.

b) V2and VBwould give.. 275ml.

G. (i) 85p. (ii) 70p. (iii) £1.30.

... 15%.

• £80.

• £84000.

'7. 32.25%... (i) 65 km/h. (ii) 128 km/h.

299

• (i) 150aC. (ii) 28 km/h. (iii) 108 mg. 71. 14.

(b) £1.90.

.. (a) 60.

51. (a) 1.1 kg.(e) £1.70.

(b) 96. (c) About i .

(c) £2.75. (d) £2.95.

• £2.72:8XI9+8XI5;£1.90: 10 x 19;85p: 2 x 15 + 3 x 19 (= 87).

". The letter Hannah sent.

52. (a) £4.70. (b) £4.15.(iii) £215.50.

(c) (i) £172. (ii) £ 11.40.82. 43.4 miles/gallon.

51. (a) 10.12 p. (b) 60.72 p. (c) 35.42 p. 13. (a) £248. (b) £342. (c) June 8th.(ii) £162.40. (iii) £730.80.

(d) (i) £203.

51. (a) Seven hundred and thirty three thousand three hundredand forty two. (b) 158000. (c) Robbery.(d) Criminal damage.

Sf. (a) + 10%. (b) 1.3 hatchback.IN. (a) £415. (b) (i) £790. (ii) 1080 francs.

(iii) £100.

85. (a) £35.40. (b) £42.32. (c) £7.72.

.. (i) 62 em.(iv) 0.81.

(ii) 250000. (iii) 5 X 106 mg . •• (i) 2791.8. (ii) 2821.50.(iii) No, it depends on how much is changed.

57. (a) (i) 42. (ii) £7.60. (b) £7.10 (c) £6. 17. (i) 49p. (ii) 20 750 yen.(iv) 4958 lire.

(iii) 108.2 rand.

... (i) 5.556. (ii) 0.6875. (iii) 6.350. (iv) tl.

• (i) 6400. (ii) about £3.10. (iii) £12 500.

• (i) £3.10.go rotten.

(ii) No, if you cannot eat them all before they

• (a) 3. (b) Queen Victoria.•• Large bag most economic but party size would give more

variety.

12. (i) 8 h 58 min. (ii) 8.26 p.m.(iv) 575. (v) 39. (vi) 8784.

". 7 min 56 s. (ii) £15.• (a) (i) £85. (ii) 2.30 a.m. (b) (i) £8.

". (i) 231 km. (ii) 48 litres.

12. 13 litres; 3 tokens; 27 weeks.

13- (a) 300. (b) £25 320.

M. £24392.

•• £200; £166.50... £63.50.

(iii) 15.

(iii) November 19th.

(c) 13.50.

(ii) Friday.

(b) 2 min 02.49s.

(b) 52 m.p.h.

(b) D. Maragh A, F. Jones B, J. Smith B.

a. (a) 0.5Is.

• (a) 2.50 p.m.

• (a) 2.

II. (i) Monday.

17. (a) 0839.

71. (i) 83 miles. (ii) 1 h 40 min.

• (a) 08.50. (b) 10 min. (c) (i) 30 min., (ii) 40 krn/h. (d) (i) 12.30. (e) (ii) 13.48.

• (a) 2003. (b) 2 h 39 min.(d) 58.4 km/h.

7Z. (a) 16 hours. (b) £40.

17. (i) £309.60, £92.40. (ii) £17 640, £1270.08... (a) £190. (b) (i) 33%. (ii) £381.71. (c) 24.(d) (i)£400, £280, £240, £200, £160.(ii) Better off by £55 if he does not claim.

•• £4176.,. (i) 17. (ii) - 5. (iii) 17.

1111. n + p + 2.

1112. lOP + 200.

1113. (i) £(E + 52e). (ii) £(e+ 5~)'

(iv) £62.59.

(c) I h 34 min.

(iii) £52.79.

(iii) £556.80.(ii) 35.

(ii) 128.75.73. (i) 125.

74. (i) 6! h.

75. £2057.08.

7& (a) (i) £150. (ii) £600. (b) £108.27. IN. 39.2 m. 0.784 m too much.

77. (a) £146.80. (b) 21. (c) £633.10. (d) £12.57. 1.. (a) 122. (b) (i) x + y. (ii)!x + y.

7& (a) £17.95.(e) £72.56.

(b) 1222. (c) £53.77. (d) £0.84.(f) £10.88. (g) £83.44.

,. (i) 7r. (ii) 12.r.(iv) 36r -:- 4r = 9.

(iii) 9.r (3x x 3x).

300

'17. (i) (4x + I) + (2x - 3) = 6x - 2.(H) (x + 10) - (x + 6) = 4.(Hi) i[(6x - 5) + (2x + 9)] = i(8x + 4) = 4x + 2.

(b) 5,1, 4.

,. (i) Ii. (ii)3i.(vi) 12.

(iii) It. (iv) 3. (v) 12.

,a (i) 6. (ii) 5.

fa (a) 699 yards.(d) 805 yards.

,a (a) 6,3.

(b) 69 yards. (c) 18.5 yards.

'32. (a) C, E, G.(b) (i) rotational order 4, also 4 axes of symmetry.(ii) rotational order 4, also 4 planes of symmetry.(c) 5,8,5.(d) a square.

,.. (3x + 2) + (x + 10) + (x + I) = 5x + 13.(i) 5x + 13 = 63 gives x = 10.(ii) Yes; 3x + 2 = x + 10 gives x = 4.

"a (i) 3x. (ii) 3x + 10. (iii) 3x + 10 = 2(x + 10)(Remember Daniel will be 10 years older). Hence x = 10, soat present, Daniel is 10, Peter is 30.

'3'. (a) P. (c) 24.

",. (i) 138. (ii) x;x - l,x + T,« + 8.(v) x would not be a whole number.

(iv) 26.,33. (a) (i) 36000 cm '.

(ii) 125.(H) 36 litres. (b) (i) 40 cm.

(a) 2 em, (b) 24 cm. (c) 101 cm '.(e) 326.7 crrr'. (f) 416 crrr'. (g) (i) 24.

"2. (a) 137. (b) x, x - 9, x - 17, x - 18, x - 19. (c) 70,61,53,52,51. (d) (i) not whole numbers, (ii) basedon 36 which is not all in the square.

"3. (i) -1.1I oC. (ii) 9C = 5(C - 32), Hence9C + 160

C = F = -40°C. (iii) F = 5

'35.

,a

(a) 1.2 rrr'.(d) 2.625 kg.

(b) 2m.

(b) 3 rn', (c) 1630 cm'.(e) 10.075 kg.

(ii) 48 .

,,4. (i) 10... V- q

(11)-4-· (iii) 4.'37. 4.07 crn '.

".. (y - 2t)(i)x=---,z--

.. (y - ak)(iii) x = ty',

1(11) X =--a-. (iv) x =y - 1.

2t(v) x = 3(y - a)

".. (a) multiply by 2. (b) 31.

"7. (a) 1,8,27, 64. (b) 1000 (cube numbers).

,a (a) (i) !1Td3 + !1Td2h.(ii) each part has 3 lengths multiplied together.(b) 1td2 + ndh.

'31. (i) 110.7 g, 3.44 cm'. (ii) 9 x 10- 5 g/cm '.

,a (ii) approx. 2260 crrr'. (iii) 490 cm.

"& (a) 8.5. (b) 8.75. (c) No; numbers are always lessthan 8, in the first sequence they are more than 8.

'42. (a) (i) v.(iii) 720°.

(H) q. (b) (i) 180°. (ii) 360°.(c) 4.7 ern. (d) 1.71 Cffi. (e) 10.42 ern.

" .. (a) 5,7,9,11,13,15.(b) line 20 PRINT NUMBER * NUMBER.

'43. (b) (i) 9.7 ern, (ii) 485 m. (c) (i) 41°.(ii) 12 1°. (iii) 30 1°. (d) 416 m.

fa (a) 81,729,6561, 59049. (b) 9 (odd power). 'ff. (a) (i) 37°. (ii) 53°. (iii) 106°. (iv) 74.(b) (i) f = 180 - 2x. (ii) 9 = 2x. (iii) all values.

'Z'. (a) (i) 1 6 15 20 15 6 1. (H) I 7 21 35 35 21 7 1.(iii) 64. (iv) 1024 (= 2 10

) . (b) (i) 13. (ii) 21.(iii) 89.

,a (a) (i) 140 km. (ii) 175 km. (b) (i) 60°.

'22. (a) 35, 5 x 7; 35, 52 + 2 x 5.(b) n x (n + 2), n2 + 2 x n (note they are the same).

,14. (a) 10 joins. (b) 15 joins. (c) 21 joins. (d) 1, 3, 6,10,15,21,28,36: difference increases by 1 each time.(e) in(n - 1) triangular numbers.

'.7. (a) (i) 30°, 60°. (ii) equilateral.(c) 812 rev/min.

(b) I!".

(c) EFGHand KLMJ.

(iii) 125 crrr'.

(H) L DEA + L EAC = 108° + 72° = 18J0

(ii) 40cm.

(b) Right angled.

,a 450.

,a (i) 1.2 km.

,e. (a) Kite.(d) 54°.

a-I(iii) n =-2-.

(iii) n = p ~ 4.

,n. (a) (i) 7, 9, 11, 41. (H) a = 2n + 1.

(b) (i) 16, 20, 24, 84. (ii) p = 4n + 4.

(iv) 19.

,a (i) 14.5. (ii) 15. (iii) 13.75.,.,. (b) (i) front. (ii) 37 front, 36 back.

,a (a) 11.4m. (b) 10. (c) 3 hours.(d) (i) 1500 kg. (ii) 1.5 tonnes. (e) £69.60.

"2. (a) y = o.(- 1.5,1).

(b) (0, - 2). (c) (- 1, I), (- 1,2),

'21. (a) 20m. (c) 23.5 m. '53. (a) (i) 90°. (iii) 150°. (b) (ii) 12.

,54. (a) 4 and 3. (b) 4 and 3, 3 and 2, 5 and 3.(c) The numbers do not have a common factor.(d) Divide the numbers by the common factor.

'55. (i) 60°. (ii) 20°, 70°, 50°. (iii) 140°.(v) 65°, 50°. (vi) 144°, 36°.

301

,112- (a) 10 em. (b) 200 em",

,a. (a) 7.10 m. (b) 61.4 rrr', (c) 410.

,114. (a) 50,80. (b) 35, 55, 75, 95. (e) 150 km.(f) (i) D.I.Y. (ii) £3. (g) C = 0.2x + 35.

I. (b) (i) £76. (ii) 47. (iii) £25. (c) (i) 36.(ii) £9.

1. (a) (i) 12l. (ii) 100 -7 35 = 2.86 galls.(b) 33.5 miles per gallon.(c) No, because it depends on the speed.

(b) (i) 12 em. (ii) 5 em (Use Pythagoras).

117. (a) 10.20. (b) 10 km. (c) (i) Stop. (ii) beginretumjourney. (d) 30 km. (e) 60 km/h,(f) (i) No. (g) 5 mins.

,.,. (a) (i) 30°. (ii) 30°.(b) (ii) a diameter.

,. (i) (4, 12). (ii) isosceles.(iv) y = -3x.

(iii) (4, -12).

,. (a) 90, 180,210. (c) 220 km. (d) C = 0.6x + 60.

(ii) 17.36 crrr'.

,14. (a) 90 - x.

,e. (i) 3.10 em.

(b) 2x. (c) 2x - 90.

,. (b) (i) 42.

,,,. (a) y = 3x.

(ii) 2. (c) t. (d) y = tx + 31.5.

3(c)y = x.

1~ (a) 100,200,400,800,1600,3200,6400.(ii) 3 h 20 mins. (c) 2.62 x 107 •

,II. (i) 56.4°. (ii) 49.3°,3.49 em. (iii) 40.0°,9.01 em.(iv) 26.8°,8.39 em. (v) 7.61 em, 3.55 em.(vi) 53°,6.95 em.

'17. (a) 5m. (b) 29°. (c) 102 rrr',

I. 36.9°,2.76 m.

1•. 58.2m.

'78. (i) 12 ft. (ii) 73.7°.

171. (a) 25.1 m. (b) 65°.

172. (i) 45.8°. (ii) 40 ern",

112. (a) 16

113. (a) 2.5 m.(b) 2.5m.

(b)

o

(b) (i) 0,4, 16,49.

I

(iii) (a) 9 m '.

(b) (i) R - r.

'74. (a) (i) 17.3 em. (ii) 73.9°.(b) (i) 73.9° -7 2 = 36.95°.

1T(R - r)2(ii) 4 .

(ii) 3.76 em.

36-rI. (a) 4xh + 2r = 72, hence h =~.(36 - xl)

(b) V= xl x 2x = 18x - !x3•

(c) 17.5, 32,27.5. (e) x = 1.8 - 1.9 em,h = 8.5 - 9.1 em.

'75. (i) 3.05 em. (iii) 85.8 cm '. ,. (a) 240 ft. (b) 28 m.p.h. (c) 250 ft.

177. (i) (a) 173.7 m. (ii) 086°, 1.25 km approx.

,n. (c) Remember x = 0 is they-axis.

17& (a) 14°. (b) 4.8m.

171. (i) 18.1 km. (ii) 128.7°. (iii) 3.7°.

1. (b) (i) 440 m. (ii) 245°. (c) 1200 hours.(d) 333 m.

,,,. (a) 4.05 em. (b) 135 cm '. (c) 6.43 em.(d) 8.15 em. (e) 75.7°.

,. (a) (i) 24 crrr'. (ii) 22 em. (b) (i) translation so thatAD is on top of CB. (ii) rotation of 180° about E.(c) (i) 20 em.(ii) 24 em.

aa (4,0,2), (4, 5,0), (1, 5,2), (4, 5,2).

.,. (a) (i) (15,4, 12), (3, 16, 12), (3, - 8, 12) for example.(ii) Sphere centre Q, radius 12.(b) (i) V31 + 41 + liz = 13. (ii) (0,0,24).(iii) 135°.

302

all. BA-, B - A, - AB, - BA. a'I. (a) 8. (b) 30. (c) 81. (d) 2.7.

... (i) 1-1=1.3

(ii) 1+ 5 =20.

22& (a) 80°, 140°, 28°, 24°, 36°.(iii) £78.75.

21& y = ix + 5.

(c) (i) 6j%.

... (a) 9. (b) i . 2.M, (c) about 37. (d) Physics: 15 : Mathematics.(e) D is higher than expected.

21'1. (i) 25• (ii) 32• (iii) 24

• (iv) 38• (v) 26 •

(vi) 4 12•

DI. (i) 5x + 8. (ii) 8t - 2. (iii) 9k - 9. (iv) 8.(v) 3ab + 2a + b.

,., j, 32.5 mins.

2117. (a) 16. (b) 4\.(c) 1 x 1 x i + 1 x f x i + 1 x 1 x f = ilJ.

24 58.. (b) 125· (c) 125· 19

(d) a delay of more than 5 minutes is 125 . DI. (i) 5, 2. (ii) -2, - 5.(v) 6, -1.

(iii) 2,4. (iv) 1.1, 0.3.

... (i)i. (ii) i. (iii) 1 - i = 1.

21.. (a) 1.(e)H.

(b) f. (c) ti.(f)rn,H.

(d) ir..a& (a) 1.6. (b) 0.6 or - 1.6.

21.. (a) 50 - 44 = 6. (b) (i) 49. (ii) 48.(c) 2388 + 50 = 47.76. (d) Yes to the nearest wholenumber. (e) is = 0.4.

,., (a) 3' is less than 50,4' is more than 50.(c) 3.68.

(b) 3.7.

111. (a) 2.(b) (1 x 17 + 2 x 31 + 3 x 24 + 4 x 28) + 100 = 2.63.

211. (a) (2 x 12.99 + 7 x 13.99 + 23 x 14.99 + 20 x 15.99+ 17 x 16.99) + 69 = £15.61. (b) 35th value is£15.99. (c) £14.99. (d) (i) mean.(e) Total number sold x mean = takings.

D7. (i) 4(x + 2y). (ii) x(x + y). (iii) 2x(2y - t).(iv) 3x(x + 3). (v) 5(5x - 3y). (vi) pq(q + p).

IJ& a + b = 8.7 and 2a = a + 2b hence a = 2b, simultaneousequations give b = 2.9, a = 5.8, P = 11.6.

,., F+ T= 35, 5F+ 20T= 445, henceF = 17, T= 18.

21«' (a) (i) 38 days. (ii) £2.80.(ii) £33.55. (iii) 23p.

(b) (i) 145.2& (a) 1.41 em.

(d) 2048 crrr',(b) 8 em", (c) 16 crrr'.

(e) 4 x 2n - 1 = 2n +1.

211. (a) frequencies are 6, 2, 5,4,4, 3, 1.(b) 13th value is 252. (c) (i) /g. (ii) H. ,... (a) 9, 14,20. (b) (i) 54. (ii) 11. (c) d = In2

- in.

21& (3 x 14 + 7 x 20) + 10 = 18.2.

217. (6 x 1.8 + 9 x 1.5) + 15 = 1.62 m.

2Gr (a) 503. (b) (i) (x - 10), (x + 1).(ii)(x-IO)(x+l) +x=r-8x-l0. (c) 36.

21& Total must be 8 x 13 = 104, hence the extra value is 27.UI. (a) 180 - x.

360(b)x·

360(c)x·

360(d)x·

21.. 39 + x = 45 hence x = 6. M4. (a) (i) 40, 50, 35,45. (ii) lOn, IOn - 5,9.(b) (i) 90 em. (iii) 4.

2JI. (b) (i) The sample might not fully reflect everybody'sopinion.(ii) Ask a certain number from each class in the school.

22.. (i) 1 + 1 + I = 3. (ii) 2 + 6 + 1 + 1 + 1 = 11.

... (i) 100. (ii) 9. (iii) 0.3 (iv) ~.

(vi) 4. (vii) l. (viii) 7. (ix) 4.5 -8

(xi) 2x'. (xii)+. (xiii) 12x-6•

(xiv) -27 m". (xv) 3x2. (xvi) 5y8.(xvii) 4q4 13. (xviii) y. (xix) 9y4 .

(v) 1.(x)~.

(xx) /,.

22J. Second year wrong, seems to make no difference. Fourthyear wrong, better test result from those nearer the back.

2M. (i) mode = 3, median = 2. (ii) 108. (iii) 2.16.(iv) tIr.

,. (a) (i) 1440. (ii) 3760. (b) 200/0.

11& (a) 570 approx. (b) £3300 approx. (c) £5.80.(d) Thursday.

,.. (i) y(a + 4y). (ii) t(3f + 2t + 5).(iii) p2q2(q - p). (iv) 4(p - 2).(v) (p + q)(y - z). (vi) (r - p)(q + s).(vii) (t - 3)(t - p). (viii) (x - y)(a + b + c).(ix) (x - 3)(x - 8). (x) (x + 3)(x - I).(xi) (x + 13)(x - 2). (xii) Impossible.(xiii) (2x + 3)(x + 5). (xiv) (2x + 1)(2x + 3).(xv) (3x - 1)(2x -I). (xvi) (9x + 2)(x - 1).(xvii) (4.r + l)(x - l)(x + I).(xviii) (2x - 5)(2x + 5). (xix) (I - x)( I + x).(xx) 1T(R - r)(R + r). (xxi) x(x - 6).(xxii) (llxy - 2)( Ilxy + 2). (xxiii) (2x - 3y)2.(xxiv) (5x - y)2. (xxv) (5 - b)2.

303

13x(iii)-.

12

2b+3(ii)--.

ab

7a+ 12b(vi)--.

8

25(iv)-.

3y

Z7Z. (a) (i) 40.5 - 45.5. (ii) 28,62,82,94,100.(b) (i) 43.5. (ii) 8. (c) 70.

(ii) £3.80.

(ii) x3 = 12. Second method convergesquicker.

(b) (i) £114.27'. (a) 3,2.

271. (a) (i) 2.29.

8x(xii) -.

5y

2x-I(vii)--.

6

b1

(xi)-.cf

2a(x)-.

b(ix) pt.(viii) 2.

x+I247. (i)-.

r4r+I

(v)--.x

... 2q(XIII)-.

3273. 278.

UI. (a) 8x. (b) 2y/r. (c) x3• (d) x-2

•

(2x + I) - 2x I(e) 2x(2x + I) = 2x(2x + I)

274. 14.6.

275. (b) 58.9,8.11. (c) 58.9 ± 16.22 gives a range42.68 - 75.12 appears to include 99%.

U& (i) 3, - I. (ii) 2,8. (iii) i, - I.(v) 1.85, - 4.85. (vi) 1.82,0.18.

(iv) t 3.

278. (i) 2.21, 0.94. (ii) 1.97, -0.21. (iii) 0.86, -0.34.

2f& (b) (ii) 2. (c) 2.73, - 0.73. 277. (a)45s. (b)0.13ms-2 •

(d) 19.3 krn/h,(c) 2.145 km.

8'. (a) 175, 315. (d) 46 m.p.h. (e) 30 m.p.h.

Z7B. (b) x ~ I, Y ~ 1, Ix+ y ~ 6. (d) 3 wine, 3 spirits.

152.

Z7f1. (a) y ~!x + 2,y + 2x ~ 10.(b) (i) 6 at (4,2). (ii) - 4 at (2,3).

153. (a) £4. (b) £7.50. (c) nc = 60.(e) straight line through the origin.

(d) B. aD. x + y ~ 1000, Y ~ 2x, x ~ 100, Y ~ 800, 333, 667.

2M. (a) 4 = 2a + ib, 1 = 4a + lb. (b) a = -!,b =r.(d) r + 12x - 28 = o. (e) x = - 14.

21'. (a) h ~ IS.

ZIIZ. 3.34 units",

(b) p> 25. (c) 45 ~ h + p < 60; 25, 30.

215. (a) multiplies by -I. (b) (i) -7.(iii) -7. (c) multiplies by -I.

(ii) -7. ZII3. 7.3.

III. (a) 5. (b) (i) -I. (ii) 9.(c) {positive integers> 3}.

(iii) 14. (v) 6.2.28& (a) 0,28,48,60,64,60,48,28,0.

(ii) 8 ms :". (iii) 672 m.(c) (i) 64 m/s.

87. (a) I + 2 + 4 + ... + t.(iii) I + P + p2 + . . . + pn

(b) (ii) I + P + p2. ZII5. (a) (i) 268 ern".(b) 3.33 em.(iii) 0.785 crrr'.

(ii) 0.0654 ern". (iii) 4096.(c) (i) 0.5 em. (ii) 3.14 crrr',

2f& (a) (x - 3)(x - 2).(3,0);.(d) -4, -5.

(b) x = 3,2. (c) (0,6), (2,0),ZI& (a) (i) 8000 cm '. (ii) 4189 cm '.

(b) Same amount.(iii) 3811 cm '.

,... (a)(x-I)2+2(a=I,b=2). 217. (d) 90°. (e) 12 em.(g) the line TQ.

(f) 30 crrr'.

• ,. (i) 20 Iitres/min hot, 32 Iitres/min cold.(ii) 3 mins 5 seconds.

... (i) 25. (ii) - 1.5, 2.5.za. (a) 72400 cm '.

(ii) 72.1 em .(b) 72 litres. (e) (i) 58 litres.

a3. (a) 2160 g/cm".

2U (i) Ii. (ii) t. (iii) 121.5.

(b) 10 em.

ZSII. (a) (i) 155 cm '.(c) 3.7 em.

(ii) 135 cm '. (b) 6.87 em.

2M (a)-i. (b) t. (e)n.21'. (a) 84.5 m. (b) 224 m.

... 62. ZIIZ. (a) 1390 cm '.(iii) 600 cm ' .

(b) (i) 251 crrr'. (ii) 5.51 em .

... (a) 19.

217. (a) 34%.

(b) 32.

(b) (ii) 24. (iii) 3.

ZI3. (a) 4 em. (b) 864 cm '.(d) 1728 crrr'.

(c) 3888 crrr'.

all. (a) 0, 1, 3,6, 10. (b) 1, 2, 3,4. (c) n. (e) 99. ZK (a) BE.r

(ii) d = 11. (c) 11~ m.

III. (a) (i) 4. (ii) 49. (iii) 484. ztII. (a) 10 em. (b) 8. (c) 2.5 em, 1.25 em. (d) 4096

304

ZIIIi. (a) (i) 52°.

2117. (a) 4.8 em. (b) 9.6 km.

(iv) 4/25. 4 -4 -8o 4 12

16 4 24).

ZIIB. (a) 9.5, 8.1, 6.2,2.7, 2.0.(d) 2.9 em/min.

(c) After 4 hours 45 mins.(f) Impossible.

(b) (i) 4.76 miles. (H) 3.14 miles. 31Z.( ! !).-I 4

3111i. (a) - 3c, 2a - c, 2c - 2a. (b) - 3c, a + c, 3a - 3c,- 3a + 3c. (c) DEFis a straight line, DE = EF.

3117. (a) 3y. (b) x + y. (c) ix + iy, ix + y. (d) iy.(e) parallel, EF = iAD.

3tI& (a) (i) 2b - 2a. (H) 6b. (Hi) 2a - 6b.(iv) a + 3b.(b) (i) 2kb - 2ka. (iii) ha + 3hb. (iv) Must haveh = 2 (I - k) and 3h = 2k. Solve as simultaneous equationsto give h = i, k = i.

(a) A to Y.(b) A to Z(2), B to X(2), Cto X(3), Cto Z(3), D to X(3),DtoZ(3).

(c) Impossible.

(1 1 1 I I 0

1).(H) I I I 0 0 I000 I I I

(b) (iii) ( g0

~ ).50

(b) (~ ~).

(e) p _~ ).

(c) (iii) (~ ~).

0).~(i

(b) (H) A shear.

(iv) S = TT (transpose).

313.

(a) !( 3 -2 )31f. -5 4 .

1 ( 9-3 ).(d) 36 6 2

1 ( 11 4 ).(f) 78 -3 6

(a) (i) ( ~3

~ ).315. 02

316.

317.

(H) 190.4 m,3IMI. (i) (a) 14.4. (b) 133.8°. (c) 0.87 km.248 m, 13.5°.

3111. 7.9 ha.

3112. p v ]. q=¥.

3113. (a) Equal. (b) (l)( =:). (d) (l).(e) Returns to A.

3IU. (a) iv3. (c) (It !v'3).

3115. 034.3°.

31.. (b) 4 units:'. (c) 72 units",(d) Enlargement by'vT8, 45° anticlockwise.(e) Reflection in y = -x.

3tII. (a) gb, ~b, -b + a. (b) ~(a - b), ~a.

(c) Trapezium. (d)~. (e) 27 crrr'.

31D. 2 x 200cos 10° = 394N.

(a) ( ). (b) (

-2 -3

).II 5 -1 14 -6311. -4 -I 4 7 -19

(c) (-5 10 ).-13 0 (d) Impossible.

31& (c) (i) (-1,0), (- 3,0), (-1,1).inyaxis. (iii) T = SR.

( -3 -3)(f) -3 3·

(H) (d) (H) reflection

ANSWERS mMENTALARITHMETIC TEST 1

ANSWERS mMENTALARITHMETIC TEST 2

305

t. 4. t. £42.

2- 1600. I. 48.

3. 13. 3. 8.

f. 10.07. f. 6.5-7 m.

.. 24. .. 27 .

I. £1.16. I. SIb.

7. 5 February. 7. 5.

& 4.38 p.m. & 8.

I. 8. I. 192 em",

,a £2.70. ,a £1.60.

n. 128805. n. 15.05.

,I. 89. ,I. 8r.

,3. 78. ,3. 2ib.

'f. £7.28. 'f. 12 crrr'.

,.. £140. ,I. f.

,I. 8 kg. ,I. 65°.

'7. 87.5 miles. '7. 45 s.

,& 1. ,& 29000.

,I. 12!%. ,I. 12 ern".

• £1.60. • 9.21 p.m.

306

ANSWERS mMENTALARITHMETIC TEST 3

ANSWERS TO MENTALARITHMmC TEST 4

t. 28 crrr', ,. 3600.

2. 37. Z. £2.64.

3- £200. ,. 240 cm '.

4- £2.40. 4- 5 ft.

•• 105008. •• 96 ern?.

.. .. 16.

7. 21°C. 7. 1000 ern".

& 32 em. & £96.

•• 61 miles. •• 151 miles.

,a 18 ern", ,II. 8em.

n. 54.7. n. 56.

'2. 110°. '2. 250°.

". 3h 14min. ". 12.30.

,4- £4.88. ,4- £3.64.

,.. 59p. ,.. 30 g.

INDEX

acceleration 16algebraic expressions 65-8algebraic fractions 194alternate segment 124angle 106-9arc 132area 85-9average 175bar chart 177, 178, 180, 181, 182bearing 108, 109, 133best buy 55bill 48bodmas 2,3calculator 19, 20, 21calendar 38chord 123circle theorems 124, 125circumference 86complement 216composite function 205, 206compound interest 61cone 99congruent 158constructions 239conversion 34, 35, 145coordinates 144correlation 177corresponding angles 107cosine ratio 129cosine rule 250, 252credit 52, 53cuboid 93cumulative frequency 221, 222cylinder 94, 99decimal places 6, 8density 98, 100depression 137determinant 262diameter 123difference of two squares 193dimensions 97directed numbers 12, 13domain 205elevation 137, 138enlargement 160, 267equations: linear 70, 71

quadratic 190, 194, 199equilateral triangle 114errors 231estimation 20exchange rates 54exterior angle 106factors 4factorise 190, 191flow charts 72, 220formulae 66formulae: rearranging: 70, 77fractions 6, 8, 9, 11frequency 172,173,174functions 205, 206

graphs: straight 145, 146, 150curved 151,152,206

gradient 171, 205H.C.F. 3, 193hexagon 108, 113histogram 221, 223incidence matrix 264income tax 47, 49indices 191inequalities 228, 230, 231insurance 59interior angle 106, 107interquartile range 221inverse function 207inverse matrix 262irrational numbers 4isosceles triangle 129iteration 219kite 115L.C.M. 3, 193like terms 191linear programming 229locus 228, 242map scales 15, 109matrices 261-5, 267-9mean 171, 173median 172, 173metric system 33, 34, 35mode 172, 173modulus 256mortgages 60mixtures 248multiple 3negative numbers 14net 92, 93, 94, 96network 263normal distribution 222number line 12number patterns 75-9parallelogram 86, 115pentagon 108, 113percentages 8, 23, 24perimeter 86pictogram 177, 179pie chart 177, 178, 181place value 7polygon 109, 110, 115prime number 2,4prism 93,99probability 164-8proportion 214pyramid 94, 99pythagoras 129quartile 221radius 123range 172rate 100ratio 12, 14, 15, 246rational number 2, 4rectangle 88, 115

reflection 267region 226, 229, 230remainder 3rhombus 115rotation 158, 267rounding 7sampling 177scale 26scale factor 160scatter diagram 183sector 123, 132, 134segment 123sets 216significant figures 8similar shapes 246, 247simple interest 58, 59simplifying 67simultaneous equations 195, 198,264sine ratio 129sine rule 250, 252speed 28,41,42,133sphere 99square 86square numbers 2, 4standard deviation 219standard form 19, 21stretch 267substitution 68surface area 95symmetry 115tables 27, 28tally chart 173tangent (circle) 122tangent ration 129tessellation 113, 117tetrahedron 92time 37timetable 37, 38transformations 157-9, 257translation 157trapezium 115trapezium rule 227, 230travel graph 41,42tree diagram 164, 167triangle 86trial and improvement 7triangular numbers 81trigonometry 129units 7, 14, 33, 34universal set 216Y.A.T. 48variation 212vectors 255velocity 227, 232velocity-time graph 227, 232Venn diagram 217vertex 92volume 55,97-100,239wages 51, 52, 56