Chsmaths.weebly.com



VOLA Mathematics Department

N5 Applications

Revision Pack for Unit Assessment

[pic]

1.1 CALCULATING the AREA of a TRIANGLE using TRIGONOMETRY

MIXED EXERCISE using TRIGONOMETRY RULES

1. Calculate the value of x in each triangle below.

2. Calculate the area of the triangle with sides measuring 12 cm, 14 cm and 20 cm.

3. (a) Calculate the length of BD.

(b) Calculate the length of AD.

(c) Calculate the area of triangle ABC

EXAM QUESTIONS using TRIGONOMETRY RULES

4. Triangle PQR has sides with lengths, in centimetres, as shown.

Show clearly that cos PQR =[pic]

5. A flagpole is attached to a wall and is supported by a wire PQ as shown in the diagram.

The wire is 3∙5 metres long and makes an angle of 55o with the vertical wall.

Given that the point P is 4∙5metres above R in the diagram, calculate the length of the flagpole.

6. A triangular sail designed for a racing yacht is shown below.

Two of its edges measure 6 metres and 3(2 metres.

Given that the sail has a perimeter of 15(5 metres, calculate the area of the sail.

EXAM QUESTIONS involving BEARINGS and TRIGONOMETRY RULES

1. The diagram below, which is not drawn to scale, represents the positions of three mobile phone masts.

Mast Q is on a bearing of 100o from mast P and is 40km away.

The bearing of mast R from mast Q is 150o.

Masts P and R are 66km apart.

(a) Use the information in the diagram to establish the size of angle PQR.

(b) Hence find the bearing of mast P from mast R.

2. A par 3 hole on a golf course the tee is a distance of 130 metres due west from the pin.

On his first shot, Bruce hits the ball 100 metres but not at the correct angle.

On his second shot he hits the ball 35 metres and gets it in the hole.

On what bearing, ao, did he hit his first stroke?

3. A helicopter sets out from its base P and flies on a bearing of 123o to point Q where it changes course to 060o and flies 18 km to point R.

When the helicopter is at point R it is 22 km from its starting point.

(a) Find the size of angle PQR.

(b) Calculate the bearing on which the helicopter must fly to return directly to its base i.e. the shaded angle in the diagram.

Give answers to the nearest whole number throughout your calculations.

4. Brampton is 70 kilometres due east of Abbott.

The bearing of Corwood from Abbott is 015o and from Brampton is 290o.

(a) Make a neat copy of the diagram and fill in all three angles inside the triangle.

(b) Calculate the distance between Corwood and Brampton, to the nearest kilometre.

2.2 WORKING with 2D VECTORS

1. Name the following vectors in 2 ways and write down the components:

2. Draw representations of the following vectors on squared paper.

(a) [pic] (b) [pic] (c) [pic] (d) [pic]

(e) [pic] (f) [pic] (g) [pic] (h) [pic]

(i) [pic] (j) [pic] (k) [pic] (l) [pic]

3. Calculate the magnitude of each of the vectors in questions 1 and 2 above leaving your answers as surds in their simplest form.

4. Find (a) [pic] (b) [pic] (c) [pic]

(d) [pic] (e) [pic] (f) [pic]

2.2 ADDITION of VECTORS using DIRECTED LINE SEGMENTS

1. (i) Draw diagrams on squared paper to illustrate u + v for each pair of vectors given.

(ii) State the components of the resultant vector and calculate its magnitude leaving your answers as a surd in its simplest form

5. The diagram shows 3 vectors x, y and z.

(i) Draw diagrams to represent:

(a) x + y (b) x + z (c) y + z (d) (x + y) + z

(e) x + ( y + z)

(ii) Calculate, correct to one decimal place:

(a) x + y (b) x + z (c) y + z (d) (x + y) + z

(e) x + ( y + z)

6. For the vectors in question 5, calculate:

(a) 2x (b) 3y (c) 0·5z (d) –2y

(e) – 4x (f) –z (g) 3x + 2y (h) 4y + 3x

2.2 SUBTRACTION of VECTORS using DIRECTED LINE SEGMENTS

1. (i) Draw diagrams on squared paper to illustrate u – v for each pair of vectors given.

(ii) State the components of the resultant vector and calculate its magnitude leaving your answers as surds in their simplest form.

4. The diagram shows 3 vectors x, y and z.

(i) Draw diagrams to represent:

(a) x – y (b) x – z (c) y – z (d) (x – y) – z

(e) x – ( y – z)

(ii) For each resultant vector, state the components and calculate its magnitude correct to one decimal place.

VECTOR JOURNEYS in 2D Part 1

1. Express each of the following displacements in terms of vectors a and b.

(a) PQ (b) QP (c) PR

(d) RQ (e) QR

2. In the diagram AB = 2DC. Express each of the following displacements in terms of vectors v and w.

(a) CD (b) CA (c) AB

(d) CB (e) BD

3. In the diagram ’M’ is the mid – point of BC.

Express each of the following displacements in terms of vectors p and q.

(a) CB (b) BC (c) BM

(d) AM

4. EFGH is a parallelogram. ‘M’ is the mid point of side HG.

Express each of the following displacements in terms of vectors a and b.

(a) FG (b) GH (c) GM

(d) FM

5. In the diagram AB is parallel to PR.

PA = 1 cm and PQ = 3 cm

Find in terms of x and/or y the vectors represented by

(a) AQ (b) QB

VECTOR JOURNEYS in 2D Part 2

1. (a) Express in terms of a and b.

(i) PS (ii) ST

(b) If QR = [pic]PQ , show that RS can be expressed as

[pic](2a – 3b) PQ = b

2. Express in terms of vectors v and w.

(a) BD (b) BC (c) AC

If v = [pic] and w = [pic], find the components of the displacement AC.

3. Express in terms of p and q.

(a) AB (b) AF (c) OF

If p = and q = find the components

of OF and hence its magnitude correct to 1 decimal place.

4. (a) Express in terms of a and b:-

(i) AB (ii) AC (iii) OC

(b) If M is the mid-point of OC show that:-

AM = =

3D Coordinates

3. State the coordinates of each vertex of the cuboid shown in the diagram.

4. A cube of side 6 units is placed on coordinate axes as shown in the diagram. Write down the coordinates of each vertex of the cube.

5. This shape is made up from 2 congruent trapezia and 2 congruent isosceles triangles.

From the information given in the diagram, write down the coordinates of each corner of the shape.

6. State the coordinates of each vertex of the square based pyramid shown in the diagram.

7. A cuboid is placed on coordinate axes as shown.

The dimensions of the cuboid are in the ratio OA : AB : BF = 4 : 1 : 2

The point F has coordinates (12, p, q) as shown.

Establish the values of p and q and write down the coordinates of all the vertices of the cuboid.

8. Write in component form (a) v = 2i + 3j – 4k (b) w = 3i – 6j + 2k

(c) u = 6i – 3k (d) a = – 3j – 4k

(e) b = 7i – 2j (f) c = 6j

USING VECTOR COMPONENTS

Adding or subtracting 2 or 3 dimensional vectors using components.

1. For each pair of vectors: (i) Write down the components of u and v.

(ii) Find the components of the resultant vector u + v

(iii) Find the components of the resultant vector v – u

(iv) Find the components of the resultant vector 2v + 3u

(v) Find the components of the resultant vector 3v – 4u

2. u, v and w are 3 vectors with components [pic] respectively.

Find the components of the following: (a) 2u + 3v (b) 3u – 6v

(c) 3w + 2v (d) 4u – 2w (e) – 3u – 4v (f) 3w – 4u

(g) 3u – 6v + 2w (h) 2u + 3v – 4w (i) 3u – 2v + w

3. Calculate the magnitude of each of these vectors giving answers to one decimal place:

(a) [pic] (b) [pic] (c) [pic] (d) [pic]

(e) [pic] (f) [pic] (g) [pic] (h) [pic]

4. u, v and w are 3 vectors with components [pic] respectively.

(i) Find the components of the following:

(a) 2u + 3v (b) 3u – 6v (c) 3w + 2v (d) 4u – 2w

(e) – 3u – 4v (f) 3w – 4u (g) 3u – 6v + 2w (h) 2u + 3v – 4w

(ii) Calculate the magnitude of each resultant vector above giving answers to 1 decimal place.

5. (i) If p = 4i + 2j – 5k and q = i – 3j + k, express the following in component form:

(a) p + q (b) p – q (c) q – 2p (d) 3p + q

(e) 3p – 2q (f) 2q – 3p (g) 3p + 4q (h) –2q – 2p

(ii) Calculate the magnitude of each resultant vector above giving answers to 1 decimal place.

6. Calculate the magnitude of these vectors , leaving you answer a surd in its in simplest form.

(a) [pic] (b) [pic] (c) t = 3i – 2j + 5k

(d) t where point T has coordinates(√3, √5, 2√2) (e) v = √3k + j – 7i

7. Given that v = 2k –3i + 4k, u = 5i + aj – k have the same magnitude, calculate the value of a if a > 0.

8. A skater is suspended by three wires with forces [pic] acting on them.

Calculate the resultant force and its magnitude correct to 3 significant figures where necessary.

9. If [pic] and [pic], solve each vector equation for x.

(a) u + x = v (b) 2u + x = 2v (c) 2x + 3v = 4u – x

10. (i) If [pic], [pic] and [pic], express these in component form:

(a) 2r + s (b) 3t – 2s (c) (r – s) + t (d) r – (s + t)

(ii) Find: (a) 2r + s (b) 3t – 2s (c) (r – s) + t (d) r – (s + t)

3.1 WORKING with PERCENTAGES

EXAM QUESTIONS

1. A gym’s membership has increased by 17% over the past year. It now has 585 members.

How many members did it have a year ago?

2. The number of school pupils not wearing school uniform has decreased by 72% since the start of last year. There are now 42 pupils not wearing school uniform.

How many pupils were not wearing school uniform at the start of last year?

3. My house has increased in value by 15% in the last two years. It is now worth £230 000.

How much was it worth 2 years ago?

4. I bought a new car in September of last year. By this September the car had

depreciated by 20% and was now worth £9600.

How much did I pay for the car last September?

5. Jane bought a painting in an auction. Unfortunately the painting depreciated in value by 7% and is now worth £4185.

How much was the painting worth when it was bought?

6. An antique chair has increased in value by 34% since it was bought. It is now worth £3 484.

What was it worth when it was bought?

EXAM QUESTIONS

1. Joseph invests £4500 in a bank that pays 6∙4% interest per annum.

If Joseph does not touch the money in the bank, how much interest will he have gained after 3 years?

Give your answer to the nearest penny.

2. Jane bought a painting in an auction for £32 250.

Unfortunately the painting depreciated in value by 7% each year.

Calculate how much the painting was worth after 2 years.

Give your answer to 3 significant figures.

3. Non calculator

Last year (2008) a company made a profit of £1 000 000. This year (2009) it expects to increase its profit by 20% and by 2010 to have increased it by a further 25%.

Calculate the profit the company expects to make in 2010.

4. A patient in hospital is given 200mg of a drug at 0900. 12% of the amount of the drug at the beginning of each hour is lost, through natural body processes, by the end of that hour.

How many mg of the drug will be lost by 1200?

5. Holly buys an antique watch costing £1200. The watch appreciates in value by 3·7% per annum.

How much will the watch be worth in 4 years time?

Give your answer to the nearest pound.

6. A local council recycles 28 000 tonnes of glass each year. After a publicity campaign they expect to increase the amount of glass recycled by 12% each year.

(a) How much glass do they expect to recycle in 3 years time?

Give your answer correct to 3 significant figures.

(b) The council aim to double the amount of glass recycled in 6 years.

If this rate is maintained, will the council meet their target?

Give a reason for your answer.

7. Non calculator

Arthur’s new car cost him £15 000. The value of it will depreciate by 20% each year.

How much will Arthur’s car be worth when he trades it in for a new one in 2 years time?

8. Barry bought a house last year costing £115 000. This year it is valued at £110 400.

(a) Calculate the percentage decrease in the value of the house.

(b) If the value of the house continues to decrease at this rate what will the house be worth in a further 3 years time?

Give your answer to 3 significant figures.

9. Marcus invested £3000 in a bank which paid 2∙5% interest per year.

(a) Calculate how much money Marcus would have in his account after 3 years.

(b) How long would it take for Marcus’ money to increase by 12%?

10. In 2007 a company made a profit of £45 000. Over the next three years its profit dropped by 3% each year due to increased manufacturing costs.

Calculate, correct to 3 significant figures, the company's profit in 2010.

3.2 WORKING with FRACTIONS

4. Express each sum as a fraction in its simplest form.

(a) [pic] (b) [pic] (c) [pic] (d) [pic]

(e) [pic] (f) [pic] (g) [pic] (h) [pic]

(i) [pic] (j) [pic] (k) [pic] (l) [pic]

(m) [pic] (n) [pic] (o) [pic] (p) [pic]

(q) [pic] (r) [pic] (s) [pic] (t) [pic]

5. Express each difference as a fraction in its simplest form.

(a) [pic] (b) [pic] (c) [pic] (d) [pic]

(e) [pic] (f) [pic] (g) [pic] (h) [pic]

6. Express each difference as a fraction in its simplest form.

(a) [pic] (b) [pic] (c) [pic] (d) [pic]

(e) [pic] (f) [pic] (g) [pic] (h) [pic]

(i) [pic] (j) [pic] (k) [pic] (l) [pic]

(m) [pic] (n) [pic] (o) [pic] (p) [pic]

7. Express each difference as a fraction in its simplest form.

(a) [pic] (b) [pic] (c) [pic] (d) [pic]

(e) [pic] (f) [pic] (g) [pic] (h) [pic]

(i) [pic] (j) [pic] (k) [pic] (l) [pic]

(m) [pic] (n) [pic] (o) [pic] (p) [pic]

8. Tom walked for 1[pic]kilometres and then walked another 2[pic]km. How far did he walk in total?

9. A rectangle has length [pic] cm and breadth [pic] cm. Calculate its perimeter.

10. Siobhan likes to go to the gym. Last week she went on Monday, Tuesday, Thursday, Friday, Saturday and Sunday. The table below shows the number of hours she trained on each of the six days.

|MON |TUES |THURS |FRI |SAT |SUN |

|[pic] |[pic] |[pic] |[pic] |[pic] |[pic] |

How many hours in total did she spend in the gym last week?

11. Billy is a long distance lorry driver. One day he drove for [pic] hours, had a break and then drove for another [pic] hours.

How long did he drive in total?

MULTIPLICATION and DIVISION

2. Express each product as a fraction in its simplest form:

(a) [pic] (b) [pic] (c) [pic] (d) [pic]

(e) [pic] (f) [pic] (g) [pic] (h) [pic]

(i) [pic] (j) [pic] (k) [pic] (l) [pic]

(m) [pic] (n) [pic] (o) [pic] (p) [pic]

(q) [pic] (r) [pic] (s) [pic] (t) [pic]

3. Express as a single fraction:

(a) [pic] (b) [pic] (c) [pic] (d) [pic]

(e) [pic] (f) [pic] (g) [pic] (h) [pic]

(i) [pic] (j) [pic] (k) [pic] (l) [pic]

(m) [pic] (n) [pic] (o) [pic] (p) [pic]

(q) [pic] (r) [pic] (s) [pic] (t) [pic]

4. Express as a single fraction:

(a) [pic] (b) [pic] (c) [pic] (d) [pic]

(e) [pic] (f) [pic] (g) [pic] (h) [pic]

(i) [pic] (j) [pic] (k) [pic] (l) [pic]

(m) [pic] (n) [pic] (o) [pic] (p) [pic]

(q) [pic] (r) [pic] (s) [pic] (t) [pic]

5. Express as a single fraction:

(a) [pic] (b) [pic] (c) [pic]

(d) [pic] (e) [pic] (f) [pic]

(g) [pic] (h) [pic] (i) [pic]

(j) [pic] (k) [pic] (l) [pic]

(m) [pic] (n) [pic] (o) [pic]

(p) [pic] (q) [pic] (r) [pic]

6. A sack of potatoes weighs 11 kgs.

(a) How many bags each weighing [pic]kgs can be filled from the bag?

(b) What weight of potatoes would be left over?

7. A twenty – one metre length of fabric is cut into [pic]metre pieces.

(a) How many pieces can be cut?

(b) What length of fabric would be left over?

8. A triangle has base [pic] cm and height [pic] cm. Calculate its area.

9. A rectangle measures [pic]metres by [pic]metres. Calculate its area.

COMPARING DATA SETS

QUARTILES AND INTERQUARTILE RANGE

1. For each of the data sets below find the median, lower quartile, upper quartile and interquartile range.

2. For each of the data sets below find the median, lower quartile, upper quartile and interquartile range.

COMPARING DATA SETS

STANDARD DEVIATION

1. Calculate the mean and standard deviation for the following sets of data.

2. A third year pupil conducting an experiment with a die got the following results

(a) Show these results in a frequency table

(b) Use your table to calculate the mean and standard deviation.

3. A company that manufactures shoelaces spot checks the length (in cm) of the laces.

Here are the results for two different production lines.

Calculate the mean and standard deviation and comment on any differences between line A and line B.

4. The running times, in minutes, of films shown on television over a week are as follows.

Calculate the mean and standard deviation.

5. The temperatures, in oC, at a seaside resort were recorded at noon over a 10-day period.

Calculate the mean and standard deviation.

EXAM QUESTIONS

MEAN and STANDARD DEVIATION

1. The weights of 6 plums are

40∙5g 37∙8g 42∙1g 35∙9g 46∙3g 41∙6g

(a) Calculate the mean and standard deviation.

The weights of 6 apples are

140∙5g 137∙8g 142∙1g 135∙9g 146∙3g 141∙6g

(b) Write down the mean and standard deviation.

2. During a recent rowing competition the times, in minutes, recorded for a 2000 metre race were

[pic] [pic] [pic] [pic] [pic] [pic]

(a) Calculate the mean and standard deviation of these times. Give both answers correct to 2 decimal places.

(b) In the next race the mean time was [pic] and the standard deviation was [pic].

Make two valid comments about this race compared to the one in part (a).

3. 6 friends joined "Super Slimmers", a weight loss class. Their weights were recorded and the results are shown below.

65kg 72kg 74kg 81kg 90kg 98kg

(a) Calculate the mean and standard deviation of the weights.

After 6 weeks the mean weight was 74kg and the standard deviation was 8·6

(b) Compare the mean and standard deviation of the friend's weights.

4. Stewart and Jenni complete a crossword puzzle every day. Here are the times (in minutes) that Stewart took to complete it each day for a week.

63 71 68 59 69 75 57

(a) Calculate the mean and standard deviation for Stewart's times.

Every day Jenni took exactly 5 minutes longer than Stewart to complete the puzzle.

(b) Write down Jenni’s mean and standard deviation.

5. The number of hours spent studying by a group of 6 student nurses over a week were

20. 23 14 21 27 24

(a) Calculate the mean and standard deviation of this data.

(b) A group of student teachers had a mean of 21·5 and a standard deviation of 6.

Make two valid comments to compare the study times of the 2 groups of students.

6. Barbara is looking for a new 'A-Pod' and searches for the best deal.

The costs of the 'A-Pod' are shown below.

£175 £185 £115 £87 £150 £230

(a) Calculate the mean and standard deviation of the above data.

(b) A leading competitor, the 'E-Pod', has a mean price of £170 and a standard deviation of 26·7. Make two valid comparisons between the 2 products.

7. In Bramley’s Toy Shop there are 6 styles of teddy bear. The price of each is shown below.

£19 £25 £17 £32 £20 £22

(a) Calculate the mean and standard deviation of these prices.

In the same shop the prices of the dolls have a mean of £22.50 and a standard deviation of 2∙3. .

(b) Compare the two sets of data making particular reference to the spread of the prices.

EXAM QUESTIONS

FORMING a LINEAR MODEL from a given SET of DATA

1. A selection of the number of games won and the total points gained by teams in the Scottish Premier League were plotted on this scattergraph and the line of best fit was drawn.

(a) Find the equation of the line of best fit.

(b) Use your equation to calculate the points gained by a team who won 27 matches.

2. The graph below shows the temperature and sales of ice cream for one week during the summer.

(a) Make a copy of the graph and draw the line of best fit on it.

(b) Find the equation of the best-fit line.

4. A group of smokers were asked how many cigarettes they smoked in a day and how many chest infections they had suffered in the last ten years. The results are shown in the scattergraph with the line of best fit drawn.

(a) Comment on the correlation between the 2 sets of data. (b) Find the equation of the line of best fit.

5. The graph below shows the relationship between the number of hours (h) a swimmer trains per week and the number of races (R) they have won.

A best fitting straight line has been drawn.

(a) Use information from the graph to find the equation of this line of best fit.

(b) Use the equation to predict how many races a swimmer who trains 22 hours per week should win.

ANSWERS

CALCULATING the AREA of a TRIANGLE using TRIGONOMETRY

1. (a) 19∙3cm2 (b) 61∙4m2 (c) 23∙6mm2 (d) 2∙7cm2 (e) 298∙8cm2 (f) 119∙4cm

2. (a) 311∙8cm2 (b) 75∙8cm2

MIXED EXERCISE using TRIGONOMETRY RULES

1. (a) 8(5 (b) 6(3 (c) 26o (d) 75(5o

2. 82(6cm2

3. (a) 11(8 (b) 8(2 (c) 110

EXAM QUESTIONS using TRIGONOMETRY RULES

4. Proof

5. 3∙8m 6. 9∙46m²

EXAM QUESTIONS involving BEARINGS and TRIGONOMETRY RULES

1. (a) 130o (b) 302o 2. 081o

3. (a) 117o (b) 256o

4. (a) Abbott: 75o Brampton: 20o Corwood: 85o (b) 68km

5. (a) 83o (b) 089o 6. 27∙5km

7. 1 643km 8. 084o 9. (a) 52o (b) 038o

10. 101o

2.2 WORKING with 2D VECTORS

1. (a) AB = u = [pic] (b) CD = v = [pic] (c) EF = w = [pic]

(d) GH = u = [pic] (e) ML = v = [pic] (f) PQ = w = [pic]

(g) RS = s = [pic] (h) WX = t = [pic] (i) PT = a = [pic]

(j) RQ = b = [pic] (k) CF = c = [pic]

2.

3. For question 1

(a) √10 (b) 2√5 (c) 6 (d) 3

(e) 2√2 (f) 5 (g) 2√5 (h) √29

(i) √37 (j) 3√5 (k) 5

For question 2

(a) 13 (b) 3√5 (c) 3√5 (d) 4√2

(e) 2√5 (f) 8 (g) √17 (h) 3

(i) √61 (j) 5 (k) 5 (l) 4

4. (a) 5 (b) 25 (c) 13 (d) 10 (e) 5 (f) 13

2.2 ADDITION of VECTORS using DIRECTED LINE SEGMENTS

1. (i)

(ii) (a) [pic]; 2√13 (b) [pic]; 6 (c) [pic]; √17

(d) [pic]; √58 (e) [pic]; 2√5 (f) [pic]; 7√2

(g) [pic]; 7 (h) [pic]; √61 (i) [pic]; √41

(ii) (a) [pic]; 7∙0 (b) [pic]; 7∙8 (c) [pic]; 6∙3

(d) [pic]; 9∙8 (e) [pic]; 9∙8

5. (i)

(ii) (a) 5∙1 (b) 7∙1 (c) 7∙2 (d) 9∙5 (e) 9∙5

6. (a) 6∙3 (b) 8∙5 (c) 2∙2 (d) 5∙7

(e) 12∙6 (f) 4∙5 (g) 13∙0 (h) 17∙7

2.2 SUBTRACTION of VECTORS using DIRECTED LINE SEGMENTS

1. (i)

(ii) (a) [pic]; 2√2 (b) [pic]; 6 (c) [pic]; 9 (d) [pic]; [pic]

(e) [pic]; 2√10 (f) [pic]; √2 (g) [pic]; √41 (h) [pic]; √13

(i) [pic]; √41

4. (i)

(ii) (a) [pic]; 3∙2 (b) [pic]; 3∙2 (c) [pic]; 2 (d) [pic]; 5∙8 (e) [pic]; 5∙1

VECTOR JOURNEYS in 2D Part 1

1. (a) b (b) – b (c) – a (d) a + b (e) – (a + b)

2. (a) – v (b) – v – w (c) 2v (d) v – w (e) w – 2v

3. (a) q – p (b) p – q (c) ½ (p – q) (d) ½ (p + q)

4. (a) b (b) a (c) ½ a (d) b + ½ a

5. (a) 2y (b) x – 2y

VECTOR JOURNEYS in 2D Part 2

1. (a) (i) b + a (ii) a – b (b) proof

2. (a) w – v (b) ¼ (w – v) (c) ¼ (w + 3v) ; [pic]

3. (a) 2p – q (b) [pic] (2p – q) (c) [pic](4p + 3q) [pic]; 8∙6

4. (a) (i) b – a (ii) ⅓ (b – a) (iii) ⅓ (2a + b) (b) Proof

2.2 WORKING with 3D COORDINATES and VECTORS

3. O(0, 0, 0); A(12, 0, 0); B(12, 4, 0); C(0, 4, 0);

D(0, 0, 6); E(12, 0, 6); F(12, 4, 6); G(0, 4, 6)

4. O(0, 0, 0); A(6, 0, 0); B(6, 6, 0); C(0, 6, 0);

D(0, 0, 6); E(6, 0, 6); F(6, 6, 6); G(0, 6, 6)

5. O(0, 0, 0); A(30, 0, 0); B(30, 14, 0); C(0, 14, 0);

D(4, 7, 8); E(26, 7, 8)

6. O(0, 0, 0); P(5, 5, 20); Q(10, 0, 0); R(10, 10, 0); S(0, 10, 0)

7. p = 3; q = 6

O(0, 0, 0); A(12, 0, 0); B(12, 3, 0); C(0, 3, 0);

D(0, 0, 6); E(12, 0, 6); F(12, 3, 6); G(0, 3, 6)

8. (a) [pic] (b) [pic] (c) [pic] (d) [pic] (e) [pic] (f) [pic]

9. (a) v = 3i + 4j + 2k (b) v = 7i – 5j + 8k

(c) v = 9i – 6j – 2k (d) v = 4i + 10k

2.3 USING VECTOR COMPONENTS

Adding or subtracting 2 or 3 dimensional vectors using components.

1. (a) (i) u =[pic]v = [pic] (ii) [pic] (iii) [pic]

(iv) [pic] (v) [pic]

(b) (i) u =[pic]v = [pic] (ii) [pic] (iii) [pic] (iv) [pic] (v) [pic]

(c) (i) u =[pic]v = [pic] (ii) [pic] (iii) [pic] (iv) [pic] (v) [pic]

(d) (i) u =[pic]v = [pic] (ii) [pic] (iii) [pic] (iv) [pic] (v) [pic]

(e) (i) u =[pic]v = [pic] (ii) [pic] (iii) [pic] (iv) [pic] (v) [pic]

(f) (i) u =[pic]v = [pic] (ii) [pic] (iii) [pic] (iv) [pic] (v) [pic]

(g) (i) u =[pic]v = [pic] (ii) [pic] (iii) [pic] (iv) [pic] (v) [pic]

(h) (i) u =[pic]v = [pic] (ii) [pic] (iii) [pic] (iv) [pic] (v) [pic]

(i) (i) u =[pic]v = [pic] (ii) [pic] (iii) [pic] (iv) [pic] (v) [pic]

2. (a) [pic] (b) [pic] (c) [pic] (d) [pic]

(e) [pic] (f) [pic] (g) [pic] (h) [pic] (i) [pic]

3. (a) 5∙4 (b) 8∙6 (c) 3∙7 (d) 5

(e) 7∙3 (f) 1∙7 (g) 3 (e) 13

4. (i) (a) [pic] (b) [pic] (c) [pic] (d) [pic]

(e) [pic] (f) [pic] (g) [pic] (h) [pic]

(ii) (a) 34∙9 (b) 44∙6 (c) 31∙2 (d) 21∙7

(e) 48∙1 (f) 23∙8 (g) 37∙7 (h) 28∙6

5. (i) (a) [pic] (b) [pic] (c) [pic] (d) [pic]

(e) [pic] (f) [pic] (g) [pic] (h) [pic]

(ii) (a) 6∙5 (b) 8∙4 (c) 14∙8 (d) 19∙3

(e) 23∙1 (f) 23∙1 (g) 20∙3 (h) 13∙0

6. (a) √38 (b) 3√3 (c) √38 (d) 4 (e) √53

7. a = √3

8. [pic]; 16∙8

9. (a) [pic] (b) [pic] (c) [pic]

10. (i) (a) [pic] (b) [pic] (c) [pic] (d) [pic]

(ii) (a) 21∙7 (b) 27∙3 (c) 8∙1 (d) 3

3.1 WORKING with PERCENTAGE

EXAM QUESTIONS

1. 500 2. 150 3. £200 000

4. £12 000 5. £4 500 6. £2 600

EXAM QUESTIONS APPRECIATION and DEPRECIATION

1. £920.48 2. £27 900 3. £1 500 000

4. 136∙3mg 5. £1 388

6. (a) 39 300 tonnes (b) just falls short of doubling

7. £9 600 8. (a) 4% (b) £97 700

9. (a) £3 230.67 (b) 5 years 10. £41 100

3.2 WORKING with FRACTIONS

4. (a) [pic] (b) [pic] (c) [pic] (d) [pic] (e) [pic] (f) [pic]

(g) [pic] (h) [pic] (i) [pic] (j) [pic] (k) [pic] (l) [pic]

(m) [pic] (n) [pic] (o) [pic] (p) [pic] (q) [pic] (r) [pic]

(s) [pic] (t) [pic]

5. (a) [pic] (b) [pic] (c) [pic] (d) [pic]

(e) [pic] (f) [pic] (g) [pic] (h) [pic]

6. (a) [pic] (b) [pic] (c) [pic] (d) [pic] (e) [pic] (f) [pic]

(g) [pic] (h) [pic] (i) [pic] (j) [pic] (k) [pic] (l) [pic]

(m) [pic] (n) [pic] (o) [pic] (p) [pic]

7. (a) [pic] (b) [pic] (c) [pic] (d) [pic] (e) [pic] (f) [pic]

(g) [pic] (h) [pic] (i) [pic] (j) [pic] (k) [pic] (l) [pic]

(m) [pic] (n) [pic] (o) [pic] (p) [pic]

8. [pic]km 9. [pic] cm 10. 7 hours

11. [pic] hours

MULTIPLICATION and DIVISION

2. (a) [pic] (b) [pic] (c) [pic] (d) [pic] (e) [pic] (f) [pic]

(g) [pic] (h) [pic] (i) [pic] (j) [pic] (k) [pic] (l) [pic]

(m) [pic] (n) [pic] (o) [pic] (p) [pic] (q) [pic] (r) [pic]

(s) [pic] (t) [pic]

3. (a) [pic] (b) [pic] (c) [pic] (d) [pic] (e) [pic] (f) [pic] (g) [pic] (h) [pic] (i) [pic] (j) [pic] (k) [pic] (l) [pic]

(m) [pic] (n) [pic] (o) [pic] (p) [pic] (q) [pic] (r) [pic]

(s) [pic] (t) [pic]

4. (a) [pic] (b) [pic] (c) [pic] (d) [pic] (e) [pic] (f) [pic]

(g) [pic] (h) [pic] (i) [pic] (j) [pic] (k) [pic] (l) [pic]

(m) [pic] (n) [pic] (o) [pic] (p) [pic] (q) [pic] (r) [pic]

(s) [pic] (t) [pic]

5. (a) [pic] (b) [pic] (c) [pic] (d) [pic] (e) [pic] (f) [pic] (g) [pic] (h) [pic] (i) [pic] (j) [pic] (k) [pic] (l) [pic]

(m) [pic] (n) [pic] (o) [pic] (p) [pic] (q) [pic] (r) [pic]

6. (a) 6 (b) [pic]kg 7. (a) 33 (b) [pic]metre

8. [pic]cm² 9. [pic]cm²

COMPARING DATA SETS

QUARTILES and INTERQUARTILE RANGE

1. 2.

STANDARD DEVIATION

1.

2. 3∙44, 1∙72

3. line A 27, 0∙55; line B 27,0∙19; line B more consistent

4. 106, 16∙7

5. 21, 3∙6

EXAM QUESTIONS

MEAN and STANDARD DEVIATION

1. (a) 40∙7g, 3∙6 (b) 140∙7g, 3∙6

2. (a) 7(55; 0(44 (b) slightly higher mean so slower times on average in 2nd race higher SD so times are less consistent than 1st race

3. (a) 80kg, 12∙2 (b) on average weight is less and less spread out

4. (a) 66; 6·56 (b) 71; 6·56

5. (a) 21·5; 4·42 (b) On average study times same but teachers are more varied

6. (a) £157, 51∙3 (b) on average E-Pod more expensive and less spread out

7. (a) £22∙50, 5∙4 (b) prices of dolls are less spread out than teddies

FORMING a LINEAR MODEL from a given SET of DATA

1. (a) [pic] (b) 6 points

2. Answers depend on line drawn

3. A – strong positive correlation.

4. (a) strong positive correlation (b) I = 1/7C + 1

5. (a) R = ½h + 4 (b) 15

-----------------------

The area of a triangle : [pic]

(d)

(c)

(b)

(a)

8

Q

R

10

12

P

P

Q

55o

3"5cm

R

6m

3(2m

P

Q

100o

40 km

N

N

N

66 km

R

150o

130 m

100 m

35 m

ao

N

N

N

N

18 km

22 km

123o

60o

P

Q

R

Corwood

Abbott

Brampton

N

N

N

70 km

P

H

D

M

v

u

(f)

(e)

3∙5cm

R

6m

3(2m

P

Q

100o

40 km

N

N

N

66 km

R

150o

130 m

100 m

35 m

ao

N

N

N

N

18 km

22 km

123o

60o

P

Q

R

Corwood

Abbott

Brampton

N

N

N

70 km

P

H

D

M

v

u

(f)

(e)

(d)

(c)

(b)

(a)

X

F

E

B

A

w

w

v

u

F

R

Q

L

G

C

(h)

T

S

c

b

a

t

(k)

(j)

(i)

(g)

C

P

W

R

s

Q

u

u

u

u

u

u

u

u

v

v

v

u

v

v

v

v

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

v

v

x

y

z

u

u

u

u

u

u

u

u

v

v

v

u

v

v

v

v

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

v

v

x

y

z

R

Q

b

a

P

D

C

v

w

B

A

B

q

M

A

p

C

M

H

G

b

a

F

E

x

y

P

R

Q

A

B

T

S

R

P

Q

a

2a

b

B

C

D

A

v

w

1

3

A

F

B

O

q

2p

3

2

[pic]

[pic]

B

2

[pic]

[pic]

b

C

1

M

a

A

O

D

G

F

(12, 4, 6)

x

z

A

B

C

E

y

D

G

F

x

z

A

B

C

E

O

y

D

x

z

A

B

C

O

y

30

14

E(26, 7, 8)

P

PT = 20

z

y

S

R

T

10

Q

O

x

C

O

A

x

F

G

(12, p, q)

E

D

y

z

B

(c)

(b)

(a)

u

u

u

v

v

v

v

(f)

(e)

(d)

u

v

v

u

u

(i)

v

(h)

(g)

u

u

u

v

v

|(a) |2 |4 |4 |6 |7 |8 |10 |14 |15 |

|1 |1 |1 |5 |1 |4 |2 |3 |4 |6 |

|1 |4 |4 |1 |5 |4 |4 |3 |6 |2 |

|5 |3 |5 |6 |3 |2 |6 |5 |5 |2 |

|3 |1 |4 |5 |2 |4 |1 |4 |4 |3 |

|Line A |26∙8 |27∙2 |26∙5 |27∙0 |27∙3 |27∙5 |26∙1 |26∙4 |27∙9 |

|95 |95 |110 |90 |110 |100 |125 |105 |90 |120 |

|19 |20 |19 |17 |21 |18 |19 |24 |25 |28 |

Wins

W

P

4

8

12

16

20

Points

10

20

30

40

50

60

70

80

Number of ice creams sold.

10

20

30

40

50

0

0

5

10

15

20

25

Temperature (Celsius)

T

N

1

2

3

4

5

6

7

8

0

5

10

15

20

25

30

35

40

45

Number of cigarettes smoked in a day (C )

Number of chest infections in last 10 years (I )

R

(

(

(

(

(

10

5

0

Number of races won

(R)

(

(

(

(

(

(

(

(

(

(

0 5 10 15

h

Number of hours training per week

(h)

(e)

C

(d)

(c)

A

D

(b)

(a)

(f)

u

F

E

B

w

v

Y

(j)

q

Q

X

r

(i)

(h)

(g)

T

S

p

(k)

(b)

(c)

(e)

(i)

u

v

u

v

v

u

u

v

u

(f)

u

(g)

u

(h)

v

u

v

v

(d)

v

v

u

u + v

u + v

u + v

u + v

u + v

u + v

u + v

u + v

u + v

x

y

(a)

x + y

x

x

z

z

x + z

y

z

y + z

(b)

(c)

x + y

(x + y) + z

(d)

y + z

x + (y + z)

(e)

(a)

(c)

(d)

(e)

(f)

(g)

(i)

u

u – v

v

u

v

u – v

u – v

(b)

u – v

u

v

u

v

v

u

u – v

u

v

u – v

u

v

u – v

u – v

(h)

u – v

u

v

u

v

x

(a)

x

z

x – z

(b)

z

(c)

x

y

x – y

(d)

y

y – z

y – z

x – (y – z)

x – y

z

(x – y) – z

| |median |Q1 |Q3 |SIR |

|(a) |7 |4 |12 |8 |

|(b) |33 |30 |37 |7 |

|(c) |22 |19 |25 |6 |

|(d) |2 |0 |3 |3 |

| |median |Q1 |Q3 |SIR |

|(a) |56 |50 |61 |11 |

|(b) |26 |16 |34 |18 |

|(c) |165 |152∙5 |169 |16∙5 |

|(d) |3 |1 |4 |3 |

| |(a) |(b) |(c) |(d) |

|mean |20∙3 |302 |14∙99 |87 |

|SD |0∙95 |3∙19 |0∙19 |1∙49 |

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

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

Google Online Preview   Download