level 9/10 pack2 - 10ticks maths · pdf filelevel 9/10 pack 2. page 1. help@ .. welcome!...

Level 9/10 Pack 2. . [email protected]

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Welcome!

This is the second in a series of teaching aids designed by teachers for teachers at level 9/10.Although level 9 and level 10 notations have now ceased to be used, we are using them forcontinuity in our series. At Key Stage 3 these levels of work would be the Extension paper and inGCSE would represent work at the Higher level. The worksheets are designed to support thedelivery of the National Curriculum in a variety of teaching and learning styles. They are notdesigned to take the pedagogy away from the teacher. The worksheets are centred around theshown level, but spiral from the level below to the level above. Consult the National NumeracyStrategy for definitive National Curriculum levels. They can be used by parents with the supportof the on-line help facility at www.10ticks.co.uk..

Contents and Teacher Notes.

Pages 3/4. Trigonometry in Three Dimensions.Questions involving solids and trigonometry.

Pages 5/6. Trigonometry 3-d Models.Cut outs to help visualise angles in 3-d solids. Stick in pupils books as an aid tosupport the questions in the previous 2 pages.

Pages 7/8. The Sine and Cosine Rules 1.Labelling diagrams correctly. Finding missing lengths in triangles using the sinerule and the cosine rule.

Pages 9/10. The Sine and Cosine Rules 2.Using the two rules together to find missing lengths in triangles.Worded questions involving the sine and cosine rules.

Pages 11/12. Areas and the Sine Rule.Finding areas of triangles using the sine rule. More worded questions.

Pages 13/14. Trigonometry, Sine and Cosine Rule. Exam Style Questions 1.Typical examination style questions involving the sine and cosine rule.

Pages 15/16. Trigonometry, Sine and Cosine Rule. Exam Style Questions 2.Typical examination style questions involving the sine and cosine rule.

Pages 17/18. Trigonometric Graphs 1.Plotting the basic trigonometric graphs. Discovering amplitude and wavelengthand the effect they have on the shape of trigonometric graphs. Using graphs tosolve problems. Sketching trigonometric graphs.

Pages 19/20. Trigonometric Graphs 2.Finding the equation of a trigonometric graph.

Pages 21/22. Trigonometric Graphs 3.Using trigonometric graphs to find solutions to equations. Finding maximum andminimum values of functions. Typical examination style questions.

Pages 23/24. Making and using a Sundial.An activity for maths clubs or activity days. Sun dials are rich in mathematicalcontent. The maths contained here barely scratches the surface of all that ispossible from the topic. Make a sundial, set it up and check its accuracy!


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Pages 25/26. Probability 1. Revision.A revision exercise to cover the style of questions that pupils should be familiarwith. Pupils should revise the level 7/8 work if they do not successfully completethese exercises.

Pages 27/28. Probability 2. Revision.As above.

Pages 29/30. Conditional Probability.Introducing dependent events through worded questions. This leads eventuallyinto tree diagrams.

Pages 31/32. Tree Diagrams (Dependent Events).Typical questions involving tree diagram and dependent events.

Pages 33/34. Probability. Exam Style Questions 1.Typical examination style questions involving all aspects of probability.

Pages 35/36. Probability. Exam Style Questions 2.Typical examination style questions involving all aspects of probability.

Page 37. Probability Investigations.Four investigations to stretch the more able mathematician. “Birthdays” looks atthe probability of 2 people having the same birth date in a class. Geometricalprobability looks at an old experiment first performed by le Comte de Buffon.“Odds” and “National Lottery” explore horse racing and lottery oddsrespectively.

Page 38. Schichi-fuku-jinA probability game/investigation adapted to a Japanese theme.

Page 39. Solving Quadratics by Factorising.Solving quadratics by factorising, progressing to rearranging equations thenfactorising to solve equations.

Page 40. The nth term of a Quadratic.Another method to find the nth term of a quadratic sequence. This method usessolving simultaneous equations.

Pages 41/42. Completing the Square.Solving quadratics by completing the square. This leads to demonstrating how theequation is derived for solving quadratics.

Pages 43/44. Solving Quadratics by Formula.Using the equation to solve quadratic equations. Worded questions involvingsolving quadratics.

Copyright in Worksheets. ©Fisher Educational Ltd. 2002.Copyright in the worksheets belongs to Fisher Educational Ltd. Each purchase of the worksheets represents alicence to use and reproduce the worksheets as set out in the Terms and conditions shown on the 10ticks website.

'10TICKS', and '10TICKS.co.uk' and/or other 10TICKS services referenced on this web site or on the Worksheetsare trademarks of Fisher Educational Ltd. in the UK and/or other countries.

Details of copyright ownership in the clip art used in these worksheets:Copyright in the clip art used entirely in this pack is owned by Nova Development Corporation, California, USA.


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Trigonometry in Three dimensions.

1). The base of a right pyramid, vertex E, is a square of side 14 cm.The length of a slant edge of the pyramid is 22 cm.Calculatea). the length of a diagonal on the base,b). the height of the pyramid,c). the volume of the pyramid,d). the angle that the slant edge EC makes with the base of the pyramid.

X is the mid point of BC.e). Find the length EX.f). Calculate the angle that EX makes with the base of the pyramid.

2). The base of a right pyramid, vertex V, is a rectangle of sides10 m and 15 m as shown in the diagram.The length of a slant edge of the pyramid is 24 cm.Calculatea). the height of the pyramid,b). the volume of the pyramid,c). the angle that the slant edge VY makes with the base of the pyramid.

A is the mid point of WZ.d). Find the length AV.e). Calculate the angle that AV makes with the base of the pyramid.

B is the mid point of YZ.f). Find the length VB.g). Calculate the angle that VB makes with the base of the pyramid.

3). The base of a right pyramid, vertex E, is a square of side 42 mm.The height of the pyramid UE is 68 mm.V is the midpoint of BC.Calculatea). the volume of the pyramid,b). the distance EV,c). ∠EVU,d). the length of one of the slant edges,e). the angle that the slant edge makes with the base of the pyramid.f). The pyramid is made of solid metal. It is melted down and recast as a cube.

Calculate the side length of the cube.

4). The base of a right pyramid, vertex V, is a rectangle of sides3 cm and 4 cm as shown in the diagram.The height of the pyramid SV is 6 cm.A is the midpoint of WZ. B is the midpoint of YZ.Calculatea). the volume of the pyramid,b). the distance VA,c). the distance VB,d). ∠VAS,e). ∠VBS,f). the length of the slant edge VZ,g). ∠VZS.h). The pyramid is made of solid metal. It is melted down and recast as a sphere.

Calculate the radius of the sphere.

BA 14 cm

E

D C

22 cm

ZW 10 m

V

X Y

24 m

15 m

BA 42 mm

E

D CU

ZW 3 cm

V

X Y

4 cmS

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Trigonometry in 3-d with a Square Based Pyramid.

Cut around the net carefully keepingto the outside of the lines.Fold into a square based pyramid.Stick the base down in your book.Unfold.

Trigonometry in 3-d with a Rectangular Based Pyramid.

Cut around the net carefully keepingto the outside of the lines.Fold into a rectangular based pyramid.Stick the base down in your book.Unfold.

Angle with face and base

Angle with edge and base

Angle with face and base

Angle with edge and base


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The Sine and Cosine Rules 1.Finish off correctly labelling these diagrams.

1). 2). 3). 4).

5). 6). 7). 8).

The Sine rule.Diagrams not to scale.

A). Find all the missing sizes in each of these diagrams. Give the answer to 1 decimal place.

1). 2). 3). 4).

5). 6). 7). 8).

9). 10). 11). 12).

13). 14). 15). 16).

B). Find all the missing angles in each of these diagrams. Give the answer to 1 decimal place.

1). 2). 3). 4).

5). 6). 7). 8).

9). 10). 11). 12).

13). 14). 15). 16).

C

A

bD

f

e Mn

o

I

H

g

P

r

q s

t u

X W

V

F G

H

6.4 cm 84˚

42˚

J

KL

38˚

55˚

15 m

X45˚

85˚10 cm

Z

Y

B115˚

15˚

20.8 cm

C

A

B 22˚19˚

54 cm

C

A

P 29˚

77˚

32 mm

R

Q

T121˚

34˚2.8 m

U

S

R Q

P

9.4 cm

64˚ 50˚

L107˚

25˚ 58 m

M

K

W21˚

46˚32 mm

U

V

N

LM

42˚

63˚

6.7 m

E

78˚

20 m

F

G

10 m

N

LM63˚

6.7 cm5.2 cm

L113˚

58 m

M

K 12 m

E13˚

16 m

F

D

12 m

5 m

A C

B

84˚

8.2 m

U T

S

7.8 cm

58˚

9.8 cm

U T

V

16.8 cm

88˚9.1 cm

X 68˚

4.1 m

Z

Y

7.4 m

W126˚

31 mm

U

V

92 mm

X 45˚

4.1 m

W

Y

4.1 m

T130˚

24 m

U

V

70 m

5.2 m

D C

E

43˚

6.9 m

N

OM

72˚

4.7 mm

2.2 mm

X

52˚

21˚5.6m

Z

Y

E

75˚

22 mm

F

G

12 mm

E 38˚ 68˚

50 m

F

G

U T

S

88 cm74˚57˚

X W

V

12 cm

75˚

45˚A C

B

8 cm

37˚ 72˚

R Q

P

12 cm

64˚

13 cm

X

32˚

5.6 m

Z

Y

10.2 m

M

n

O


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Areas and the Sine Rule.

Find the perimeter and the area of each of these triangles. Leave the answerto an appropriate degree of accuracy.

1). 2). 3). 4).

5). 6). 7). 8).

9). 10). 11). 12).

13). 14). 15). 16).

Examination Style Questions.1). In ∆ABC, AC = 12.4 cm, ∠BAC = 29˚, ∠ACB = 65˚.

The length of AB is x cm. ∠ABC is ø˚.a). Write down the value of ø.b). Hence calculate the value of x.

2). The diagram shows a church clock at 1240 hours.The hour hand is 0.8 m long and the minute hand is 1.4 m long.a). Calculate the angle the hour hand has moved through since 1200.b). Calculate the distance, in cm, between the tips of the hands

at 1240 hours.

3). A helicopter leaves a heliport H and, its measuringinstruments show that it flies 3.5 Km on a bearing of 128˚to a checkpoint C. It then flies 5.2 Km on a bearing of 068˚to its base B.a). Show that ∠HCB is 120˚.b). Calculate the direct distance from the heliport H to the base B.

4). A ship sails south west from A at 16 Kmh-1 .At midnight it is at B when flashes from a lighthouse are seen on a bearing of 230˚.90 minutes later, flashes from the same lighthouse are seen due west of the ship.a). Sketch a diagram showing this information, clearly indicating North.b). Calculate the distance between the lighthouse and the ship at 01.30 am.

E

75˚ 18 mm

F

G

12 mm

E38˚

24.5 m

19 m

F

G

Z65˚

9.7 m

W

Y

7.9 m

H12.4 cm

F

G

8.6 cm126˚

Q

5.4 cm

7.2 cmP R

49˚

Q

OP

72˚ 12 mm9 mm

W121˚

46 mm

X

Y37 mm

N

OM42˚

6 mm

8 mm

Q

0.14 m

0.83 mP R

79˚

L129˚

108 m

M

N

113 m

U T

S

8 cm74˚

4.2 cm

25 m

D F

E

47 m

72˚

F19.5 m

E

G

9.8 m136˚

W100˚

4.6 cm

X

Y37 mm

H

F28˚

34.5 m

29 m

GQ

0.54 m0.33 m

P R

65˚

x cm

A C

B

12.4 cm

65˚

ø˚

29˚

H 68˚

3.5 Km

N

B

C

5.2 Km128˚


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Trigonometry, Sine and Cosine Rule. Exam Style Questions 1.

1). Two metal bars are hinged at X. Y and Z are attachedto a vertical pole. XZ is horizontal.a). Calculate YZ (to 2 s.f.).b). Find ∠ YXZ to the nearest degree.c). Find ∠ XYZ to the nearest degree.d). XZ is examined and it is found not to be horizontal, because YZ = 74 cm.

Now calculate ∠ XZY to the nearest degree.

2). a). Find the height of the tree in the diagram,and then the distance AB.

b). Find the distance XY in thisdiagram showing the height ofa building.Find the height of the building XZ.

3). A van unloads at a factory. Its trailer dooris opened onto a loading bay as shown.From the diagrama). find the vertical distance between C and B,b). find the angle AB makes with the horizontal,c). find the horizontal distance between B and the

vertical wall, CD, of the factory loading bay.

4). At 5.00 pm a ship is observed at X, 9 Km due south of a lighthouse L.The ship travels at a constant speed on a bearing of 054˚ and reaches Y, a point due eastof the lighthouse, at 5.50 pm. Z is a point on the ships path which is nearest to L.a). Draw a sketch showing X, L, Y and Z. Calculate LZ to 2 s.f..b). Find XZ.c). Find the speed of the ship.d). Find the time at which the ship reaches Z.

5). A tennis ball is on a table, with a ruler resting on it as shown.DE = 22 cm and ∠ EDG = 18˚.Calculate the radius of the tennis ball.

6). A woman stands on a cliff at X. She looksout to sea and can see a power boat at C.The angle of depression from where shestands to the boat is 52˚. Calculatea). the height of the cliff,b). CX to 3 s.f..C sails in a straight line away from Y to D.It takes 16 minutes to travel this distance.The angle of depression from X is now 25˚.c). Calculate the average speed in Km/h.

Y

X Z150 cm

165 cm

8 m

61˚ A

B

YW

X

Z42 m

21˚ 36˚

1.10m

2.8m

0.65m0.4m

A

D

C

B

E

DG H

22 cm

18˚

F

X

Y DC

9.8 Km

25˚52˚


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Trigonometric Graphs 1.For the following questions you will need A4 graph paper placed landscape.

1). Label your axes as follows:y axis: 1≤ y ≤ -1, 8 cm = 1 unit, θ axis (x axis): 0˚≤ θ ≤ 540˚, 4 cm = 90˚.

a). Plot the graph of y = Sin θ for this range.b). On the same axes plot the graph of y = Cos θ for this range.

2). Use your graphs, or otherwise, to find the two possible solutions for θ between0˚ and 360˚ to the nearest degree for the following:

a). Cos θ = 0.5 b). Sin θ = 0.5 c). Sin θ = 0.7 d). Cos θ = 0.7e). Cos θ = 0.2 f). Cos θ = 0.8 g). Sin θ = 0.8 h). Cos θ = 0.25i). Sin θ = 0.9 j). Sin θ = 0.3 k). Cos θ = -0.5 l). Sin θ = -0.5m). Cos θ = -0.1 n). Sin θ = 0.95 o). Sin θ = -0.95 p). Cos θ = 0.4q). Sin θ = 0.6 r). Sin θ = -0.6 s). Cos θ = 0.6 t). Cos θ = -0.6

3). Label your axes as follows:y axis: 4 ≤ y ≤ -4, 2 cm = 1 unit, θ axis (x axis): 0˚≤ θ ≤ 540˚, 4 cm = 90˚.

a). Copy and complete the table below for Tan θ

θ 0 10 20 30 40 50 60 70 80 90Tanθ 0 0.58

b). Continue this table to 540˚.c). Copy and complete the table below for Tan θ

θ 81 83 85 87 89 91 93 95 97 99Tanθ

d). What happens to Tan θ as θ approaches 90˚?e). Plot the graph of Tan θ.

4). For each of the following questions each pair of graphs should be plotted on thesame set of axes. The scales for the axes are shown.

a). y axis: 4 ≤ y ≤ -4, 1 cm = 1 unit, θ axis (x axis): 0˚≤ θ ≤ 540˚, 4 cm = 90˚.i). 2 Sinθ, ii). 4 Cosθ.

b). y axis: 4 ≤ y ≤ -4, 1 cm = 1 unit, θ axis (x axis): 0˚≤ θ ≤ 540˚, 4 cm = 90˚.i). 3 Cosθ, ii). 1.5 Sinθ.

c). y axis: 2 ≤ y ≤ -2, 2 cm = 1 unit, θ axis (x axis): 0˚≤ θ ≤ 540˚, 4 cm = 90˚.i). 1/

2 Sinθ, ii). 2 Cosθ.

d). y axis: 2 ≤ y ≤ -2, 2 cm = 1 unit, θ axis (x axis): 0˚≤ θ ≤ 540˚, 4 cm = 90˚.i). 8/

5 Cosθ, ii). 1/

3 Sinθ.

e). What effect does the coefficient have upon the shape of the graph ?f). Find a definition for ‘amplitude’ and write it in your book.

5). Use your graphs for the question above, or otherwise, to find the possible solutionsto the nearest degree in the range 0˚≤ θ ≤ 360˚ for the following:

a). 2 Sinθ = 1.0 b). 4 Cosθ = 3.4 c). 3 Cosθ = -2 d). 2 Sinθ = 0.4


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Read the scales!

15). 16).

13). 14).

11). 12).

9). 10).

50 100 150 200 250 300 350

50 100 150 200 250 300 350

50 100 150 200 250 300 350

50 100 150 200 250 300 350 50 100 150 200 250 300 350

50 100 150 200 250 300 350

50 100 150 200 250 300 350

50 100 150 200 250 300 350

-3

-2

-1

1

2

3

-6

-4

-2

2

4

6

-3

-2

-1

1

2

3

-3

-2

-1

1

2

3

-3

-2

-1

1

2

3

-6

-4

-2

2

4

6

-0.6

-0.4

-0.2

0.2

0.4

0.6

-0.3

-0.2

-0.1

0.1

0.2

0.3


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Making and Using a Sundial

12 111 210

9 3

8 4

7 5

66

5 7

TEMPUS FUGIT

flapflap

Gnomon

Sundial Face

The 12 noon line must be positioned in line with True orPole Star north.True or Pole Star north is 8˚ east of magnetic north.

Positioning the sundial.

True north

Making the sundial.

Cut out the sundial face and the gnomon.Cut down the length of the thick line on the sundial face.Fold over the gnomon and glue together.The angle the gnomon makes to the sundial should be the latitudeof where it is to be used. Find out the latitude of your town or cityin an atlas, or check the table overleaf to see if it is there.Fold up the flap at the appropriate angle for your latitude.Slide in to the sundial face and glue the flaps of the gnomon to theunderside of the sundial face.Your sundial is complete and ready for positioning.

Gnomon Sundial

12

51˚53˚55˚


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Probability 1. Revision.

1). Shahida is playing "heads or tails" with her friend.She spins a fair coin four times and gets four heads.a). What is the probability that she gets a tail with her next spin of the coin ?b). If Shahida spins the coin 500 times in succession, approximately how many

times would she expect a tail to come up ?

2). The probability that Jenny will beat Gill at tennis is 2/5.

The probability that Jenny will beat Gill at both tennis and snooker is 1/3.

What is the probability that Jenny will beat Gill at snooker ?

3). The probability that it will rain tomorrow is r.The probability that it will be windy tomorrow is w.The probability that it will rain and be windy tomorrow is b.Both r and w are marked on the number line.Draw it and mark on a possible position for b.

4). In football, a team scores 3 points for a win, 1 point for a draw and 0 points if it loses.The probability that my team win their next match is 0.2. The probability that it will losethe next match is 0.35.What is the probability that it will gain at least 1 point in its next match ?

5). The diagram shows two fair spinners.Both are spun and the scores are added together.What is the probability that the sum of the scoresis at least 7 ?

6). In a board game a player is given two choices to get out of jail.A - throw a coin three times. Get 2 heads and 1 tail.B - throw a dice and a coin. Get a number less than 5 and a tail.

The dice is numbered 1 - 6. Which of the two choices should the player choose and why ?

7). a). When Steven ‘Gusty Wind’ Harris plays snooker the probability that he scores100 point or more is 1/

21.

i). What is the probability he will score less than 100 points ?The probability that he will score 50 point or less is 2/

3.

ii). Calculate the probability he scores more than 50 but less than 100 points.b). Snooker is played with 15 red balls, 6 other coloured balls and 1 white ball.

All 22 are kept in a box.i). A ball is taken out at random. What is the probability that it is red ?ii). This ball is put back in the box. A second ball is now taken out.

Calculate the probability that both balls taken out are red.

8). Two fair, six-sided dice are thrown. A double occurs when the samenumber appears on both dice.a). How many ways can a double occur ?b). What is the probability of scoring a double ?c). What is the probability of scoring a total of 7 on the 2 dice ?d). What is the probability of scoring either a double or a total of 7 on the two dice ?

123

23 4

12

525

1

3

2

0 1 2r w


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Conditional Probability.

When the probability of an event is dependent on the outcome of another eventthen this is described as conditional probability.

Typical examples would be sampling without replacement. Taking 2 items from a bag at the sametime would also be conditional probability as it is impossible to choose the same item twice.

E.g. A bag contains 6 counters, 4 blue and 2 white.A counter is taken out and not replaced. A second counter is then taken out.Find the probability thata). the first counter is blue,b). if the first counter is blue, the second counter is blue,c). both counters are blue.

The diagram shows the counters available to be picked.

First pick

If first pick is blue

Second pick

if first pick is white

a). P (the first counter is blue) = 4/6

= 2/3

b). P (if the first counter is blue, the second counter is blue) = 3/5

c). P (both counters are blue) = 4/6 x 3/

5= 2/5.

1). A box contains 5 red balls and 3 yellow balls.One ball is taken out and not replaced and then a second ball is taken out.a). If the first ball taken out is red, find the probability that the second ball is

i). red, ii). yellow.b). If the first ball taken out is yellow, find the probability that the second ball is

i). red, ii). yellow.

2). A bag contains 10 green counters and 5 yellow counters.One counter is taken out and not replaced and then a second counter is taken out.a). If the first counter taken out is green, find the probability that the second counter is

i). green, ii). yellow.b). If the first counter taken out is yellow, find the probability that the second counter is

i). green, ii). yellow.


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Tree Diagrams (Dependent Events).

1). Laura has 6 white socks and 10 black socks of the samekind in a drawer. In the dark she takes two socks atrandom, one after the other, out of the drawer.a). Copy and complete the tree diagram.b). Calculate the probability that she takes out

i). a pair of white socks,ii). a pair of black socks,iii). two socks of a different colour,iv). at least one of them is black.

2). A bag contains 6 red balls and 4 blue balls. Hazel removestwo balls from the bag without replacing the first.a). Copy and complete the tree diagram.b). Calculate the probability that she takes out

i). 2 red balls,ii). 2 blue balls,iii). one of each colour,iv). at least one of them is blue.

3). In a cage are 4 blue budgies, 5 red budgies and 3 greenbudgies. The door of the cage is left open and two fly out.a). Copy and complete the tree diagram.b). Calculate the probability that

i). 2 blue budgies fly out,ii). 2 red budgies,iii). a green and red budgie fly out,iv). at least one of them is blue,v). no green budgies escape.

4). Samantha has 9 white socks and 6 black socks of the same kind in a drawer. In the dark shetakes two socks at random, one after the other, out of the drawer.a). Draw a tree diagram and mark on all the appropriate probabilities

and outcomes.b). Calculate the probability that she takes out

i). a pair of white socks,ii). a pair of black socks,iii). two socks of a different colour,iv). at least one sock that is black,v). no black socks.

5). A bag contains 7 red fruit drops and 3 yellow fruit drops. One fruit drop is taken out andeaten. A second fruit drop is then taken out and eaten.a). Draw a tree diagram and mark on all the appropriate probabilities and outcomes.b). Calculate the probability that

i). both fruit drops are red,ii). both fruit drops are yellow,iii). one of each colour is eaten,iv). at least one yellow fruit drop is eaten,v). no red fruit drops are eaten.

First sock

Second sock

White

Black

616

9 15

White

White

Black

Black

First ball

Second ball

Red

Blue

610

3 9

Red

Red

Blue

Blue

Firstescapee

Second escapee

Blue

Red

GreenBlue

Blue

Blue

Red

Red

Red

Green

Green

Green


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Probability. Exam Style Questions 1.

1). In an experiment a fair dice (numbered 2, 2, 3, 4, 6, 6) and a five sided spinner(numbered 1, 1, 3, 5, 5) are thrown. Find the probabilities that

a). the sum is 5,b). the product is 6,c). both are odd numbers.

2). Maisey passes 2 traffic lights on her journey to work. As she arrives at each traffic lightthe probability of it being red is 3/4. Calculate the probability that

a). the first light is not red,b). the first light is not red and the second light is red,c). the first light is red and the second is not red,d). one light is red and the other is not red,e). they are both not red.

3). Andrew and Bella either cycle to school or walk. One day the probability that Andrewcycles is 0.7, and the probability that Bella cycles is 0.5. The two probabilities areindependent of each other.

a). Copy and completethe tree diagram.

b). Use the tree diagram to find the probabilities thati). Andrew will cycle and Bella will walk,ii). at least one of them cycles to school.

c). Neither Andrew nor Bella was absent during 80 days. On how many of these dayswould you expect them both to have cycled to school ?

d). Andrew and Bella find that on a particular day, they both cycle. The probability ofthem meeting when cycling is 0.2. If one cycles and the other one walks theprobability of them meeting is 0.4. If they both walk the probability of meeting is 0.9.By extending your tree diagram or otherwise, find the probability that they will meeton the way to school.

4). When 3 coins are tossed in the air the possible outcomes are below.

3 Heads HHH2 Heads 1 Tail HHT HTH THH a). What is the probability that all 3 coins1 Heads 2 Tails HTT THT TTH land on heads when thrown ?3 Tails TTT

b). Construct similar tables showing outcomes wheni). 2 fair coins are thrown,ii). 4 fair coins are thrown.

c). State the probabilities of all the coins showing heads wheni). 2 fair coins are thrown,ii). 4 fair coins are thrown.

Andrew Bella

Cycles

Walks

0.7

0.5

Cycles

Cycles

Walks

Walks


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Who is the luckiest ? You decide!Place a Counter on S (Start). Throw a coin. If it lands on heads move the counter to the left,

if it lands on tails move the counter to the right.After 6 throws you should reach one of the Gods.

Record who it is.

Is this a fair way of deciding ?Repeat the experiment and record the outcomes.

What would you expect and why ?

Schichi-fuku-jinThe Schichi-fuku-jin is the seven Gods of luck in Japanese folklore.

They are comical deities often portrayed together riding on a treasure ship (takarabune).They carry various magical items such as an invisible hat, a lucky rain hat

and an inexhaustible purse.

sHEADS TAILS

Daikoku Bishamon Ebisu Fukurokuju Jurojin Hotei Benzaiten


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Solving Quadratics by Factorising.

A. Solve these equations.1). x(x - 7) = 0 2). f(f + 9) = 0 3). 2a(a - 4) = 04). 3a(a - 1) = 0 5). s(4s + 2) = 0 6). p(6p + 3) = 07). 3y(2y + 8) = 0 8). 5p(3p + 12) = 0 9). 3g(2g - 7) = 010). 5b(4b + 9) = 0 11). 4t(3t + 15) = 0 12). 4a(5a - 9) = 013). 2x(7x - 10) = 0 14). 2z (5z + 19) = 0 15). 7m(2m + 21) = 0

B. Factorise the following to solve each quadratic equation.1). x2 - 6x = 0 2). a2 + 4a = 0 3). b2 - 7b = 04). 2n2 + 4n = 0 5). 6k2 - 8k = 0 6). 9x2 + 12x = 07). 4d2 + 20d = 0 8). 35p2 - 5p = 0 9). 4v2 - 10v = 010). 9q2 - 15q = 0 11). 35x2 + 15x = 0 12). 35a2 + 21a = 013). 18f2 - 42f = 0 14). 16k2 + 40k = 0 15). 21e2 + 49e = 0

C. Solve these equations.1). (x + 5)(x + 4) = 0 2). (f - 6)(f - 3) = 0 3). (a + 2)(a - 5) = 04). (a - 2)(a + 7) = 0 5). (2s - 10)(s - 4) = 0 6). (p - 2)(3p - 9) = 07). (4y - 8)(y + 3) = 0 8). (3p - 2)(p + 9) = 0 9). (2g - 11)(g - 3) = 010). (4b + 4)(3b - 5) = 0 11). (4t + 2)(5t - 7) = 0 12). (3a + 6)(2a - 5) = 013). (6x - 4)(3x + 2) = 0 14). (5z - 7)(4z + 1) = 0 15). (2m - 3)(3m + 9) = 0

D. Factorise first, then solve these equations.1). r2 + 4r - 21 = 0 2). h2 + 4h - 12 = 0 3). c2 + 7c - 60 = 04). f2 - 15f + 36 = 0 5). g2 - 14g - 15 = 0 6). g2 - 15g + 54 = 07). d2 + 10d + 24 = 0 8). p2 + 18p + 81 = 0 9). 2x2 + 5x + 3 = 010). 3b2 + 5b + 2 = 0 11). 2a2 + 7a + 5 = 0 12). 3p2 - 19p + 28 = 013). 5n2 + 26n + 5 = 0 14). 7h2 - 38h + 15 = 0 15). 3e2 - 13e - 10 = 0

These are harder!16). 4y2 - 20y + 25 = 0 17). 6u2 + u - 12 = 0 18). 8h2 - 22h + 15 = 019). 9f2 + 18f + 8 = 0 20). 4p2 + 12p - 7 = 0 21). 3e2 - 13e - 10 = 022). 4n2 + 3n - 7 = 0 23). 6f2 + 33f - 63 = 0 24). 8a2 + 10a - 3 = 025). 15p2 + 2p - 1 = 0 26). 25e2 + 40e + 16 = 0 27). 10n2 + 11n - 6 = 028). 16f2 - 8f + 1 = 0 29). 18n2 +9n + 1 = 0 30). 28e2 - 85e + 63 = 0

E. First rearrange, then factorise to finally solve these equations.1). x2 + 18x = -45 2). a2 + 8a = 9 3). b2 - 6b = 404). f2 = 10f - 16 5). g2 = 6g + 16 6). g2 = 13g - 127). d2 = -14d - 13 8). p2 = 44 - 7p 9). x2 = 24 - 10x10). 9b + 36 = b2 11). 14 = 13a + a2 12). 20 + 8p = p2

These are harder!13). 2n2 = 13n - 11 14). 3h2 = 5 - 14h 15). 3 - e = 2e2

16). 7y2 = 8y - 1 17). 11u - 15 = 2u2 18). 5h2 = 9h + 219). 7 = 4f + 3f2 20). 18 + 5p = 2p2 21). 5e2 = 7 - 2e22). 6n2 = 5n + 4 23). 7 - 12f = 4f2 24). 8a2 = 2a + 1525). 12p2 =35 + 16p 26). 15 + 8e = 12e2 27). 28n - 49 = 4n2

28). 12f2 + 7 = 31f 29). 19n = 2 + 24n2 30). 25e2 + 4 = 20e


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The nth Term of a Quadratic.

We have already used the difference method to find a quadratic function from a table inLevel 7/8 Pack 1 Page 39. When looking at sequences always see if it is a simple function beforeattempting the ‘rote’ methods. The difference method is useful to discover if a sequence is linear orquadratic. Remember, if the first difference is constant the sequence is a linear function, if the seconddifference row is constant then the function is a quadratic.

Here is another method to find a quadratic function from a quadratic sequence.

Find the nth term of the sequence 3, 12, 25, 42, 63.

3 12 25 42 63 .... 9 13 17 21 1st difference row 4 4 4 2nd difference row is constant.

We know that the nth term of a quadratic sequence is of the form

an2 + bn +c where a, b and c are numbers.

If we put the general equation and sequence together

For the first term i.e. n = 1, a + b + c = 3For the second term i.e. n = 2, 4a + 2b + c = 12For the third term i.e. n = 3, 9a + 3b + c = 25

By subtracting these we can create two simultaneous equations

4a + 2b + c = 12 _ 9a + 3b + c = 25 _ a + b + c = 3 4a + 2b + c = 123a + b = 9 5a + b = 13

These can be solved to give 2a = 4, i.e. a = 2 and hence by substitution, b and c can be found.In this case b = 3 and c = -2.

These can now be put back into the general equation for the nth term.

The nth term of the sequence 3, 12, 25, 42, 63... is 2n2 + 3n - 2

For each of the following a). find the next two terms, b). find the nth term for the quadratic sequences.(See if the sequence is a simple quadratic function before attempting the rote methods.)

1). 5, 8, 13, 20, 29... 2). 0, 3, 8, 15, 24... 3). 2, 8, 18, 32, 50...4). 0.5, 2, 4.5, 8, 12.5... 5). 2, 6, 12, 20, 30... 6). 4, 10, 18, 28, 40...7). 7, 14, 25, 40, 59... 8). 0, 7, 16, 27, 40... 9). -6, -6, -4, 0, 6...10). 0, 13, 32, 57, 88... 11). 11, 11, 9, 5, -1... 12). 3, 12, 31, 60, 99...13). 4, -10, -32, -62, -100... 14). 1.5, 5, 9.5, 15, 21.5... 15). -4, 18, 54, 104, 168...16). 3, 12, 25, 42, 63... 17). -3.5, -1.5, 6.5, 20.5, 40.5..18). 5.5, 11, 20.5, 34, 51.5...19). -4, -9, -20, -37, -60... 20). 6, 5, 3, 0, -4... 21). 2.25,6.5,12.25,19.5, 28.25..


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Completing the Square.A). Solve the equations.

e.g. (f - 3)2 = 9, square root both sides,f - 3 = ± 3,

so f = 3 + 3 = 6 or -3 + 3 = 0.

1). x2 = 64 2). d2 = 100 3). x2 = 49 4). h2 = 105). (c - 2)2 = 9 6). (h - 7)2 = 4 7). (x + 3)2 = 4 8). (p + 2)2 = 259). (x - 1)2 = 2 10). (q + 4)2 = 3 11). (w - 3)2 = 5 12). (a + 2)2 = 213). (h - 6)2 = 36 14). (c - 4)2 = 10 15). (x + 3)2 = 49 16). (m - 1)2 = 717). (t - 8)2 = 3 18). (y + 7)2 = 6 19). (h + 9)2 = 3 20). (c + 10)2 = 8

B). Expand the brackets. The expressions are all perfect squares.

1). (x + 4)2 2). (a - 6)2 3). (r + 7)2 4). (e - 5)2

5). (p - 3)2 6). (t + 2)2 7). (b - 8)2 8). (c - 9)2

9). (k - 12)2 10). (u + 11)2

C). Add the term that will make each expression a perfect square, then factorise it.

e.g. To find the number we halve the coefficient of x and then square it. x2 + 6x Halve 6 and then square it. i.e. 9.

Therefore we have x2 + 6x + 9 = (x + 3)2

1). a2 + 8a 2). b2 + 10b 3). c2 - 4c 4). d2 - 6d5). x2 + 5x 6). y2 - 3y 7). z2 - 7z 8). m2 + 2m9). n2 - n 10). n2 - 12n 11). x2 + 9x 12). k2 + 11k13). b2 - 13b 14). v2 - 1 v 15). j2 + 1 j

2 4D). Solve these equations by completing the square. (Some will factorise, use this as a check).

e.g. x2 - 8x + 3 = 0,x2 - 8x = -3, Add 16 to both sides to complete the square.x2 - 8x + 16 = 13,(x - 4)2 = 13,

Therefore x - 4 = ± √13, x = 4 ± √13, x = 0.39 or x = 7.61 (2 d.p.)

1). a2 + 4a - 21 = 0 2). b2 - b - 12 = 0 3). n2 + 4n + 4 = 04). d2 - 5d + 6 = 0 5). g2 + 5g + 4 = 0 6). x2 - 10x + 25 = 07). q2 + 10q + 22 = 0 8). t2 - 6t + 9 = 0 9). m2 + 6m + 7 = 010). y2 - 3y + 1 = 0 11). d2 + 2d - 2 = 0 12). x2 - 4x - 2 = 013). k2 - 5k + 2 = 0 14). f2 - f - 1 = 0 15). x2 + 3x - 2 = 016). p2 - 10p + 15 = 0 17). v2 + 9v + 19 = 0 18). u2 - 14u - 3 = 0

E). Solve these equations by completing the square. (Some will factorise, use this as a check).

1). c2 + 2c = 3 2). d2 - 4d = 5 3). x2 - 3x = -2 4). x2 - 2x = 45). y2 + 2y = 1 6). h2 + 6h = 5 7). n2 - 2m = 2 8). f2 - 4f = -19). w2 - 8w = -13 10). d2 + 6d = -7 11). e2 = 6e - 4 12). f2 = 11 - 4f13). j2 = -4j - 2 14). n2 = 2n + 1 15). h2 = h + 5 16). d2 = 12d - 3517). h2 + h = 8 18). e2 - 6e - 3 = 0 19). x2 + 5x = 15 20). e2 = 3e + 11


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Solving Quadratics by Formula.

x = - b ± √ b2 - 4ac 2a

A). Solve, with the above formula, the quadratics which are in the form ax2 + bx + c.These quadratics will factorise so you can use this as a check.

1). x2 + 5x + 6 = 0 2). x2 - 5x + 4 = 0 3). x2 - 4x - 5 = 04). x2 - 6x + 5 = 0 5). x2 + 2x - 8 = 0 6). x2 - 10x + 21 = 07). 2x2 + 5x + 3 = 0 8). 2x2 + 5x - 3 = 0 9). 3x2 - 4x + 1 = 010). 3x2 - 5x - 2 = 0 11). 6x2 + 7x + 2 = 0 12). 3x2 - 13x - 10 = 013). 6x2 - 11x -10 = 0 14). 5x2 - 3x - 2 = 0 15). 4x2 + 7x - 2 = 016). 6x2 + 13x + 6 = 0 17). 5x2 + 6x + 1 = 0 18). 8x2 - 6x + 1 = 0

B). Solve the following quadratics using the above formula to 2 decimal places.

1). x2 + 6x + 4 = 0 2). x2 + 5x + 3 = 0 3). x2 + 2x - 5 = 04). x2 - 10x + 8 = 0 5). x2 + 12x + 10 = 0 6). x2 + 7x + 4 = 07). 2x2 - 5x - 4 = 0 8). 2x2 + 9x + 3 = 0 9). 3x2 + 7x + 3 = 010). 2x2 - 3x - 1 = 0 11). 5x2 + 9x + 2 = 0 12). 3x2 + 2x - 3 = 013). 5x2 + x - 2 = 0 14). 5x2 + 8x + 2 = 0 15). 2x2 - 11x + 7 = 016). 3y2 + 6y - 7 = 0 17). 4p2 + 7p - 6 = 0 18). 5a2 + 9a + 2 = 019). b2 + 2b - 5 = 0 20). q2 - 15 q + 8 = 0 21). 2t2 - t - 4 = 022). 3w2 - 5w - 8 = 0 23). 2x2 + 11x + 8 = 0 24). 3r2 + 8r + 3 = 025). 5k2 + 9k + 2 = 0 26). x2 + 3x + 1 = 0 27). x2 - 2x - 4 = 028). 2x2 + 7x - 3 = 0 29). 3x2 - 5x - 3 = 0 30). 5x2 + 3x - 3 = 0

C). Simplify and solve the following quadratics to 2 decimal places.

1). x2 + 10x = 5 2). 2x2 + 5 = 9x 3). 2k2 + 4k = 34). 4c2 + 9c = 3 5). 2x2 - x = 7 6). x2 + 6 = 8x7). 4x2 + 3x = 5 8). 5x2 = 7x - 1 9). 3x2 + 5 = 9x10). 3x2 = 7x - 3 11). x2 - 8x = 7 12). x2 + 5x = 213). 2x2 = 7x + 3 14). 4x2 - 9x = 3 15). x2 = 8x - 1116). 5x2 - 2 = 6x 17). 5 - 7x = 2x2 18). 14x = 3x2 + 819). 6x + 3 = 5x2 20). 2 = 5x2 + 8x 21). 12x = 3x2 + 1022). 8x = 3x2 + 2 23). 16x = 4x2 + 3 24). 2x2 = 7x - 3

D). Simplify and solve the following quadratics to 2 decimal places.

1). x(x - 2) = 5 2). 2x(x + 3) + 3 = 0 3). (x - 2)(x - 4) =54). (x + 1)(x - 1) = 1 5). (x + 1)(x - 5) = 3 6). (2x + 1)(x - 1) = 17). (2x - 5)(x + 3) = 8 8). 3x(x - 1) = 5 9). (x + 3)2 = 210). 2x(x + 4) = 1 11). 3x(x - 2) = 2 12). x(x + 8) = 2x + 113). 5x(x + 3) = 3(8x - 1) 14). (x + 3)(x + 1) = 4 15). (x - 3)(2x - 5) = 216). 4 = 5x(4 - 5x) 17). (5x - 3)(3x + 1) = 1 18). 5x(2x - 3) = 7(3 - 2x)19). x(x + 4) = 6(x + 4) 20). 2(2x + 1)2 + 5(2x + 1) = 1

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