IGCSE Foundation Sheets 4 -...

For use only in Whitgift School IGCSE Foundation Sheets 4

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IGCSE Foundation

Sheet F4-1 F4-10-1 Similar Shapes Sheet F4-2 F4-10-2 Similar Shapes Sheet F4-3 F4-10-3 Similar Shapes Sheet F4-4 F4-10-4 Similar Shapes Sheet F4-5 F4-10-5 Similar Shapes Sheet F4-6 F5-02a-c-01 Rotations Sheet F4-7 F5-02a-c-02 Centre of Rotations Sheet F4-8 F5-02a-c-03 Centre of Rotations Sheet F4-9 F5-02a-c-04 Rotations and Reflections Sheet F4-10 F5-02d-f-01 Reflections Sheet F4-11 F5-02d-f-02 Reflections Sheet F4-12 F5-02d-f-03 Rotations and Reflections Sheet F4-13 F5-02g-h-01 Translations and Enlargements Sheet F4-14 F5-02j-l-01 Centre of Enlargements Sheet F4-15 F5-02j-l-02 Centre of Enlargements Sheet F4-16 F5-02j-l-03 Centre of Enlargements Sheet F4-17 F5-02j-l-04 Transformations Sheet F4-18 F5-02j-l-05 Enlargements Sheet F4-19 F6-02-1 Averages Sheet F4-20 F6-02-2 Grouped Mean Sheet F4-21 F6-02-3 Grouped Mean Sheet F4-22 F6-03-1 Probability-Introduction Sheet F4-23 F6-03-2 Probability-Estimation Sheet F4-24 F6-03-3 Probability-Mutually Exclusive



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Sheet F4-1 F4-10-1 Similar Shapes

1. In each of the following, shape P is enlarged to a similar shape Q. Find the scale factor, k, for each enlargement. Find also the value of x.

(a) (b) (c) (d) (e)

2. A photocopier is set to reduce the lengths of copies to 23

of the original size. If the original

measured 12cm by 15cm what will be the dimensions of the copy? 3. A photography shop produces enlargements of photos. A 15cm x 10cm photo was enlarged

so that its longest side was 24cm. What was the length of the shorter side?

4. A map is reduced to 35

of its original size. A field on the original measured 25mm x 35mm.

What will its dimensions on the image be?

PTO

P Q 5cm

4cm

8cm

xP Q 5m 12m

2m x

6mm 8mm

x 5mm P Q

NB Q is the larger rectangle.

Q

P

1m 2m

1.5m

x

5cm

3cm

Q

7cm

NB Q is the larger triangle.

x

P



Sheet F4-1 F4-10-1 Similar Shapes (cont.)

5. A rectangle P is enlarged to a rectangle Q. The dimensions of P are 5m by 12m. The shortest side of Q is 6m. (a) What is the scale factor of enlargement? (b) What is the length of the largest side of Q?

6. A right-angled triangle P is enlarged to a triangle Q. The hypotenuse of P is 12cm and the

hypotenuse of Q is 15cm. (a) What is the scale factor of enlargement? (b) If the shortest side of P is 8cm find the shortest side of Q.

7. A map measures 24cm by 30cm and it is reduced to 3

2 of its original size. What are the dimensions of the reduced map?

8. In each of the following, shape P is enlarged to a shape Q. Find the scale factor, k, for each

enlargement. Find also the value of x. (a) (b) 9. A document is reduced to 5

3 of its original size. If the reduced document has dimensions 12cm by 15cm then what was the size of the original document?

10. A photo has width 10cm and an area of 150 2cm . Its length and width are enlarged by the

same factor so that its width is 12cm. What is the area of the enlarged photo? 11. A photocopier is to reduce documents so that the area of the copy is ¼ of the area of the

original. If the original had dimensions 112mm by 142mm what will the dimensions of the copy be?

P Q 9cm 3cm

6cm

x

P

Q x

12mm

4mm

5mm




1. Two rectangles A and B are similar. Rectangle A measures 3cm by 4cm. If the longer side of B is 5cm then find the shorter side, x.

2. Find x in the following:

(a) (b)

(c) (d)

3. Two right angled triangles A and B are similar. The hypotenuse of A is 5cm and the hypotenuse of B is 8cm. If the shortest side of A is 3cm then find the shortest side of B.

4. A cone has height 10cm and radius 4cm. The cone is placed upright with its point on a table and water is poured into the cone up to a height of 4cm. Find the radius of the surface of the water in the cone.

5. A photo measures 12cm by 15cm. It is put in a frame which measures 15cm by 18cm. Explain whether or not the frame is similar to the picture.

6. A cable goes from the top of a building which is 48m tall to a point X on the ground which is 15m away from the foot of the building. What is the distance between the cable and a point which is directly below the cable and which is 10m from the foot of the building.

7. Ben and Sarah want to measure the height of a building. Ben knows that he is 1.8m tall and Sarah suggests that he stands next to the building and compare the shadows. She measures his shadow to be 2.4m long and the shadow of the building to be 16m long. How tall was the building?

PTO

3cm 4cm

5cm

x A B

24mm

A x

12cm 8cm

10cm

x

20cm 15cm

12cm

14cm 11cm

60mm

32mm

x

18mm

x 21mm

16mm

20mm 25mm

40mm

A

A A

A

A

A A

B B B

B

B

B

B B

C C

C

C

C

C

C

C D

D



Sheet F4-2 F4-10-2 Similar Shapes (cont.)

8. Find x and y in the following (first draw the triangles ABC and ADE separately and mark on all the lengths):

(a) (b)

(c) (d)

9. The above diagram shows the cross section of a swimming pool. Water has been added so that it is at a height of 1.5m. Find the length, l, of the surface of the water.

27m

1.2m

4.8m

1.5m

l

A

B

C

D

E

x

35cm

15cm

6cm 20cm

y

A

B

C

D

E 14cm

10cm

5cm x

y

18cm 2+x

x

7cm

8cm

18cm A

B

C

E

D

A

B

C

D

E

18cm

24cm

15cm

y

x 20cm

y




1. In the diagram shown below BC 24cm, ED 30cm, BE 18cm and AE 40cm= = = = . CD is parallel to BE, CD and BCx y= = .

(a) Draw the two triangles that are similar to one another.

(b) Find x and y.

2. In the diagram shown below AB = 32mm, CD = 63mm, DE = 54mm and AE = 42mm . AB is parallel to CE, CE = x and BC = y.

(a) Draw the two triangles that are similar to one another.

(b) Find x and y.

3. A light L is placed 80cm in front of a vertical object PQ, of height 52cm. The object PQ casts a shadow RS on a vertical wall 320cm away from L.

(a) To which triangle is triangle LPQ similar? (b) Calculate the height in metres of the shadow RS. (c) Use Pythagoras’ theorem to find LS (to 3sf). (d) Hence, or otherwise, find LQ (to 3sf).

PTO

42mm

y

63mm

A D E

B

54mm

32mm C

x

40cm

32cm 18cm

x

A D E

B

30cm

y C

S

R

P

Q

L

80cm

320cm

52cm



Sheet F4-3 F4-10-3 Similar Shapes (cont.) 4. A picture measuring 8cm by 12cm has a border of width 2cm put around it. State, with

reasons, whether or not the picture with the border is similar to the picture without the border.

5. Find x and y in the following (first draw the triangles ABC and ADE separately and mark on

all the lengths):

(a) (b)

6.

(a) Draw the triangle ABC (without marking on the point D) shown above, marking on all lengths and angles.

(b) Copy and complete the triangle BDC shown below, marking on all lengths.

(c) Explain why ABC and BDC are similar.

(d) Use the fact that they are similar to find x where AB = x.

(e) Find also y where AD = y.

C

D

β α

B

A

B

C D

α

α

12cm

9cm

8cm y

x

β

A

B

C

D

E

18cm

24cm

12cm

x

y

5cm

A

B

C

D

E

24cm

44cm

21cm

x

y

7cm



Sheet F4-4 F4-10-4 Similar Shapes (Harder)

1. In the triangle shown below CD = 18mm, AD = 24mm, AC = x, AB = y and BD = z.

(a) Write down the value of the angle CBA. (b) Find x by using Pythagoras’ theorem on the triangle ADC. (c) Draw the triangle ABC (without marking on the point D) shown above, marking on all lengths and angles. (d) Find the angle CAD. (e) Copy and complete the triangle DAC shown below, marking on all lengths and angles. (f) Explain why DAC and ABC are similar. (g) Write down an equation involving y. (h) Solve this to find y. (i) Calculate the value of z.

2.

(a) What is the ratio of BC: ED?

(b) What is the ratio of AC: AD?

(c) Use these to explain why ABC is not similar to AED (letters in that order).

(d) To which triangle is ABC similar?

(e) Draw, in the same orientation, the triangles ABC (without E and D marked on) and the triangle in (d). Mark on all the lengths.

(f) Use these to find x (where AE = x) and y (where BE = y).

PTO

D A

C

A

C

D

B y

z 24mm

x

18mm53.1

C

A

B

D

E 15cm

25cm

9cm

14cmx

y

50

50



Sheet F4-4 F4-10-4 Similar Shapes (Harder) (cont.) 3. In the triangle shown below BD = 8cm, AB = 10cm, AD = 6cm, AC = x and CD = y

(a) Draw the two triangles ABC and DBA in the same orientation and mark on all their angles.

(b) Hence explain why ABC and DBA are similar.

(c) Mark all the lengths on the two triangles.

(d) Write down an equation involving x.

(e) Solve this to find x.

(f) Calculate the value of y.

4. A photo is 8cm tall and 10cm long. A border is put around the photo and the photo is framed along with the border. The border is 2cm high along the bottom and the top of the photo and wcm wide on the left and right. Find w if the framed photo is similar to the photo itself.

5.

In the above diagram DCB CAB θ∠ = ∠ = , DB = 8cm, DC = 12cm and CB = 10cm.

(a) To which triangle is ABC similar?

(b) Draw ABC and the triangle of part (a) so that they have the same orientation and mark each side clearly.

(c) Find the length AB.

(d) Find also the length AC.

6. A cone of radius 6cm and height 15cm has a cone of height 9cm cut from the top. What is the radius of this cone?

A

C

D

B 10cm

8cm 6cm

x

y

θ

A B

C

D

θ

θ

10cm 12cm

8cm



Sheet F4-5 F4-10-5 Similar Shapes (Harder) 1.

In the diagram above ABCD is a trapezium, AB = 12cm, EB = 5cm, EC = 8cm, DC = 15cm, BDC α∠ = and ACD β∠ = .

(a) State which other angle is equal to α .

(b) State which other angle is equal to β .

(c) Hence state to which triangle DEC is similar. Draw both triangles with the same orientation and mark each length.

(d) Hence find AE.

(e) Find also the length DE.

2.

In the shape shown above AD = 8cm, DB = 5cm, AC = ycm, CB = xcm and θ=∠CAB .

(a) Find ABC∠ in terms of θ and explain why θ=∠DCB

(b) Explain why ABC is similar to CBD by drawing both triangles with the same orientation, marking each angle and length.

(c) Use this to find a value for 2x .

(d) Explain why ABC is similar to ACD and draw both triangles with the same orientation , marking each length.

(e) Use this to find a value for 2y .

(f) Use (c) and (e) to show that 222 13=+ yx .

(g) What theorem does this demonstrate? PTO

A

C

D

B y

x

8cm

5cm

θ

12cm

C D

A B

E α β

15cm

8cm 5cm



Sheet F4-5 F4-10-5 Similar Shapes (Harder) (cont.) 3. (a) Chris says “any two squares must be similar to one another”. Explain whether or not

he is right to say this.

(b) Chris goes on to say “in fact any two rectangles must be similar to one another since all their corresponding angles are equal”. Explain whether or not he is right to say this.

4. Which of the following types of triangles must always be similar to one another:

(a) right angled triangles

(b) equilateral triangles

(c) isosceles triangles.

5.

(a) Draw two triangles from the diagram that are similar to one another, marking on all the lengths

(b) Find x .

(c) Show that mm55=y , using Pythagoras’ theorem.

(d) Hence find z by using similar triangles.

6. A rectangular sheet of paper measures 10cm by xcm where 10 20x< < . The largest possible square is cut out and the rectangle that remains is similar to the initial rectangle.

Draw the initial rectangle, marking clearly the lengths of its sides.

(a) Draw the final rectangle, explaining why its sides are 10 and 10x − .

(b) By using the fact that the two rectangles are similar, write down an equation involving x.

(c) Show that 555 +=x satisfies this equation.

(d) Hence find, to 3sf, the ratio of the larger to the smaller side of two rectangles (this is called the Golden Ratio).

7. The side length of a cone is 13cm and its diameter is 10cm. The top of the cone is cut off to form a smaller cone which has perpendicular height 4.8cm.

(a) Find the perpendicular height of the initial cone.

(b) Find the diameter of this smaller cone.

10cm

x cm

44mm

42mm

x

A D

E

B

33mm C

y

z



Sheet F4-6 F5-02a-c-01 Rotations

Draw a set of axes with x and y both going from -3 to 3 with 2cm per unit on both axes. 1. (a) Draw the rectangle with vertices (1, 1), (3, 1), (3, 2) and (1, 2).

(b) Rotate 1R by °90 in an anticlockwise direction about the origin. Label this 2R . (c) Rotate 1R by 180° about the origin. Label this 3R . (d) Rotate 1R by °90 in a clockwise direction about the origin. Label this 4R .

Draw a set of axes with x and y both going from -4 to 4 with 2cm per unit on both axes. 2. (a) Draw the triangle 1T with vertices (1, 1), (4, 1), (4, 3).

(b) Rotate 1T by °90 in an anticlockwise direction about the point (1, 0). Label this 2T . (c) Rotate 1T by 180° about the point (1, 0). Label this 3T . (d) Rotate 1T by °90 in a clockwise direction about the point (1, 0). Label this 4T .

3. (a) Copy and complete the axis shown below, using 1cm per unit. Copy also and label the flags F1 and F2 that are drawn on the axis.

−6 −4 −2 2 4 6

−4

−3

−2

−1

1

2

3

4

x

y

(b) Mark the flag F3 on the axis which is a rotation of F1 about the point (1, -2) through °90 anticlockwise. (c) F2 is a rotation of F1. Find the angle and centre of this rotation.

PTO

F1

F2



Sheet F4-6 F5-02a-c-01 Rotations (cont.) 4. (a) Copy and complete the axis shown below, using 1cm per unit. Copy also and label the flag F1 that is drawn on the axis.

(b) Mark the flag F2 on the axis which is a rotation of F1 about the point (-1, 0) through °90 anticlockwise. (c) Mark the flag F3 on the axis which is a rotation of F1 about the point (1,1) through 180° . (d) Mark the flag F4 on the axis which is a rotation of F1 about the point (0,-1) through °90 clockwise. (e) Under a rotation the points (1, 1), (1, 2), (1, 4) and (2, 3) are mapped to (3, 3), (4, 3), (6, 3) and (5, 2) respectively. Draw the image of F1 under this rotation and label it F5. (f) Find the centre and the angle of rotation of F1 to F5. 5. (a) Draw a set of axes with x and y both going from -8 to 8 with 1cm per unit on both axes. (b) Mark the following points A(1, 2), B(5, 2), C(3, 3) and D(4, 6).

(c) Join up AB, BC, CD and DA and label this Q1. (b) Rotate Q1 by °90 in an anticlockwise direction about the point (-1, 1). Label this Q2. (c) Q1 is rotated onto Q3 which has vertices (4, -1), (8, -4), (5, -3) and (4, -5). Draw and label Q3. (d) Find the centre and the angle of rotation.

F1



Sheet F4-7 F5-02a-c-02 Centre of Rotations The diagram below shows two triangles T1 and T2. T2 is a rotation of T1. Copy and complete the diagram, including the triangles, using a scale of 2cm per unit on both axes. 1. (a) Construct (with a compass and a ruler) the perpendicular bisectors of A1 A2 and B1B2.

(b) Hence find the centre of the rotation that maps T1 to T2. (c) Verify that the perpendicular bisector of C1C2 passes through this point.

The diagram below shows two triangles T1 and T2. T2 is a rotation of T1. Copy and complete the diagram, including the triangles, using a scale of 2cm per unit on both axes. 2. (a) Use the diagram below to construct (with a compass and a ruler) the perpendicular

bisectors of A1 A2 and B1B2. (b) Hence find the centre of the rotation that maps T1 to T2. (c) Verify that the perpendicular bisector of C1C2 passes through this point.

PTO

A1

B1

C1 A2

B2

C2 T2

T1

A1

B1

C1

A2

B2

C2

T1

T2



Sheet F4-7 F5-02a-c-02 Centre of Rotations (cont.) The diagram below shows two shapes S1 and S2. S2 is a rotation of S1. Copy and complete the diagram, including the shapes, using a scale of 2cm per unit on both axes. 3. (a) Use the diagram below to construct (with a compass and a ruler) the perpendicular

bisectors of A1 A2 and B1B2. (b) Hence find the centre of the rotation that maps S1 to S2. (c) Verify that the perpendicular bisectors of C1C2 pass through this point.

The diagram below shows two shapes S1 and S2. S2 is a rotation of S1. Copy and complete the diagram, including the shapes, using a scale of 2cm per unit on both axes. 4. (a) Construct (with a compass and a ruler) the perpendicular bisectors of A1 A2 and B1B2.

(b) Hence find the centre of the rotation that maps S1 to S2. (c) Verify that the perpendicular bisectors of C1C2 pass through this point.

C1

A1

B1

D1 A2

B2 C2

D2

S2

S1

C1

A1

B1

D1

A2

C2 D2

B2

S1

S2



Sheet F4-8 F5-02a-c-03 Centre of Rotations The diagram below shows two triangles T1 and T2. T2 is a rotation of T1. Copy the diagram with the triangles using a scale of 1cm per unit on both axes. 1. (a) Construct (with a compass and a ruler) the perpendicular bisectors of A1 A2 and B1B2.

(b) Hence find the centre of the rotation that maps T1 to T2. (c) Verify that the perpendicular bisector of C1C2 passes through this point. (d) T3 is a rotation of T1 by 180° about the point (-1, 2). Draw and label T3, using the

letters A3, B3 and C3. (e) T4 is a rotation of T1 by 90− ° (i.e. 90° in a clockwise direction) about the point (0,

1). Draw and label T4, using the letters A4, B4 and C4.

PTO

A1

B1

C1

T1

A2

B2

C2

T2



Sheet F4-8 F5-02a-c-03 Centre of Rotations (cont.)

The diagram below shows two shapes S1 and S2. S2 is a rotation of S1. Copy the diagram with the shapes using a scale of 1cm per unit on both axes. 2. (a) Use the diagram below to construct (with a compass and a ruler) the perpendicular

bisectors of A1 A2 and C1C2. (b) Hence find the centre of the rotation that maps S1 to S2. (d) S3 is a rotation of S1 by 90− ° (i.e. 90° in a clockwise direction) about the point (1,

1). Draw and label S3, using the letters A3, B3, C3 and D3.

A2

B2

C2

D2

S2

C1

A1

B1

D1

S1



Sheet F4-9 F5-02a-c-04 Rotations and Reflections

1. Copy and complete the diagram shown below, using 1cm per unit. Copy also and label the triangles T1, T2 and T3 that are drawn on the diagram.

−2 2 4 6 8

2

4

6

x

y

(a) Find the equation of the line about which T1 is reflected to T2. (b) Mark and label the triangle T3 which is a rotation of T1 about the point (2, 2) through

180° . (c) T1 is rotated 180° to the triangle T4. Find the centre of rotation.

2. Copy and complete the diagram shown below, using 1cm per unit. Copy also and label

the shapes S1, S2 and S3 that are drawn on the diagram.

(a) Find the equation of the line about which S1 is reflected to S2. (b) Find the equation of the line about which S1 is reflected to S3. (c) Draw the shape S4 which is a reflection of S1 about the line 2y = .

T1

T4

T2

S1

S3

S2



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Sheet F4-10 F5-02d-f-01 Reflections 1. (a) Copy and complete the diagram shown below, using 1cm per unit. Copy also and

label the triangle T1 that is drawn on the diagram.

(b) Mark the triangle T2 on the diagram which is a reflection of T1 in the y-axis. (c) Mark the triangle T3 on the diagram which is a reflection of T1 in the line 1y = . (d) Draw on the line y x= − . (e) Mark the triangle T4 on the diagram which is a reflection of T3 in the line y x= − . (f) Mark the triangle T5 on the diagram which is a reflection of T4 in the line 2x = − .

Draw a set of axes with x going from -6 to 6 and y going from -4 to 4 with 1cm per unit on both axes. 2. (a) Draw the triangle T1 with vertices (0, 0), (-2, 0) and (-2, 2).

(b) Reflect T1 in the line 2x = and label this shape T2. (c) Reflect T1 in the line 1y = − and label this shape T3. (d) Reflect T1 in the line y x= and label this shape T4. (e) Reflect T3 in the line y x= and label this shape T5. (f) About what line is T4 reflected to the shape T5?

PTO

T1



Sheet F4-10 F5-02d-f-01 Reflections (cont.) 3. The diagram below shows four shapes.

(a) About what line is S1 reflected to S2? (b) About what line is S1 reflected to S3? (c) About what line is S1 reflected to S4?

S1S2

S4S3



Sheet F4-11 F5-02d-f-02 Reflections 1. (a) Copy and complete the axis shown below, using 1cm per unit. Copy also and label

the flag F1 that is drawn on the axis.

(g) Mark the flag F2 on the axis which is a reflection of F1 in the x-axis. (h) Mark the flag F3 on the axis which is a reflection of F1 in the y-axis. (i) Draw the line y x= − . (j) Mark the flag F4 on the axis which is a reflection of F1 in line y x= − .

Draw a set of axes with x going from -8 to 8 and y going from -10 to 8 with 1cm per unit on both axes. 2. (a) Draw the triangle T1 with vertices (3, 1), (6, 2) and (5, 5).

(b) Reflect the shape T1 in the line 1x = and label this shape T2. (c) Reflect the shape T1 in the line 2y = − and label this shape T3. (d) Reflect the shape T1 in the line y x= and label this shape T4. (e) Reflect the shape T1 in the line y x= − and label this shape T5.

Draw a set of axes with x going from -4 to 10 and y going from -6 to 8 with 1cm per unit on both axes. 3. (a) Draw the triangle T1 with vertices (1, 2), (6, 2) and (5, 0).

(b) Reflect the shape T1 in the line 1y x= − and label this shape T2. (c) Reflect the shape T1 in the line 2y x= and label this shape T3. (d) Draw the triangle T4 with vertices (8, -5), (8, 0) and (6, -1). (e) Describe in full how T1 is mapped to T4.

PTO

F1



Sheet F4-11 F5-02d-f-02 Reflections (cont.)

4. (a) Copy and complete the axis shown below, using 1cm per unit. Copy also and label the triangle T1 with vertices (2, 1), (9, 2) and (5, 5).

(b) Reflect the shape T1 in the line 1y = − and label this shape T2. (c) Reflect the shape T1 in the line 3y x= and label this shape T3. (d) Reflect the shape T2 in the line 1x = and label this shape T4. (e) Describe in full how T1 is mapped to T4.

T1



Sheet F4-12 F5-02d-f-03 Rotations and Reflections

1. (a) Draw a set of axes with x and y both going from -8 to 8 with 1cm per unit on both axes and label the triangle 1T which has vertices (2, 2), (5, 2) and (5, 4) (b) (i) Draw and label the triangle 2T which is the image of 1T under a rotation of

°90 anticlockwise about the point (7, 2) (ii) Draw and label the triangle 3T which is the image of 2T under a reflection in

the line 4−=y . (c) (i) Draw and label the triangle 4T which is the image of 1T under a rotation of

°90 clockwise about the point (-4, 2) (ii) Draw and label the triangle 5T which is the image of 4T under a reflection in

the y-axis. (iii) Draw and label the triangle 6T which is the image of 1T under a rotation of

°90 anticlockwise about the point (-2, 0).

(d) Show that the vector 3

1−⎛ ⎞⎜ ⎟⎝ ⎠

translates 3T onto 5T .

(e) Find the vector which translates 2T onto 6T .

2. (a) Draw a set of axes with x and y both going from -8 to 8 with 1cm per unit on both

axes and label the parallelogram 1P which has vertices (2, 2), (5, 2), (3, 4) and (6,4). (b) (i) Draw and label the parallelogram 2P which is the image of 1P under a rotation of °180 about the origin.

(ii) Draw and label the parallelogram 3P which is the image of 2P under a reflection in the line 1=x .

(c) Draw and label the parallelogram 4P which is the image of 1P under reflection in the y-axis. (d) Find the vector which translates 3P onto 4P . (e) To what point does (2, 2) get mapped under the transformation described in (b) (i) followed by the transformation described in (b) (ii) followed by the transformation described in (d)? (f) To what point does (2, 2) get mapped under the transformation described in (c)?

PTO



Sheet F4-12 F5-02d-f-03 Rotations and Reflections (cont.) 3. Copy and complete the diagram shown below, using 1cm per unit. Copy also and label the

triangle T1 that is drawn on the diagram.

−6 −4 −2 2 4 6

−5

−4

−3

−2

−1

1

2

3

4

x

y

(d) Draw and label the triangle T2 on the diagram which is a reflection of T1 in the line

2y x= . (e) Draw and label the triangle 3T which is the image of 1T under a rotation of

°90 anti-clockwise about the point (-1, -4). (f) Draw and label the triangle 3T which is the image of 2T under a rotation of

180° about the point (0, -1).

T1



Sheet F4-13 F5-02g-h-01 Translations and Enlargements

Draw a set of axes with x and y both going from -8 to 8 with 1cm per unit on both axes. 1. (a) Draw and label the following shapes:

(i) 1T which has vertices (-6, 6), (-5, 7) and (-2, 6) (ii) 2T which has vertices (2, -3), (3, -2) and (6, -3) (iii) 1R which has vertices (-6, 2), (-6, 4), (-3, 4) and (-3, 2) (iv) 2R which has vertices (1, 4), (4, 4), (4, 6) and (1, 6) (v) 1S which has vertices (6, 6), (7, 7), (8, 6) and (7, 5) (vi) 2S which has vertices (-2, 0), (-1, -1), (-2, -2) and (-3, -1)

(b) What is the vector of the following translations: (i) 1T onto 2T (ii) 2R onto 1R (iii) 1S onto 2S

(c) Draw and label the triangle 3T which is the image of 1T under the translation

described by the vector ⎟⎟⎠

⎞⎜⎜⎝

⎛−1310

.

(d) Draw and label the square 3S which is the image of 2S under the translation

described by the vector ⎟⎟⎠

⎞⎜⎜⎝

⎛−−

54

.

(e) What is the vector of the translation of 1S onto 3S ? (f) What is the connection between the vectors in (b)(iii), (d) and (e)?

Draw a set of axes with x and y both going from -8 to 8 with 1cm per unit on both axes. 2. (a) Draw and label the triangle 1T which has vertices (1, 1), (1, 2) and (4, 1)

(b) Enlarge 1T by scale factor 2 with the following centre of enlargements to get the following shapes:

(i) Centre of enlargement (0, 0) to give 2T (ii) Centre of enlargement (7, 1) to give 3T

(c) What is vector of the translation of 2T to 3T ? (d) Draw and label the triangle 4T which is the image of 2T under the translation given by

the vector ⎟⎟⎠

⎞⎜⎜⎝

⎛−−

77

.

(e) Find the centre of the enlargement which maps 1T to 4T .



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Sheet F4-14 F5-02j-l-01 Centre of Enlargements

For Q1-3 draw a set of axes with x and y both going from 0 to 14 with 1cm per unit on both axes. 1. (a) Mark the following points A(4, 2), B(7, 2), C(6, 3) and D(7, 5).

(b) Join up AB, BC, CD and AD. (c) Mark the point P (0, 1). (d) Enlarge the shape ABCD by a scale factor 2 with P as the centre of enlargement and

label it A B C D′ ′ ′ ′ . 2. (a) Draw the triangle EFG with vertices E(1, 0), F(5, 0) and G(9, 2)

(b) Mark the point Q (1, 8). (c) Enlarge the shape EFG by a scale factor ½ with Q as the centre of enlargement and

label it E F G′ ′ ′ . 3. (a) Draw the square JKLM where J(6, 9), K (9, 9), L(9, 12), M(6, 12)

(b) Mark the point R (6, 0) (c) Enlarge the shape JKLM by a scale factor 3

2 with R as the centre of enlargement and label it J K L M′ ′ ′ ′ .

For Q4-5 draw a set of axes with x and y both going from 0 to 10 with 1cm per unit on both axes. 4. (a) Draw the rectangle ABCD with vertices A(1, 4), B(5, 4), C(5, 6) and D(1, 6).

(b) Draw the rectangle EFGH with vertices E(1, 6), F(7, 6), G(7, 9) and H(1, 9). (c) Find the centre of the enlargement which maps ABCD to EFGH. (d) Find the scale factor of this enlargement. (e) Find the scale factor of the enlargement which maps EFGH to ABCD.

5. (a) Draw the triangle PQR with vertices P(3, 2), Q(5, 2) and R(5, 3).

(b) Draw the triangle XYZ with vertices X(6, 1), Y(10, 1) and Z(10, 3). (c) Find the centre of the enlargement which maps PQR to XYZ. (d) Find the scale factor of this enlargement.

PTO



Sheet F4-14 F5-02j-l-01 Centre of Enlargements (cont.)

For Q6-8 draw a set of axes with x and y both going from -8 to 8 with 1cm per unit on both axes. 6. (a) Draw the triangle ABC with vertices A(4, 6), B(8, 6) and C(6, 8).

(b) Mark the point P (2, 2). (c) Enlarge the shape ABC by a scale factor ½ with P as the centre of enlargement,

labelling it CBA ′′′ . (d) Draw the triangle E(5, 5), F(7, 5), G(6, 6). (e) Find the scale factor of the enlargement which maps ABC to EFG. (f) Find the centre of this enlargement.

7. (a) Draw the triangle LMN with vertices L(-6, 2), M(-3, 2) and N(-3, 8).

(b) Mark the point Q (3, -4). (c) Enlarge the shape LMN by a scale factor 3

1 with Q as the centre of enlargement, labelling it NML ′′′ .

(d) Draw the triangle R(-3, 3), S(-1, 3), T(-1, 7). (e) Find the scale factor of this enlargement which maps LMN to RST. (f) Find the centre of the enlargement.

8. (a) Draw the parallelogram HIJK with vertices H(-7, -4), I(-5, -2), J(-1, -4) and K(-3, -6).

(b) Mark the point X (7, -6). (c) Enlarge the parallelogram HIJK by a scale factor of ½ with centre X and write down

the co-ordinates of the new shape, which you should label KJIH ′′′′ .



Sheet F4-15 F5-02j-l-02 Centre of Enlargements

For Q1 and 2, draw a set of axes with x and y both going from -3 to 10 with 1cm per unit on both axes. 1. (a) Draw the square ABCD with vertices A(1, 2), B(3, 2), C(3, 4) and D(1, 4).

(b) Mark the point P (2, 0). (c) Enlarge the shape ABCD by a scale factor 2 with P as the centre of enlargement. Label this A B C D′ ′ ′ ′ .

2. (a) Draw the triangle EFG with vertices E(5, 3), F(7, 3) and G(9, 5).

(d) Mark the point Q (9, 6). (e) Enlarge the shape EFG by a scale factor 3 with Q as the centre of enlargement. Label

this E F G′ ′ ′ . For Q3-5, draw a set of axes with x and y both going from -1 to 12 with 1cm per unit on both axes. 3. (a) Draw the rectangle ABCD with vertices A(2, 5), B(5, 5), C(5, 7) and D(2, 7).

(b) Mark the point P (1, 2). (c) Enlarge the shape ABCD by a scale factor 2 with P as the centre of enlargement.

Label this A B C D′ ′ ′ ′ . 4. (a) Draw the triangle EFG with vertices E(3, 2), F(4, 2) and G(4, 3).

(b) Mark the point Q (2, 3). (c) Enlarge the shape EFG by a scale factor 3 with Q as the centre of enlargement. Label

this E F G′ ′ ′ . 5. (a) Draw the triangle HIJ with vertices H(5, 4), I(8, 4) and J(8, 7). (b) HIJ is an enlargement of EFG. Find the centre and the scale factor of enlargement. 6. (a) Find the scale factor of enlargement of the small triangle to the large triangle shown below. (b) Calculate the centre of enlargement.



Sheet F4-16 F5-02j-l-03 Centre of Enlargements 1. (a) Draw a set of axes with x going from -8 to 8 and y going from -5 to 8 with 1cm per unit on both axes. (b) Mark the following points A(1, -1), B(5, -1), C(4, 0) and D(5, 2).

(c) Join up AB, BC, CD and AD. (d) Enlarge the shape ABCD by a scale factor 2 with (3, -4) as the centre of

enlargement. Label this A B C D′ ′ ′ ′ . (e) Enlarge the shape by a scale factor -2 with (1, 1) as the centre of enlargement. Label

this A B C D′′ ′′ ′′ ′′ . 2. (a) Copy and complete the axis shown below, using 1cm per unit. Copy also and label the triangle T1 that is drawn on the axis.

(b) Enlarge T1 by scale factor -1 about the point (-1, 5) and label this T2. (c) Enlarge T1 by scale factor -2 about the point (4, 4) and label this T3. (d) Enlarge T1 by scale factor -1 about the point (0, 2) and label this T4. (e) Draw and label the triangle T5 with vertices (0, -1), (8, -1) and (8, -5). (f) Find the centre and scale factor of the enlargement which maps T1 to T5.

T1



Sheet F4-17 F5-02j-l-04 Transformations

Draw a set of axes with x and y both going from -8 to 8 with 1cm per unit on both axes. 1. (a) Draw and label the following shapes: (i) 1T which has vertices (-3, 2), (0, 2) and (2, 4)

(vii) 2T is a translation of 1T by the vector ⎟⎟⎠

⎞⎜⎜⎝

⎛−56

(viii) 1R which has vertices (-7, -5), (-5, -5), (-5, -8) and (-7, -8) (ix) 2R which has vertices (4, 7), (6, 7), (6, 4) and (4, 4)

(b) Find the vector which translates 1R onto 2R . (c) Find the vector which translates 2R onto 1R . (d) Enlarge the shape 1T by scale factor 2 (with centre of enlargement (1, 1). Label this

3T . (e) Enlarge the shape 2T by scale factor 2 (with centre of enlargement (8, 1). Label this

4T . (f) Find the vector which translates 3T onto 4T .

Draw a set of axes with x and y both going from -8 to 8 with 1cm per unit on both axes. 2. (a) Draw and label the following shapes:

(i) 1T which has vertices (1, 3), (3, 1) and (5, 7) (ii) 2T which is an enlargement of 1T with scale factor ½ with centre (-5, 5)

(iii) 3T which is a translation of 2T with vector ⎟⎟⎠

⎞⎜⎜⎝

⎛− 75

(iv) 4T which is a reflection of 3T in the line 2=x (v) 5T which an enlargement of 4T with scale factor 2 with centre (2, -3) (vi) 6T which is a reflection of 1T in the line 1−=x

(b) Find the vector which translates 5T onto 6T . Draw a set of axes with x and y both going from -8 to 10 with 1cm per unit on both axes. 3. (a) Draw the triangle T with vertices A(2, 4), B(6, 4) and C(1, 1).

(b) Draw the triangle 1T with vertices 111 and , CBA which is a reflection of T in the line xy −= .

(c) Draw 2T with vertices 222 and , CBA which is an enlargement of T with scale factor -2 and centre of enlargement (3, 1).

(d) Draw 3T with vertices A3(-3, 5), B3 (-3, 1) and C3 (-6, 6). (e) Find the centre and angle of the rotation which maps T to 3T . (f) Draw 4T with vertices A4(2, 6), B4 (6, 6) and C4 (1, 9). (g) Find the equation of the line through which T is reflected onto 4T .



Sheet F4-18 F5-02j-l-05 Enlargements

1. A map has a scale of 1:50000. How long (in mm) is the field on the map whose actual length is 120m?

2. A map has a scale of 1:100000. How long (in km) is a road which measures 5cm on the

map? 3. A map has a scale of 1:20000. What is perimeter (in m) of a field whose perimeter is 25mm

on the map? 4. (a) The shape below (not to scale) has a total perimeter of 12cm. If it is enlarged by a

factor of 300 then what will its perimeter be?

(b) If one of the angles in the star is °60 then what will it be in the enlarged shape? 5. A triangle has coordinates A(3, 2), B(6, 2) and C(3, 6).

(a) Calculate the three side lengths of the triangle. The triangle is enlarged by scale factor 3.

(b) What is the perimeter of the new triangle? 6. A field has length 2cm on a map and 500m is reality. What is the scale of the map? 7. A road has length 4cm on a map and is 20km in reality. What is the scale of the map? 8. A triangle has coordinates A(1, 5), B(6, 5) and C(6, 17).

(a) Calculate the three side lengths of the triangle. The triangle is enlarged to points A′ (7, 8), B′ (17, 8), C ′ (17, 32).

(b) What is the perimeter of the new triangle? (c) What is the scale factor of enlargement? (d) Write down the angles ABC and A B C′ ′ ′ .



Sheet F4-19 F6-02-1 Averages 1. Find the mean, median, mode and range of the following sets of numbers by first writing

down these numbers in ascending order: (a) 3 5 8 1 2 3 13 (b) 13 19 22 25 29 11 20 22 34 28 (c) 1.03 1.07 1.21 1.05 1.29 0.95 (d) 1 7 3 9 10 13 13 (e) 42 41 45 46 48 51 51 52 (f) 119 113 108 109 112 114 103 119 120 (g) 89 81 82 85 87 88 90 91 92 95 88

2. (a) In 1(c) what would the mean and median be if the biggest value, i.e. 1.29 is replaced

with 4.29? (b) What can you say about the effect on the mean and on the median when the

largest number in a set of data is replaced with a very large number?

3.

x 1 2 3 4 5 6 f 3 8 13 10 6 5

(a) Find the mean of the above frequency distribution (to 3sf). (b) Find the median of the above frequency distribution. (c) Write down the mode and the range of the above frequency distribution.

4. The following table shows the number of days on which pupils were late to school in a week.

Number of days 0 1 2 3 4 5 Frequency 75 31 28 21 10 2

(a) What is the mean number of days on which a pupil was late in that week (to 3sf)? (b) What is the mode? (c) What is the median number of days late?

5. The following shows how many minutes (to the nearest minute) it took pupils to complete a test:

Minutes 35 36 37 38 39 40Frequency 11 12 17 11 9 1

Calculate the mean and median for the above data (to 3sf).

PTO



Sheet F4-19 F6-02-1 Averages (cont.) 6. The following shows how many grams (to the nearest 5g) a group of small animals weighed

Weight (g) 100 105 110 115 120 125 Frequency 8 11 14 9 6 2

(a) Calculate the mean and median for the above data (to 3sf). (b) Calculate the range of weights in this group. (c) Calculate the modal weight. (d) What can you say about the mean, median and mode?

7. x 3 4 5 6 7 f 5 8 3 2 8

Calculate the mean and the median of the above distribution without a calculator. 8.

(i) x 11 13 15 17 19 21 f 15 12 14 9 3 2

(ii) x 100 110 120 130 140 150 f 6 6 5 4 2 1

(iii) x 2.1 2.2 2.3 2.4 2.5 2.6 f 35 27 58 67 38 24

(a) Find the range and the median for the above sets of data. (b) Find the mean for the above sets of data.



Sheet F4-20 F6-02-2 Grouped Mean 1. The table below shows the weights of 150 of a certain insect:

Weight, x (g) 120- 130- 140- 150- 160- 170-180 Frequency 25 19 23 16 12 5

(a) Copy and complete the following table with midpoints against frequency.

Weight, x (g) 125 175 Frequency 25 19 23 16 12 5

(b) Use this table to calculate an estimate of the mean (to 1dp). (c) Find the class interval in which the median lies.

2. The table below shows the weights of 175 children:

Weight, x (g) 40- 45- 50- 55- 60- 65-70 Frequency 43 31 39 28 19 15

(a) Find the midpoint of the “40-” class interval. (b) Draw a table with midpoints against frequency. (c) Use this table to find an estimate of the mean (to 3sf). (d) Find the class interval in which the median lies.

3. The table below shows the time it took for a group of 100 commuters to get to work:

Time taken, x (min) 55-59 60-64 65-69 70-74 75-79 80-84 Frequency 21 17 22 16 13 11


Time taken, x (min) 62 77 Frequency 21 17 22 16 13 11

(a) Draw a table with midpoints against frequency. (b) Use this table to find an estimate of the mean (to 3sf).

4. The table below shows the ages of 200 people applying for tickets for a sports match.

Age, x (year) 11-15 16-20 21-25 26-30 31-35 36-40 Frequency 46 34 51 29 22 18

(a) Find the midpoint of the “11-15” class interval. (b) Draw a table with midpoints against frequency. (c) Use this table to find an estimate of the mean (to 3sf). (d) Find the class interval in which the median lies.

PTO



Sheet F4-22 F6-02-2 Grouped Mean (cont.)

5. Find the mean of the following data: (a)

x (g) 30- 40- 50- 60- 70- 80- 90-100 f 25 19 29 18 13 10 6

(b)

x (cm) 40-44 45-49 50-54 55-59 60-64 65-69 f 53 64 72 81 51 42



Sheet F4-21 F6-02-3 Grouped Mean 1. At a girls’ school, a random sample of 130 pupils was taken and each pupil recorded her

intake of milk (in ml) during a given day. The results are shown below:

Milk intake 10- 30- 60- 100- 150- 200- 300-500 No. of students 3 7 25 55 23 15 2


Milk intake (Midpoint) 20 175 No. of students 3 7 25 55 23 15 2

(b) Use this table to find an estimate of the mean (to 3sf). (c) Find the class interval in which the median lies. (d) Write down the modal class.

2. Summarised below are the prices of the goods (to the nearest £) sold by an electrical shop on a certain day.

Price of good (£)

Frequency

less than 20 2 20- 8 30- 19 40- 37 50- 62 60- 51 70- 29 90- 9

130-150 2

(e) Draw a table with midpoints against frequency. (f) Use this table to find an estimate of the mean (to 3sf).

3. The table below shows the results of a multiple choice exam:

Number of Correct Answers (x) 11-15 16-20 21-25 26-30 31-35 36-40 Frequency 2 11 31 37 24 12

(a) Find the midpoint of the “11-15” class interval. (b) Draw a table with midpoints against frequency. (c) Use this table to find an estimate of the mean (to 3sf). (d) Explain why this is only an estimate of the mean. (e) To which value does the median correspond? (f) Find the class interval in which the median lies. (g) Write down the modal class.

PTO



Sheet F4-21 F6-02-3 Grouped Mean (cont.) 4. The table below shows how many points were scored by a group of rugby players in a

season:

Number of Points (x) 5-9 10-14 15-19 20-24 25-29 30-34 Frequency 18 12 7 5 2 3

(a) Find the midpoint of the “5-9” class interval. (b) Draw a table with midpoints against frequency. (c) Use this table to find an estimate of the mean (to 3sf). (d) Find the class interval in which the median lies. (e) Write down the modal class.

5. The table below shows the heights of a group of school children:

Height, h (cm) 140- 145- 150- 155- 160- 165-170 Frequency 40 45 94 97 85 57

(a) Find the midpoint of the “140-” class interval. (b) Draw a table with midpoints against frequency. (c) Use this table to find an estimate of the mean (to 3sf). (d) State the modal class.

6. The table below shows the ages of 300 young people injured in car accidents in a certain

month:

Age, x (year) 1-5 6-10 11-15 16-20 21-25 26-30 Frequency 87 71 51 43 26 22

(a) Explain why the class interval for “1-5” is can be written as 61 <≤ x . (b) Find the midpoint of the “1-5” class interval. (c) Find an estimate of the mean. (d) Find the class interval in which the median lies.



Sheet F4-22 F6-03-1 Probability-Introduction 1. A fair die is tossed. Find, as fractions, the probability of getting :

(a) A six (b) An even number (c) A number greater than 4 (d) A prime number (1 is not prime) (e) 4 or an odd number.

2. A bag contains 7 red balls, 3 yellow balls and 2 blue balls. A ball is selected at random.

Find, as fractions, the probability of the ball being: (a) a red ball (b) a yellow ball (c) a yellow or a blue ball (d) a red or a blue ball.

3. A die is biased so that the numbers 1 to 5 appear with the same probability but the number 6 appears less often. The probability of getting a 6 is only 0.1.

(a) Find the probability of getting a number other than 6. (b) Show that the probability of getting the number 5 is 0.18. (c) Find the probability of getting a number greater than 3.

4. A normal pack of 52 cards contains 13 cards of each suit (Spades,

Clubs, Diamonds and Hearts). In each suit there is an ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen and King. I choose a card at random from the pack. What is the probability that: (a) The card is a Heart. (b) The card is a King. (c) The card is the Ace of Spades (d) The number on the card is 7 or 8.

5. The first day of 2003 was a Wednesday and, as a non leap- year, the year 2003 had 365 days. A pupil was chosen at random from a class. What is the probability that, in 2003, her birthday fell on a Thursday? 6. In a rugby team of 15 players, five players were 17 years old, seven players were 18 and the rest were 16. A player is chosen at random. Find the probability that he was:

(a) 16 (b) 16 or 17

7. The lottery basket contains 49 balls numbered from 1 to 49. What is the probability of getting:

(a) A single digit number. (b) An even number. (c) A multiple of 7. (d) A multiple of 5 or a multiple of 17.

PTO



Sheet F4-22 F6-03-1 Probability-Introduction (cont.)

8. A circular spinner has an arrow in the middle. The circle is split into four sections as shown below. Find the probability that when the arrow is spun, the number of the section on which it lands is:

(a) 4. (b) 3. (c) at least 2. (d) an even number.

9. A spinner similar to that in question 8 has 4 sections, numbered 1, 2, 3 and 4. The area for 2

is twice the area for 1, the area for 3 is three times the area for 1 and the area for 4 is four times the area for 1. If p is the probability of getting a 1 then: (a) write down, in terms of p, the probabilities of getting 2, 3 and 4. (b) Find p. (c) Find the probability of getting an even number.

10. The probabilities associated with the scores on a biased die are shown below:

Score 1 2 3 4 5 6 Probability 0.1 0.3 0.05 0.25 0.15

(a) Find the probability of getting a 4 when the die is rolled. (b) Find the probability of getting an odd number when the die is rolled. (c) Find the probability of getting at least 4 when the die is rolled.

1

2

3

4

120

120



Sheet F4-23 F6-03-2 Probability-Estimation

1. (a) A coin is tossed twice and there are no heads. Would you say

that the coin was biased? (b) A coin is tossed two hundred times and there are no heads. Would you say that the coin was biased?

2. A fair coin is tossed one hundred times. The number of heads is recorded.

(a) How many heads would you expect? (b) Which of the following values would be very unlikely? (i) 54 (ii) 85

(iii) 48 3. A fair die is rolled six hundred times. How many times would you expect to get:

(a) an even number? (b) the number six (c) a number greater than 3?

4. About 10% of the population is left handed. How many left handed pupils would you

expect to find in a school of 1300 pupils? 5. A firm employs 1400 employees. How many would you expect to have been born on a

Sunday? 6. A boy picks out a marble from a bag of 20 coloured marbles. He records its colour and then

puts it back. He does this fifty times. Ten of the marbles he takes out are red. How many of the 20 marbles in the bag would you expect to be red?

7. The spinner shown is spun 400 times. How many times would you

expect the spinner to land on: a. The sector numbered 1? b. The sector numbered with an even number? c. A sector with a number of more than 3?

8. A pack of 52 playing cards is dealt to four players so that each player gets 13 cards. This is done forty times. How many cards in total from the spades suit would one player expect to have been given from these forty deals?

9. A dart is dropped onto a board by a machine four thousand times, the machine

moves in such a way that it is equally likely to drop the dart onto any point of the board. The radius of the board is 5cm. The radius of the darker coloured centre is 1cm. How many times would you expect the dart to land in the darker coloured centre?

1 2

3 4



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Sheet F4-24 F6-03-3 Probability-Mutually Exclusive 1. Sarah gets the train to work. The probability that she goes from

Windsor is 0.3 and the probability that she goes from Slough is 0.65. What is the probability that she does not go from either Slough or Windsor?

2. A girl walks to school using one of three routes. She walks along the High

Street with probability 0.3, she walks across the fields with probability 0.15 and the rest of the time she walks along the tow path. Find the probability that the girl walks to school: (a) Along the tow path (b) Either across the fields or along the tow path. (c) Either along the High Street or across the fields.

3. A postman delivers letters to the Allen’s house every day. The probability that he delivers

them at certain times is shown below:

Time Probability Between 7.00am and 7.30am 0.15 Between 7.30am and 8.00am 0.3 Between 8.00am and 8.30am 0.35

(a) Find the probability that he delivers before 7.00am or after 8.30am. It is also known that he is as likely to deliver the letters before 7.00am as he is to deliver them after 8.30am. (b) Find the probability that he delivers before 7.00am . (c) Find the probability that he delivers a card before the Allen children went to school at

8.30am. (d) Find the probability that he delivered after 7.30am.

4. A milkman always delivers the milk between 6.00am and 9.00am each morning. He is as

likely to deliver the milk at any time in those three hours. Find the probability that he delivers the milk: (a) Before 7.00am. (b) Between 6.15am and 8.45am.

5. Mr Taylor drives to work each morning. The probability that he parks his car at the front of the building is 0.4. The probability that he parks his car at the side of the building is 0.15. The rest of the time he parks it at the back of the building. (a) What is the probability that he will park at the back of the building on any particular

morning? (b) What is the probability that he will park either at the back or at the side of the building

on any particular morning? (c) In the next year, he works for 220 days. On approximately how many days will Mr

Taylor park either at the back or at the side of the building? PTO



Sheet F4-24 F6-03-3 Probability-Mutually Exclusive (cont.)

6. Tom watches only one programme out of Yellow County, Who Wants to be a Mathematician? and 25 on a certain evening each week. He watches Who Wants to be a Mathematician? twice as often as he watches Yellow County and he is three times as likely to watch Yellow County as he is to watch 25.

If p is the probability that Tom watches 25 then find: (a) The probability (in terms of p) that he watches Yellow County (b) The probability (in terms of p) that he watches Who Wants to be a Mathematician? (c) Use these to show that the probability that he watches 25 is 0.1. (d) Hence find the probability that he watches either 25 or Who Wants to be a

Mathematician? (e) Over a ten week period how many times do you expect him to watch Yellow County?

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