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Mathematics Extended Practice Book Example Practice Papers

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Example Practice Papers for Cambridge IGCSE® Mathematics Extended Practice Book

Example Practice Paper 2

Mark scheme for Paper 2

Example Practice Paper 4

Mark scheme for Paper 4

Cambridge IGCSE Mathematics Extended Practice Book Example Practice Paper 2

PLEASE NOTE: this example practice paper contains exam-style questions only

READ THESE INSTRUCTIONS FIRST Answer all questions

Working for a question should be written below the question

If the answer is not exact but a degree of accuracy has not been provided,

- for π,

use either your calculator value or 3

The number of marks is given in brackets [ ] next to each question or part question

The total of the marks for this paper is 67

PLEASE NOTE: this practice examination paper has been written in association with the below publication and is not an official exam paper:

Paperback 9781107672727

(a) For the diagram above write down (i) the order of rotational symmetry,

Answer(a)(i) ………………………………

(ii) the number of lines of symmetry

Answer(a)(ii) ……………………………… [1] (b) The prism below has 6 square faces and a regular hexagonal cross-section

Write down the number of planes of symmetry for the prism

Answer(b) ………………………………

Calculate the value of

(a) writing down all the figures in your calculator answer,

Answer(a) ………………………………

(b) writing you answer correct to 3 decimal places

Answer(b) ……………………………… 2

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Find the midpoint of the line joining the points A(4,

Expand the brackets and simplify

Answer ……………………………………

Zagreb changed $600 into euros at an exchange rate of $1 = €1

He later changed all of the euros back into dollars at an exchange rate of $1 = €1

How many dollars did he receive

Answer $ ………………………………

Solve the simultaneous equations

Answer x = ……………………………… Answer y = ………………………………

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The dimensions of a rectangle are 13 cm by 7 cm,

Find the smallest possible area of the rectangle

Answer ……………………………… cm2

The intensity of radiation from the Sun,

is inversely proportional to the square of the distance from the Sun,

When d'= 2 ×108,

Find d'when R = 500

Answer d'= ………………………………

Shade the region required in each Venn diagram

(A∩B)∪C

A ∩ B' [2]

Calculate the final amount she had after 5 years

Answer $ ……………………………… 4

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Work out (a) A2

Answer(a)

⎛ ⎜ ⎜ ⎜ ⎝

⎞ ⎟ ⎟ ⎟ ⎠

Answer(b)

⎛ ⎜ ⎜ ⎜ ⎝

⎞ ⎟ ⎟ ⎟ ⎠

A conference table is made of two quarter circles and two identical triangles

The radius of the quarter circles is 0

Calculate the surface area of the top of the table

Answer ………………………………… m2 [3]

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5− x 6

Answer x = ………………………………

The length of the longer diagonal of the parallelogram is 11 cm

AB is a side of the parallelogram

Using a straight edge and a compasses only,

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20 15 10 5

The graph above shows the journey of two cars,

A and B

(a) Work out the acceleration of car A during the first 10 seconds

Answer(a) …………………………… m/s2

(b) Calculate how far car B travels before coming to rest

Answer(b) ……………………………… m

(c) State which car experiences the highest deceleration

Answer(c) ………………………………

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Answer(a) ………………………………

Answer(b) ………………………………

÷ 2−2 x 5

write the following as a single fraction

x 2 + 6 x − 16 −1 x 2 + 5 x − 14

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2 x −3

(b) Write the letter R in the region defined by the three inequalities below

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Show all your working and give your answers correct to 2 decimal places

20 f(x) =

Answer x = ………… or x = ……………

Answer(a) ………………………………

Answer(b) ………………………………

Answer(c) ………………………………

(b) Find fg(x) in its simplest form

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D 42° 62° O B

C and D'lie on the circle,

The line PCQ is a tangent to the circle at C

Angle AOD = 62°,

angle BAC = 42° and angle DCQ = 78°

Find (a) angle ODA

Answer(a) Angle ODA = …………………

Answer(b) Angle ACD = …………………

Answer(c) Angle BAD = …………………

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Cambridge IGCSE Mathematics Extended Practice Book Example Practice Paper 2 (Extended) Mark Scheme Key:

A – Accuracy marks awarded for a correct answer seen

M – Method marks awarded for clear attempt to apply correct method

“ ” – allow M marks for methods that include wrong answers from previous results

A1 A1 A1

M1 A1 A1

600 × 1

25 ÷ 1

20 $625

Clear attempt at elimination or substitution method x = −7,

M1 A1 A1

M1 M1 A1

1 (x 2 1

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300 × 1

075 420

⎛2 ⎜ ⎝ −1 ⎛2 ⎜ ⎝ −1

M1 ⎛ 0

π × 0

Two arcs at 6 cm from A and B Arc at 11 cm from A or B Arc at 8 cm from intersection of 11 cm and 6 cm arc,

M1 M1 A1

26 m3 13

⎛ 5 −20 ⎞ ⎜ ⎟ Any 2 correct ⎝1 0 ⎠ ⎛ 5 −20 ⎞ ⎜ ⎟ All 4 correct ⎝1 0 ⎠

M1 M1 A1

⎛ x10 ⎞ 5 ⎛ x 2 ⎞ ⎜ ⎟ =⎜ ⎟ ⎝ 32 ⎠ ⎝ 2 ⎠

M1 A1 A1

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( x − 2)( x + 8) −1 ( x − 2)( x + 7)

x +8 x +7 − x+7 x+7 x +8− x −7 x+7 1 x+7

M1 M1 A1 14

A1 A1 A1 A1

2 x −3

R labelled in triangle formed by lines

M1 A1 A1 A1 M1

rearrange to y = … x= y f −1 ( x) =

5 1− x

A1 M1 A1 3

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A1 A1 A1 Total: 67

Cambridge IGCSE Mathematics Extended Practice Book Example Practice Paper 4

PLEASE NOTE: this example practice paper contains exam-style questions only

READ THESE INSTRUCTIONS FIRST Answer all questions

Working for a question should be written below the question

If the answer is not exact but a degree of accuracy has not been provided,

- for π,

use either your calculator value or 3

The number of marks is given in brackets [ ] next to each question or part question

The total of the marks for this paper is 130

PLEASE NOTE: this practice examination paper has been written in association with the below publication and is not an official exam paper:

Paperback 9781107672727

One way to measure the height of a flag pole from the ground is to stand in two different positions and measure the angle of inclination of the top of the pole,

as well as the difference between the two positions

This is shown below

Diagram 1

NOT TO SCALE

20° 5m

In Diagram 1,

angle DAC = 20° and angle DBC = 30°

The length AB = 5 m

Answer(a)(i) ………………………………

Answer(a)(ii) ……………………………… [1] (iii) the length BD,

Answer(a)(iii) …………………………… m [3] (iv) the height of the flag pole,

Answer(a)(iv) …………………………… m [3]

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Another way to measure the height of the flag pole is to use two short poles of a known height and line them up so that their tops aim towards the flag

This is shown in Diagram 2

Diagram 2

NOT TO SCALE

In Diagram 2,

BC = 2 m,

DE = 3 m,

BD = 4 m and DF = 120 m

(b) (i) By considering the similar triangles ABC and ADE,

Answer(b)(i) …………………………… m [3] (ii) By considering the similar triangles ABC and AFG,

find the height of the flagpole,

Answer(b)(ii) …………………………… m [2]

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−8 −7 −6 −5 − −4 −3 − −2 −11 − −1

Describe fullyy the single transformatio t on which maaps (a) D'(ii) triangle A onto C,

Answeer(a)(i) …… ……………… ……………… ……………… ……………… ……………… ……………

Answeer(a)(ii) …… ……………… ……………… ……………………………………… ……………… …

Answeer(a)(iii) …… ………………………… …………… ……………… ……………… ……………… …

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Answer(a)(iv) …………………………………………………………………………………

(b) Find the matrix representing the transformation which maps (i) triangle A onto C,

Answer(b)(i)

⎛ ⎜ ⎜ ⎜ ⎝

⎞ ⎟ ⎟ ⎟ ⎠

Answer(b)(ii)

⎛ ⎜ ⎜ ⎜ ⎝

⎞ ⎟ ⎟ ⎟ ⎠

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Diagram 1 26 cm

NOT TO SCALE B

The diagram shows a rectangle with a width of x cm and a height of y cm

(a) (i) If the perimeter of the rectangle is 68 cm,

Answer(a)(i)

that x satisfies the equation x2 – 34x + 240 = 0

Answer(a)(ii)

Answer(a)(iii) ……………………………… [2] (iv) Solve the equation x2 – 34x + 240 = 0

Answer(a)(iv) x = ……… or x = ………

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Diagram 2 1 cm NOT TO SCALE B

Diagram 2 shows a different rectangle

The line EF cuts ABCD into two rectangles

(b) (i) Rectangle ABCD is similar to rectangle DEFC

Show that x2 + x – 1 = 0

Answer(b)(i)

giving your answers correct to 3 decimal places

Answer(b)(ii) x = ……… or x = ………

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(a) The table shows some values for the equation y =

x4 + x3

(i) Write the missing values of y in the empty spaces

5 x −6

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(b) Use your graph to solve the equation

work out the gradient of the graph where x = −1

……………………………

Answer(c)(ii) ……………………………

Answer(c)(i) (ii) Write down the gradient of the graph where x = 0

draw the line y = −2x – 10

(ii) Use your graphs to solve the equation

Answer(d)(ii) x = ……… or x = ………

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Bag A contains 5 red beads and 5 green beads

Bag B contains 2 red beads and 3 green beads

A bead is taken at random from bag A,

then a bead is taken at random from bag B

(a) Complete the tree diagram below,

showing the probabilities of each outcome

Red Red

Red Green Green

(b) Calculate the probability that (i) two red beads are picked,

Answer(b)(i)

……………………………

Answer(b)(ii)

……………………………

(ii) exactly one red bead is picked

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(c) All the beads are returned to the bags

A bead is taken from bag A,

A bead is now taken from bag B

(i) Complete the tree diagram to show the new probabilities

Red Red

Red Green Green

Calculate the probability that (ii) two red beads are picked,

Answer(c)(ii)

……………………………

Answer(c)(iii)

……………………………

(iii) at least one green bead is picked

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Small mug

Large mug

A small cylindrical mug has a diameter of 8 cm,

(a) Calculate the height of the small mug

Answer(a)

(b) (i) Work out how many cm3 there are in 1 m3

Answer(b)(i)

……………………………

(ii) Work out how many small mugs would be filled by 1 m3 of water

Answer(b)(ii)

……………………………

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(c) The large mug holds 1000 cm3 of water

(i) Work out the scale factor for volumes between the small and large mug

……………………………

Answer(c)(i)

(ii) Work out the scale factor for lengths between the small and large mug

Answer(c)(ii)

……………………………

Answer(c)(iii)

(iii) Work out the height of the large mug

(d) Calculate the volume of the largest sphere which would fit inside the large mug

Answer(d)

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The heights of 120 trees in an orchard are measured

The results are used to draw this cumulative frequency diagram

Cumulative frequency

Height (cm)

(a) Find (i) the median height,

Answer(a)(i)

Answer(a)(ii)

Answer(a)(iii)

……………………………

(iii) the interquartile range,

(iv) the number of trees with a height greater than 316 cm

Answer(a)(iv)

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(b) The frequency table shows the information about the 120 trees that were measured

Height (h cm)

Frequency

(i) Use the cumulative frequency diagram to complete the table above

(ii) Construct a histogram to represent this information

Frequency density

Height (cm)

(c) Calculate an estimate of the mean height of the 120 trees

Answer(c)

……………………………

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Answer(a) x = ……………………………

Find the value of y when x = 2

Answer(b)(i) y = …………………………… [2]

Answer(b)(ii) ……………………………… [2]

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Answer(b)(iii) x = …………………………… [3] (c)

Find c'in terms of a,

Answer(c) c'= …………………………… [3]114 17 Written specifically for the publication ‘Cambridge IGCSE Mathematics Extended Practice Book’

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The diagram shows the triangle PQS

T is the midpoint of PS and R divides QS in the ratio 1 : 3

JJJG JJJG PT = a and PQ = b

(a) Express in terms of a and/or b,

JJJG PS Answer(a)(i)

JJJG PS = ………………………… [1]

JJJG (ii) QS JJJG Answer(a)(ii) QS = ………………………… [2] JJJG (iii) PR

JJJG Answer(a)(iii) PR = ………………………… [2] JJJG 1 (b) Show that RT = (2a − 3b ) 4 Answer(b)

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Diagram 1

Diagram 2

Diagram 3

Diagram 4

Diagram 5

The diagram shows a pattern of triangles of dots

Diagram number

Number of triangles

Number of dots

Answer (b) Number of triangles = ……………… ,

Number of dots = ………………… [2] (c) Write down an expression for the number of triangles in the nth diagram

Answer(c) ……………………………

(d) The number of dots in the nth diagram is k (n 2 + 3n + 2)

Find (i) the value of k,

Answer(d)(i) k = …………………………… [2] (ii) the number of dots in diagram 100

Answer(d)(ii) ………………………………… [1]

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Cambridge IGCSE Mathematics Extended Practice Book Example Practice Paper 4 (Extended) Mark Scheme Key:

A – Accuracy marks awarded for a correct answer seen

M – Method marks awarded for clear attempt to apply correct method

“ ” – allow M marks for methods that include wrong answers from previous results

Rotation,

Stretch in y-direction,

about x-axis as invariant line

⎛1 0⎞ ⎜ ⎟ ⎝

⎞ ⎜ ⎟ ⎝ 0 −1 ⎠ ⎛1 ⎜ ⎝

0⎞ ⎟

A1 A1 M1 M1 A1 M1 M1 A1 M1 M1 A1 M1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1

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x + (34 − x) = 26 (a)(iii) (a)(iv) (b)(i)

M1 A1 M1 2

M1 A1 A1 A1 M1 M1 A1

M1 A1 A1 A1 A1 A1

Shape A1

Points accurate A1 A1

Smooth curve A1 Domain correct A1

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A1 A1 M1 M1 A1 A1

P(R) = 2/5,

P(G) = 3/5 5 2 × 10 5 0

P(R) = 2/6,

P(G) = 4/6 5 3 × 10 6 0

25 oe 0

A1 A1 A1 A1 A1 M1 A1 M1 A1 A1 A1 A1 M1 A1 M1 A1

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(a) (b)(i) (b)(ii) (c)(i) (c)(ii) (c)(iii) (d)

(a)(i) (a)(ii) (a)(iii) (a)(iv) (b)(i)

26 “9

26” 12

26” = 5

M1 A1 M1 A1 A1 A1 M1 A1 M1 A1 M1 M1 A1

A1 A1 A1 M1 A1 A1 A1

Widths A1 0

Relative heights A1 A1

Correct FDs A1

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M1 M1 A1

Finding LCM of 6 and 9 (18) 3( x − 5) + 2( x + 2) = −4 18 3 x − 15 + 2 x + 4 = −72 61 x=− oe 5 5 4 − −1 6 −5 23 oe

M1 A1 M1 A1 M1

x + 32 1 = ( x − 3)( x + 4) x

x 2 + 32 x = x 2 + x − 12 leading to 31x = −12 12 x=− 31 a (c + d') = b

M1 A1 M1 M1

A1 M1 A1 M1 A1

PR + RT = a,

a − (¾b + ½a) = ½a − ¾b

− 34 b + 12 a leading to result given

Triangles 16,

Dots 45 n2 k (12 + 3 × 1 + 2) = 3 k = 0

A1 A1 A1 A1 A1 M1 A1 A1 Total: 130 5

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Cambridge IGCSE Study Guide for Physics

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