PDF- solution of extended mathematics for -Extended Mathematics Cambridge Igcse Past Papers Ebook - Cambridge Igcse Mathematics Extended Practice Book Example Practice Papers

Description

Mark scheme for Paper 2

Mark scheme for Paper 4

- 1 hour 30 minutes

- give the answer as follows:
- - to three significant figures for all values,
- - to one decimal place for degrees

- for π,

use either your calculator value or 3

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,

(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

- 2 3 + 74 (2 − 3)2

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

(b) writing you answer correct to 3 decimal places

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- 3) and B(−3,
- ( …………… ,
- ………… )

- 1 (9 x − 3) − 5( x − 3) 3

Answer ……………………………………

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

How many dollars did he receive

Solve the simultaneous equations

- x + 4y = −19 3y – 5 = 2x

- 3 Written specifically for the publication ‘Cambridge IGCSE Mathematics Core Practice Book’

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

- correct to the nearest cm

Answer ……………………………… cm2

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

Answer d'= ………………………………

A ∩ B' [2]

- 10 A woman invested $300 for 5 years at 7% per year compound interest

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- 11 ⎛ 3 −4 ⎞ A= ⎜ ⎟ ⎝1 2 ⎠

Answer(a)

⎞ ⎟ ⎟ ⎟ ⎠

⎛ ⎜ ⎜ ⎜ ⎝

- (b) A−1,
- the inverse of A

The radius of the quarter circles is 0

Calculate the surface area of the top of the table

- 5 Written specifically for the publication ‘Cambridge IGCSE Mathematics Core Practice Book’

- 13 Make x the subject of y =

5− x 6

Answer x = ………………………………

- 14 The lengths of the sides of a parallelogram are 6 cm and 8 cm

AB is a side of the parallelogram

Using a straight edge and a compasses only,

- construct the parallelogram

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- 15 40 35 Car A
- 30 25 Speed (m/s)

20 15 10 5

- 25 30 35 Time (seconds)

(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

- 7 Written specifically for the publication ‘Cambridge IGCSE Mathematics Core Practice Book’

- 16 Simplify ⎛ x10 ⎞ (a) ⎜ ⎟ ⎝ 32 ⎠
- −3 −2

- 17 Simplifying as much as possible,

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

- (a) Draw the lines y = 12,
- 6x + 2y = 12 and y – 6x = 0 on the grid above

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

- y ≤ 12
- 6x + 2y ≥ 12
- y – 6x ≥ 0 [1]
- 9 Written specifically for the publication ‘Cambridge IGCSE Mathematics Core Practice Book’

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- 19 Solve the equation
- x2 – 12x + 30 = 0

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

20 f(x) =

- x−5 x

Answer(b) ………………………………

Answer(c) ………………………………

- (a) Work out f(3)

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

- (c) Find f−1(x)

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- 21 A NOT TO SCALE

- centre O

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

- (b) angle ACD
- (c) Angle PCB = 42°,
- find BAD
- 11 Written specifically for the publication ‘Cambridge IGCSE Mathematics Core Practice Book’

© Cambridge University Press,

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

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

- oe – Or Equivalent

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

- (a)(i) (a)(ii) (b)

A1 A1 A1

- (a) (b)
- 64365994… (accept more figures) 52
- + x2 ) or 12 ( y1 + y2 ) (0

- 3 x − 1 − 5 x + 15 14 – 2x oe

600 × 1

25 ÷ 1

20 $625

- y = −3

- 5 (rounding down) 81
- 500 ÷ 20 = 25 2 × 108 ÷ "25" (allow methods that involve finding a formula) 4 × 107 oe

1 (x 2 1

- 1 Written specifically for the publication ‘Cambridge IGCSE Mathematics Core Practice Book’

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300 × 1

- 07n 300 × 1

075 420

- 4⎞ 1 ⎟ seen or 3⎠ 10 4⎞ 1
- ⎟ seen and 3⎠ 10

- 4 ⎞ accept ⎜ ⎟ ⎝ −0

- 6y = 5 − x
- x = 5 − 6y
- x = (5 − 6 y )2

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,

- and fully correct answer

- 30 ÷ 10 3 m/s2
- 5 × 20 × 45 + 0
- 5 × 20 × 5 or
- 500 m Car A (it has the steeper downward gradient) 3

26 m3 13

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

- x6 8 x6 16 x 4
- 1 × 50 × 20 (finding area under car B curve) 2

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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

2 x −3

- 12 ± 144 − 120 2 8

- (a) (b)
- 2 3 1 −5 x2

M1 A1 A1 A1 M1

- 1 − 5x 2 y −5 exchange letters,

rearrange to y = … x= y f −1 ( x) =

5 1− x

A1 M1 A1 3

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- (a) (b) (c)
- 59° (isosceles triangle) 31° (angle at centre = 2 × angle at circumference) 120° (opposite angles in a cyclic quadrilateral)

Written specifically for the publication ‘Cambridge IGCSE Mathematics Core Practice Book’

© Cambridge University Press,

- 2 hours 30 minutes

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

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

- give the answer as follows:
- - to three significant figures for all values,
- - to one decimal place for degrees

- 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

20° 5m

In Diagram 1,

angle DAC = 20° and angle DBC = 30°

- (a) Find (i) angle ABD,

- (ii) angle ADB,

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

- using the sine rule,

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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

Diagram 2

In Diagram 2,

BC = 2 m,

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

- find the length of AB

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

find the height of the flagpole,

- 3 Written specifically for the publication ‘Cambridge IGCSE Mathematics Extended Practice Book’

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−8 −7 −6 −5 − −4 −3 − −2 −11 − −1

- − −2 − −3 − −4 C
- − −5 − −6

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

- (iii) triangle C onto D,

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

- (iiii) triangle D'onto E,

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

Writtten specifically fo or the publicationn ‘Cambridge IGC CSE Mathematicss Core Practice B Book’

Cambridge Innternational Exam minations does noot take responsibiility for this conte ent or the associaated answers

- (iv) triangle Bon to A

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

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

- (ii) triangle B onto A
- 5 Written specifically for the publication ‘Cambridge IGCSE Mathematics Extended Practice Book’

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,

- show that y = 34 – x

- [2] (ii) The diagonal of the rectangle is 26 cm
- using Pythagoras’ theorem,

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

Answer(a)(ii)

- [3] (iii) Factorise x2 – 34x + 240

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

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

- [3] (ii) Solve the equation x2 + x – 1 = 0,

giving your answers correct to 3 decimal places

- 7 Written specifically for the publication ‘Cambridge IGCSE Mathematics Extended Practice Book’

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

x4 + x3

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

- (ii) On the grid,
- draw the graph of y =
- x4 + x3 for −5
- 5 ≤ x ≤ 2

5 x −6

- −20 [5]

Written specifically for the publication ‘Cambridge IGCSE Mathematics Core Practice Book’

(b) Use your graph to solve the equation

- x4 + x3 = 5
- 5 Answer(b) x = ……… or x = ………
- (c) (i) By drawing a tangent,

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

……………………………

Answer(c)(ii) ……………………………

- (d) (i) On the grid,

draw the line y = −2x – 10

(ii) Use your graphs to solve the equation

- x4 + x3 + 2 x + 10 = 0

Answer(d)(ii) x = ……… or x = ………

- 9 Written specifically for the publication ‘Cambridge IGCSE Mathematics Extended Practice Book’

Bag A contains 5 red beads and 5 green beads

then a bead is taken at random from bag B

(a) Complete the tree diagram below,

showing the probabilities of each outcome

Red Red

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

……………………………

……………………………

(ii) exactly one red bead is picked

(c) All the beads are returned to the bags

- its colour noted,
- and placed in bag B

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

Answer(c)(ii)

……………………………

Answer(c)(iii)

……………………………

(iii) at least one green bead is picked

- 11 Written specifically for the publication ‘Cambridge IGCSE Mathematics Extended Practice Book’

Small mug

- and a holds 500 cm3 of water

(a) Calculate the height of the small mug

Answer(a)

- ………………………cm

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

……………………………

(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

……………………………

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

……………………………

Answer(c)(iii)

- ………………………cm

(iii) Work out the height of the large mug

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

- ……………………cm3
- 13 Written specifically for the publication ‘Cambridge IGCSE Mathematics Extended Practice Book’

© Cambridge University Press,

The results are used to draw this cumulative frequency diagram

Cumulative frequency

- 120 110 100 90 80 70 60 50 40 30 20 10 0 140

Height (cm)

(a) Find (i) the median height,

- ………………………cm

- ………………………cm

- ………………………cm

……………………………

- (ii) the lower quartile,

(iii) the interquartile range,

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

Answer(a)(iv)

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

- 140 ≤ h ≤ 200

- 200 ≤ h ≤ 220
- 220 ≤ h ≤ 260
- 260 ≤ h ≤ 300
- 300 ≤ h ≤ 380

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

(ii) Construct a histogram to represent this information

Frequency density

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

Answer(c)

……………………………

- (a) Solve the equation
- x−5 x+2 + = −4 6 9

Answer(a) x = ……………………………

- (b) (i)
- 5 4 − x−3 x+4

- (ii) Write
- 5 4 as a single fraction
- − x−3 x+4

Written specifically for the publication ‘Cambridge IGCSE Mathematics Core Practice Book’

- (iii) Solve the equation
- 5 4 1 − = x−3 x+4 x

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

- b and d

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

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

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

- as simply as possible,
- the vectors (i)

JJJG PS Answer(a)(i)

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

Diagram 1

Diagram 4

- (a) Complete the table below

- [2] (b) Work out the number of triangles and the number of dots in the 8th diagram

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

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

- 19 Written specifically for the publication ‘Cambridge IGCSE Mathematics Extended Practice Book’

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

- oe – Or Equivalent

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

- (a)(i) (a)(ii) (a)(iii)
- (a)(iv)
- (b)(ii) 2
- (a)(i) (a)(ii)
- 150° 10° BD 5 = sin 20° sin10° 9
- 85 m CD sin 30° = BD CD = sin 30° × “9
- 848…” 4
- 92 m 3 4 + AB = 2 AB 3 AB = 8 + 2 AB AB = 8 m 132 FG = 8 2 FG = 33 m Reflection,
- in x-axis 1 Enlargement,
- factor ,
- centre (6,
- (a)(iii)

Rotation,

- about (3,
- (a)(iv)

Stretch in y-direction,

- scale factor 3,

about x-axis as invariant line

- (b)(ii)

- ⎠ ⎛

⎞ ⎜ ⎟ ⎝ 0 −1 ⎠ ⎛1 ⎜ ⎝

- ⎜ ⎝0

0⎞ ⎟

- ⎞ ⎟ 3⎠

- 1 Written specifically for the publication ‘Cambridge IGCSE Mathematics Core Practice Book’

- 2 x + 2 y = 68 x + y = 34
- (a)(ii)
- x 2 + y 2 = 262 2

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

- (b)(ii)

- 2 x 2 + 68 x + 34 2 − 262 = 0 proceeding to result ( x − 10)( x − 24) x = 10 or 24 AD DC = oe AB FC 1+ x 1 = oe 1 x x(1 + x) = 1 leading to result

- −1 ± 5 2 x = 0

M1 A1 A1 A1 A1 A1

- 4 (accept 4

- (a)(ii)

Smooth curve A1 Domain correct A1

- (b) (c)(i) (c)(ii)
- 6 (allow ± 0
- 5 squares from their graph) Correct tangent drawn Correct triangle used to calculate gradient Gradient = approx
- 2 Written specifically for the publication ‘Cambridge IGCSE Mathematics Core Practice Book’

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- (d)(ii) 5
- (b)(ii)
- (c)(ii) (c)(iii)
- 2 (allow ± 0
- 5 squares) From top to bottom of tree diagram: P(G) = 3/5,

P(R) = 2/5,

- 2 oe ⎛ 5 3⎞ ⎛ 5 2⎞ ⎜ × ⎟+⎜ × ⎟ ⎝ 10 5 ⎠ ⎝ 10 5 ⎠ 0
- 5 oe From top to bottom of tree diagram: P(G) = 3/6,

- 25 oe 1 − 0

25 oe 0

- 3 Written specifically for the publication ‘Cambridge IGCSE Mathematics Core Practice Book’

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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)

- 500 π × 42 9
- 95 cm 1003 = 1 000 000 oe 1 000 000 = 2000 500 1000 =2 500 3 2 1

26 “9

- 95” × “1

26” 12

- 5 cm r = 4 × “1

26” = 5

- 04… 4 V = π(5
- )3 3 536 cm3

- 270 cm 236 cm 300 – 236 = 64 cm Reading correctly at 316 cm 120 – 100 = 20 trees 10,

A1 A1 A1 M1 A1 A1 A1

Correct FDs A1

- (b)(ii)
- x 0 140
- 10 × 170 + 10 × 210 + 30 × 240 + 40 × 280 + 30 × 340 = 32400 "32400" 120 270 cm
- 4 Written specifically for the publication ‘Cambridge IGCSE Mathematics Core Practice Book’

© Cambridge University Press,

- (b)(ii)
- (b)(iii)

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

- x + 32 ( x − 3)( x + 4)
- (allow expanded denominator)

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

- c= (a)(i) (a)(ii)
- 5( x + 4) − 4( x − 3) ( x − 3)( x + 4)
- b −d a

M1 A1 M1 M1

- (a)(iii)
- 2a JJJG JJJG JJJG QS = QP + PS −b + 2a JJJG JJJG JJJG PR = PQ + QR leading to b + 14 (−b + 2a)

- 3 b + 12 a 4 JJJG JJJG JJJG JJJG RT = RQ + QP + PT

PR + RT = a,

- so RT = a − PR
- − 14 (−b

a − (¾b + ½a) = ½a − ¾b

- (a) (b) (c) (d)(i) (d)(ii)
- + 2a) − b + a

− 34 b + 12 a leading to result given

Triangles 16,

- 25 Dots 15,
- 21 Triangles 64,

- 5(1002 + 3 × 100 + 2) = 5151

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

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