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Maths Jan 2019 IGCSE Math Sheet their Numbers, algebra, statistics, and geometry Questions available

1.

2.
International GCSE Mathematics

Formulae sheet – Higher Tier

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Arithmetic series 1

n Area of trapezium = (a + b)h

Sum to n terms, Sn = [2a + (n – 1)d] 2

2

The quadratic equation a

The solutions of ax2 + bx + c = 0 where

a ¹ 0 are given by: h

−b ± b2 − 4ac

x=

2a b

Trigonometry In any triangle ABC

C a b c

Sine Rule = =

sin A sin B sin C

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b a Cosine Rule a2 = b2 + c2 – 2bccos A

1

Area of triangle = ab sin C

A B 2

c

1 2 Volume of prism

Volume of cone = πr h = area of cross section × length

3

Curved surface area of cone = πrl

l cross

h section

length

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Volume of cylinder = πr2h 4 3 DO NOT WRITE IN THIS AREA

Curved surface area Volume of sphere = πr

3

of cylinder = 2πrh

Surface area of sphere = 4πr2

r

r

h

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Formulae sheet – Higher Tier

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Arithmetic series 1

n Area of trapezium = (a + b)h

Sum to n terms, Sn = [2a + (n – 1)d] 2

2

The quadratic equation a

The solutions of ax2 + bx + c = 0 where

a ¹ 0 are given by: h

−b ± b2 − 4ac

x=

2a b

Trigonometry In any triangle ABC

C a b c

Sine Rule = =

sin A sin B sin C

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b a Cosine Rule a2 = b2 + c2 – 2bccos A

1

Area of triangle = ab sin C

A B 2

c

1 2 Volume of prism

Volume of cone = πr h = area of cross section × length

3

Curved surface area of cone = πrl

l cross

h section

length

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Volume of cylinder = πr2h 4 3 DO NOT WRITE IN THIS AREA

Curved surface area Volume of sphere = πr

3

of cylinder = 2πrh

Surface area of sphere = 4πr2

r

r

h

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3.
Answer ALL TWENTY ONE questions.

Write your answers in the spaces provided.

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You must write down all the stages in your working.

1 (a) Factorise fully 4p + 6pq

(2)

(b) Expand and simplify (e + 3)(e – 5)

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(2)

2y + 1

(c) Solve y =

5

Show clear algebraic working.

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y=

(3)

(Total for Question 1 is 7 marks)

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Write your answers in the spaces provided.

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You must write down all the stages in your working.

1 (a) Factorise fully 4p + 6pq

(2)

(b) Expand and simplify (e + 3)(e – 5)

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(2)

2y + 1

(c) Solve y =

5

Show clear algebraic working.

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y=

(3)

(Total for Question 1 is 7 marks)

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4.
y

10

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A

8

6

B

4

2

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–4 –2 O 2 4 6 8 x

(a) Describe fully the single transformation that maps triangle A onto triangle B.

(3)

2

(b) On the grid, translate triangle A by the vector

−5

Label the new triangle C.

(1)

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4

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10

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A

8

6

B

4

2

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–4 –2 O 2 4 6 8 x

(a) Describe fully the single transformation that maps triangle A onto triangle B.

(3)

2

(b) On the grid, translate triangle A by the vector

−5

Label the new triangle C.

(1)

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4

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5.

6.

7.
4 The table gives information about the price of gold.

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1st February 2016 1st March 2016

Price of one ounce of

1126.50 1236.50

gold (dollars)

(a) Work out the percentage increase in the price of gold between 1st February 2016 and

1st March 2016

Give your answer correct to 3 significant figures.

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%

(3)

The price of one ounce of gold on 1st February 2016 was 1126.50 dollars.

The price of gold increased by 19% from 1st February 2016 to 1st July 2016

(b) Work out the price of one ounce of gold on 1st July 2016

Give your answer correct to the nearest dollar.

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dollars

(3)

(Total for Question 4 is 6 marks)

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1st February 2016 1st March 2016

Price of one ounce of

1126.50 1236.50

gold (dollars)

(a) Work out the percentage increase in the price of gold between 1st February 2016 and

1st March 2016

Give your answer correct to 3 significant figures.

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%

(3)

The price of one ounce of gold on 1st February 2016 was 1126.50 dollars.

The price of gold increased by 19% from 1st February 2016 to 1st July 2016

(b) Work out the price of one ounce of gold on 1st July 2016

Give your answer correct to the nearest dollar.

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dollars

(3)

(Total for Question 4 is 6 marks)

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8.
D

Diagram NOT

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C

(30x – 5)° accurately drawn

(4x + 15)°

B

E

(20x + 45)°

F

A

BCD and AFE are straight lines.

Show that BCD is parallel to AFE.

Give reasons for your working.

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(Total for Question 5 is 5 marks)

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Diagram NOT

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C

(30x – 5)° accurately drawn

(4x + 15)°

B

E

(20x + 45)°

F

A

BCD and AFE are straight lines.

Show that BCD is parallel to AFE.

Give reasons for your working.

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(Total for Question 5 is 5 marks)

8

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9.
6 (a) Complete the table of values for y = x2 – 5x + 6

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x 0 1 2 3 4 5

y 6 0 0 2

(1)

(b) On the grid, draw the graph of y = x2 – 5x + 6 for 0 x 5

y

7

6

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5

4

3

2

1

O 1 2 3 4 5 x

–1

–2

(2)

(c) By drawing a suitable straight line on the grid, find estimates for the solutions of the

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equation

x2 – 5x = x – 7

(3)

(Total for Question 6 is 6 marks)

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x 0 1 2 3 4 5

y 6 0 0 2

(1)

(b) On the grid, draw the graph of y = x2 – 5x + 6 for 0 x 5

y

7

6

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5

4

3

2

1

O 1 2 3 4 5 x

–1

–2

(2)

(c) By drawing a suitable straight line on the grid, find estimates for the solutions of the

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equation

x2 – 5x = x – 7

(3)

(Total for Question 6 is 6 marks)

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10.
7 The table shows the volumes, in km3, of four oceans.

Ocean Volume (km3)

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Arctic Ocean 1.88 ´ 107

Atlantic Ocean 3.10 ´ 108

Indian Ocean 2.64 ´ 108

Southern Ocean 7.18 ´ 107

(a) Write 7.18 ´ 107 as an ordinary number.

(1)

(b) Calculate the total volume of these four oceans.

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km3

(2)

The volume of the South China Sea is 9 880 000 km3

(c) Write 9 880 000 in standard form.

(1)

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Ocean Volume (km3)

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Arctic Ocean 1.88 ´ 107

Atlantic Ocean 3.10 ´ 108

Indian Ocean 2.64 ´ 108

Southern Ocean 7.18 ´ 107

(a) Write 7.18 ´ 107 as an ordinary number.

(1)

(b) Calculate the total volume of these four oceans.

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km3

(2)

The volume of the South China Sea is 9 880 000 km3

(c) Write 9 880 000 in standard form.

(1)

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11.
8 The diagram shows an isosceles triangle.

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Diagram NOT

accurately drawn

x cm x cm

5 cm

The area of the triangle is 12 cm2

Work out the perimeter of the triangle.

Give your answer correct to 3 significant figures.

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cm

(Total for Question 8 is 4 marks)

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Diagram NOT

accurately drawn

x cm x cm

5 cm

The area of the triangle is 12 cm2

Work out the perimeter of the triangle.

Give your answer correct to 3 significant figures.

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cm

(Total for Question 8 is 4 marks)

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12.
9 The table shows information about the speeds of 60 cycles.

Speed (s km/h) Frequency

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0 < s 10 3

10 < s 20 16

20 < s 30 24

30 < s 40 10

40 < s 50 5

50 < s 60 2

(a) Complete the cumulative frequency table.

Cumulative

Speed (s km/h)

frequency

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0 < s 10

0 < s 20

0 < s 30

0 < s 40

0 < s 50

0 < s 60

(1)

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Speed (s km/h) Frequency

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0 < s 10 3

10 < s 20 16

20 < s 30 24

30 < s 40 10

40 < s 50 5

50 < s 60 2

(a) Complete the cumulative frequency table.

Cumulative

Speed (s km/h)

frequency

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0 < s 10

0 < s 20

0 < s 30

0 < s 40

0 < s 50

0 < s 60

(1)

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13.
(b) On the grid, draw a cumulative frequency graph for your table.

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60

50

40

Cumulative

frequency 30

20

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10

0

0 10 20 30 40 50 60

Speed (km/h)

(2)

(c) Use your graph to find an estimate for the interquartile range of the speeds.

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km/h

(2)

(Total for Question 9 is 5 marks)

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60

50

40

Cumulative

frequency 30

20

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10

0

0 10 20 30 40 50 60

Speed (km/h)

(2)

(c) Use your graph to find an estimate for the interquartile range of the speeds.

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km/h

(2)

(Total for Question 9 is 5 marks)

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14.
10 Here is triangle ABD.

B

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Diagram NOT

8 cm accurately drawn

C

20°

A D

13 cm

The point C lies on BD.

AD = 13 cm BC = 8 cm angle ADB = 90° angle CAD = 20°

Calculate the size of angle BAC.

Give your answer correct to 1 decimal place.

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°

(Total for Question 10 is 5 marks)

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B

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Diagram NOT

8 cm accurately drawn

C

20°

A D

13 cm

The point C lies on BD.

AD = 13 cm BC = 8 cm angle ADB = 90° angle CAD = 20°

Calculate the size of angle BAC.

Give your answer correct to 1 decimal place.

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°

(Total for Question 10 is 5 marks)

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15.
5 x+2

11 Express − as a single fraction in its simplest terms.

3 2x

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(Total for Question 11 is 3 marks)

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11 Express − as a single fraction in its simplest terms.

3 2x

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(Total for Question 11 is 3 marks)

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16.
1

12 The curve C has equation y = x3 − 9 x + 1

3

dy

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(a) Find

dx

dy

=

dx

(2)

(b) Find the range of values of x for which C has a negative gradient.

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(3)

(Total for Question 12 is 5 marks)

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12 The curve C has equation y = x3 − 9 x + 1

3

dy

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(a) Find

dx

dy

=

dx

(2)

(b) Find the range of values of x for which C has a negative gradient.

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(3)

(Total for Question 12 is 5 marks)

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17.
13 All the students in Year 11 at a school must study at least one of Geography (G), History (H)

and Religious Studies (R).

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In Year 11 there are 65 students.

Of these students

15 study Geography, History and Religious Studies

21 study Geography and History

16 study Geography and Religious Studies

30 study Geography

18 study only Religious Studies

37 study Religious Studies

(a) Using this information, complete the Venn diagram to show the number of students in

each region of the Venn diagram.

E

G H

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R

(3)

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A student in Year 11 who studies both History and Religious Studies is chosen at random.

(b) Work out the probability that this student does not study Geography.

(2)

(Total for Question 13 is 5 marks)

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and Religious Studies (R).

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In Year 11 there are 65 students.

Of these students

15 study Geography, History and Religious Studies

21 study Geography and History

16 study Geography and Religious Studies

30 study Geography

18 study only Religious Studies

37 study Religious Studies

(a) Using this information, complete the Venn diagram to show the number of students in

each region of the Venn diagram.

E

G H

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R

(3)

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A student in Year 11 who studies both History and Religious Studies is chosen at random.

(b) Work out the probability that this student does not study Geography.

(2)

(Total for Question 13 is 5 marks)

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18.
14 T is directly proportional to the cube of r

T = 21.76 when r = 4

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(a) Find a formula for T in terms of r

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(3)

(b) Work out the value of T when r = 6

(1)

(Total for Question 14 is 4 marks)

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T = 21.76 when r = 4

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(a) Find a formula for T in terms of r

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(3)

(b) Work out the value of T when r = 6

(1)

(Total for Question 14 is 4 marks)

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19.
15 The total surface area of a solid hemisphere is equal to the curved surface area of a cylinder.

The radius of the hemisphere is r cm.

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The radius of the cylinder is twice the radius of the hemisphere.

Given that

volume of hemisphere : volume of cylinder = 1 : m

find the value of m.

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m=

(Total for Question 15 is 4 marks)

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The radius of the hemisphere is r cm.

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The radius of the cylinder is twice the radius of the hemisphere.

Given that

volume of hemisphere : volume of cylinder = 1 : m

find the value of m.

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m=

(Total for Question 15 is 4 marks)

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20.
a + 4b

16 (a) Rationalise the denominator of where a is an integer and b is a prime number.

a − 4b

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Simplify your answer.

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(3)

−5

y xm

(b) Given that = where x ≠ y

x ym

find the value of m.

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m=

(1)

(Total for Question 16 is 4 marks)

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16 (a) Rationalise the denominator of where a is an integer and b is a prime number.

a − 4b

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Simplify your answer.

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(3)

−5

y xm

(b) Given that = where x ≠ y

x ym

find the value of m.

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m=

(1)

(Total for Question 16 is 4 marks)

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21.
17 Here is triangle ABC.

C

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5.3 cm

B Diagram NOT

accurately drawn

110°

4.1 cm

x°

A

Calculate the value of x.

Give your answer correct to 3 significant figures.

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(Total for Question 17 is 5 marks)

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C

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5.3 cm

B Diagram NOT

accurately drawn

110°

4.1 cm

x°

A

Calculate the value of x.

Give your answer correct to 3 significant figures.

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(Total for Question 17 is 5 marks)

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22.
18 The graph of y = f(x) is shown on the grid.

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y

4

2

−8 −6 −4 −2 O 2 4 6 8 x

−2

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−4

−6

−8

1

(a) On the grid above, sketch the graph of y = f x

2

(2)

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y

4

2

−8 −6 −4 −2 O 2 4 6 8 x

−2

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−4

−6

−8

1

(a) On the grid above, sketch the graph of y = f x

2

(2)

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23.
The graph of y = f(x + k) is shown on the grid below.

y

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4

2

−8 −6 −4 −2 O 2 4 6 8 x

−2

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−4

−6

−8

(b) Write down the value of k

(1)

(Total for Question 18 is 3 marks)

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y

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4

2

−8 −6 −4 −2 O 2 4 6 8 x

−2

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−4

−6

−8

(b) Write down the value of k

(1)

(Total for Question 18 is 3 marks)

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24.
19 g is the function with domain x –3 such that g(x) = x2 + 6x

(a) Write down the range of g–1

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(1)

(b) Express the inverse function g–1 in the form g–1 : x

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g–1 : x

(4)

(Total for Question 19 is 5 marks)

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(a) Write down the range of g–1

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(1)

(b) Express the inverse function g–1 in the form g–1 : x

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g–1 : x

(4)

(Total for Question 19 is 5 marks)

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25.
20 A bowl contains n pieces of fruit.

Of these, 4 are oranges and the rest are apples.

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Two pieces of fruit are going to be taken at random from the bowl.

1

The probability that the bowl will then contain (n – 6) apples is

3

Work out the value of n

Show your working clearly.

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(Total for Question 20 is 6 marks)

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Of these, 4 are oranges and the rest are apples.

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Two pieces of fruit are going to be taken at random from the bowl.

1

The probability that the bowl will then contain (n – 6) apples is

3

Work out the value of n

Show your working clearly.

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(Total for Question 20 is 6 marks)

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26.
21 (2x + 23), (8x + 2) and (20x – 52) are three consecutive terms of an arithmetic sequence.

Prove that the common difference of the sequence is 12

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(Total for Question 21 is 4 marks)

TOTAL FOR PAPER IS 100 MARKS

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*P59017A02628*

Prove that the common difference of the sequence is 12

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(Total for Question 21 is 4 marks)

TOTAL FOR PAPER IS 100 MARKS

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*P59017A02728*

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*P59017A02828*

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