Algebra- Maths exercise

Algebra

Question 1

3

© UCLES 2010 0580/23/M/J/10 [Turn over

For

Examiner's

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5 Amalie makes a profit of 20% when she sells a shirt for $21.60.

Calculate how much Amalie paid for the shirt.

Answer $ [2]

6 3x

! 94

= 3n

.

Find n in terms of x.

Answer n = [2]

7 Shade the required regions in the Venn diagrams below.

A B

C

A B

C

[2]

8 Write as a single fraction in its simplest form

1

3 2

x x !

+ .

Answer [2]

Question 2 5


For

Examiner's

Use

12 Expand and simplify 2(x – 3)2

– (2x – 3)2

.

Answer [3]

13 (a) Write down the number of lines of symmetry for the diagram below.

Answer(a) [1]

(b) Write down the order of rotational symmetry for the diagram below.

Answer(b) [1]

(c) The diagram shows a cuboid which has no square faces.

Draw one of the planes of symmetry of the cuboid on the diagram.

[1]

Question 36

© UCLES 2010 0580/23/M/J/10

For

Examiner's

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14 Solve the equation 9

2

)4(3 =+!

y

y .

Answer y = [3]

15

G

O

H

N

g

h

NOT TOSCALE

In triangle OGH, the ratio GN : NH = 3 : 1.

= g and = h.

Find the following in terms of g and h, giving your answers in their simplest form.

(a)

Answer(a) = [1]

(b)

Answer(b) = [2]

1

Question 4

7


For

Examiner's

Use

16 Make y the subject of the formula.

5

)2( +

=

yr

A

Answer y = [3]

17

40°O L

K

5.6 cm

NOT TOSCALE

OKL is a sector of a circle, centre O, radius 5.6 cm.

Angle KOL = 40°.

Calculate

(a) the area of the sector,

Answer(a) cm2

[2]

(b) the perimeter of the sector.

Answer(b) cm [2]

2

Question 59


For

Examiner's

Use

20

y

x

4

3

2

1

0 1 2 3 4

Find the three inequalities which define the shaded region on the grid.

Answer

[5]

3

Question 6

3


For

Examiner's

Use

4 Helen measures a rectangular sheet of paper as 197 mm by 210 mm, each correct to the nearest

millimetre.

Calculate the upper bound for the perimeter of the sheet of paper.

Answer mm [2]

5

0

y

x

NOT TOSCALE

The sketch shows the graph of y = axn

where a and n are integers.

Write down a possible value for a and a possible value for n.

Answer a =

n = [2]

6 (a) Write 16 460 000 in standard form.

Answer(a) [1]

(b) Calculate 7.85 ÷ (2.366 ! 102

), giving your answer in standard form.

Answer(b) [2]

Question 74

© UCLES 2011 0580/23/M/J/11

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7 (a) Find the value of x when

x

27

24

18

= .

Answer(a) x = [1]

(b) Show that

3

2

÷ 1

6

1

=

7

4

.

Write down all the steps in your working.

Answer(b)

[2]

8 Solve the simultaneous equations.

x + 2y = 3

2x – 3y = 13

Answer x =

y = [3]

9 Eva invests $120 at a rate of 3% per year compound interest.

Calculate the total amount Eva has after 2 years.

Give your answer correct to 2 decimal places.

Answer $ [3]

4

Question 8

4

© UCLES 2011 0580/23/M/J/11

For

Examiner's

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7 (a) Find the value of x when

x

27

24

18

= .

Answer(a) x = [1]

(b) Show that

3

2

÷ 1

6

1

=

7

4

.

Write down all the steps in your working.

Answer(b)

[2]

8 Solve the simultaneous equations.

x + 2y = 3

2x – 3y = 13

Answer x =

y = [3]

9 Eva invests $120 at a rate of 3% per year compound interest.

Calculate the total amount Eva has after 2 years.

Give your answer correct to 2 decimal places.

Answer $ [3]

Question 95


For

Examiner's

Use

10 The cost of a cup of tea is t cents.

The cost of a cup of coffee is (t + 5) cents.

The total cost of 7 cups of tea and 11 cups of coffee is 2215 cents.

Find the cost of one cup of tea.

Answer cents [3]

11 The volume of a solid varies directly as the cube of its length.

When the length is 3 cm, the volume is 108 cm3

.

Find the volume when the length is 5 cm.

Answer cm3

[3]

5

Question 10

6

© UCLES 2011 0580/23/M/J/11

For

Examiner's

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12 Federico changed 400 euros (!) into New Zealand dollars (NZ$) at a rate of !1 = NZ$ 2.1 .

He spent x New Zealand dollars and changed the rest back into euros at a rate of !1 = NZ$ d.

Find an expression, in terms of x and d, for the number of euros Federico received.

Answer ! [3]

13

0

y

x

NOT TOSCALE

The diagram shows the lines y = 1, y = x + 4 and y = 4 – x .

On the diagram, label the region R where y [ 1, y [ x + 4 and y Y 4 – x . [3]

Question 11

7


For

Examiner's

Use

14

0 3

y

x1

13

NOT TOSCALE

The diagram shows the straight line which passes through the points (0, 1) and (3, 13).

Find the equation of the straight line.

Answer [3]

15 A cylinder has a height of 12 cm and a volume of 920 cm3

.

Calculate the radius of the base of the cylinder.

Answer cm [3]

6

Question 12 8

© UCLES 2011 0580/23/M/J/11

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16 Write

2

3

2

2

+

+

! xx

as a single fraction.

Give your answer in its simplest form.

Answer [3]

17

9 cm

20 cm

d cm

10 cm

NOT TOSCALE

The diagrams show two mathematically similar containers.

The larger container has a base with diameter 9 cm and a height 20 cm.

The smaller container has a base with diameter d cm and a height 10 cm.

(a) Find the value of d.

Answer(a) d = [1]

(b) The larger container has a capacity of 1600 ml.

Calculate the capacity of the smaller container.

Answer(b) ml [2]

Question 133


For

Examiner's

Use

4 Solve the inequality.

3y + 7 Y 2 – y

Answer [2]

5

9 cm

12 cm

5 cm5 cm NOT TOSCALE

The diagram shows a quadrilateral.

The lengths of the sides are given to the nearest centimetre.

Calculate the upper bound of the perimeter of the quadrilateral.

Answer cm [2]

6

C

AB

28°

9 cm

15 cm

NOT TOSCALE

Calculate the area of triangle ABC.

Answer cm2

[2]

Question 14

4

© UCLES 2012 0580/23/M/J/12

For

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7

Height (h cm) 0 < h Y 10 10 < h Y 15 15 < h Y 30

Frequency 25 u 9

Frequency density 2.5 4.8 v

The table shows information about the heights of some flowers.

Calculate the values of u and v.

Answer u =

v = [2]

8 During her holiday, Hannah rents a bike.

She pays a fixed cost of $8 and then a cost of $4.50 per day.

Hannah pays with a $50 note and receives $10.50 change.

Calculate for how many days Hannah rents the bike.

Answer days [3]

9 Make w the subject of the formula.

t = 2 – a

w3

Answer w = [3]

7

Question 15

6

© UCLES 2012 0580/23/M/J/12

For

Examiner's

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12 Without using your calculator, work out the following.

Show all the steps of your working and give each answer as a fraction in its simplest form.

(a)

12

11

!

3

1

Answer(a) [2]

(b)

4

1

÷

13

11

Answer(b) [2]

13 (a) Find the value of 7p – 3q when p = 8 and q = O5 .

Answer(a) [2]

(b) Factorise completely.

3uv + 9vw

Answer(b) [2]

Question 16

7


For

Examiner's

Use

14 Simplify the following.

(a) ( )32

4pq

Answer(a) [2]

(b) ( )4

1

8

16

!

x

Answer(b) [2]

15 Solve the equation 2x2

+ 6x – 3 = 0 . Show your working and give your answers correct to 2 decimal places.

Answer x = or x = [4]

8

Question 1711


For

Examiner's

Use

20 Simplify fully.

xxx

xx

2510

20

23

2

+!

!!

Answer [5]

Question 21 is printed on the next page.

9

Question 18 16

© UCLES 2010 0580/43/M/J/10

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Examiner's

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9 (a) The cost of a bottle of water is $w.

The cost of a bottle of juice is $j.

The total cost of 8 bottles of water and 2 bottles of juice is $12.


Find the cost of a bottle of water and the cost of a bottle of juice.

Answer(a) Cost of a bottle of water = $

Cost of a bottle of juice = $ [5]

(b) Roshni cycles 2 kilometres at y km/h and then runs 4 kilometres at (y – 4) km/h.

The whole journey takes 40 minutes.

(i) Write an equation in y and show that it simplifies to y2

! 13y + 12 = 0.

Answer(b)(i)

[4]

10

16

© UCLES 2010 0580/43/M/J/10

For

Examiner's

Use

9 (a) The cost of a bottle of water is $w.

The cost of a bottle of juice is $j.



Find the cost of a bottle of water and the cost of a bottle of juice.

Answer(a) Cost of a bottle of water = $

Cost of a bottle of juice = $ [5]

(b) Roshni cycles 2 kilometres at y km/h and then runs 4 kilometres at (y – 4) km/h.

The whole journey takes 40 minutes.

(i) Write an equation in y and show that it simplifies to y2

! 13y + 12 = 0.

Answer(b)(i)

[4]

11

17


For

Examiner's

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(ii) Factorise y2

! 13y + 12.

Answer(b)(ii) [2]

(iii) Solve the equation y2

! 13y + 12 = 0.

Answer(b)(iii) y = or y = [1]

(iv) Work out Roshni’s running speed.

Answer(b)(iv) km/h [1]

(c) Solve the equation

u2

! u – 4 = 0.

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

Answer(c) u = or u = [4]

Question 19

12

4

© UCLES 2011 0580/43/M/J/11

For

Examiner's

Use

3

(x + 5) cm

2x cm

x cm

NOT TOSCALE

The diagram shows a square of side (x + 5) cm and a rectangle which measures 2x cm by x cm.

The area of the square is 1 cm2

more than the area of the rectangle.

(a) Show that x2

– 10x – 24 = 0 .

Answer(a)

[3]

13

5


For

Examiner's

Use

(b) Find the value of x.

Answer(b) x = [3]

(c) Calculate the acute angle between the diagonals of the rectangle.

Answer(c) [3]

14

Question 20 18

© UCLES 2012 0580/43/M/J/12

For

Examiner's

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10 (a) Rice costs $x per kilogram.

Potatoes cost $(x + 1) per kilogram.

The total cost of 12 kg of rice and 7 kg of potatoes is $31.70 .

Find the cost of 1 kg of rice.

Answer(a) $ [3]

(b) The cost of a small bottle of juice is $y.

The cost of a large bottle of juice is $(y + 1).

When Catriona spends $36 on small bottles only, she receives 25 more bottles than when she

spends $36 on large bottles only.

(i) Show that 25y2

+ 25y O 36 = 0 .

Answer(b)(i)

[3]

(ii) Factorise 25y2

+ 25y O 36 .

Answer(b)(ii) [2]

(iii) Solve the equation 25y2

+ 25y O 36 = 0 .

Answer(b)(iii) y = or y = [1]

(iv) Find the total cost of 1 small bottle of juice and 1 large bottle of juice.

Answer(b)(iv) $ [1]

15

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Algebra- Maths exercise