15
Algebra Question 1 Write as a single fraction in its simplest form 1 3 2 x x - + . Answer [2] Question 2 Expand and simplify 2(x – 3) 2 – (2x – 3) 2 . Answer [3] Question 3 Solve the equation 9 2 ) 4 ( 3 = + - y y . Answer y = [3] 1

Algebra- Maths exercise

Embed Size (px)

DESCRIPTION

Algebra Maths

Citation preview

Algebra

Question 1

3

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

For

Examiner's

Use

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

Calculate how much Amalie paid for the shirt.

6 3x

! 94

= 3n

.

Find n in terms of x.

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 !

+ .

Question 2 5

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

For

Examiner's

Use

12 Expand and simplify 2(x – 3)2

– (2x – 3)2

.

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

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

(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

For

Examiner's

Use

14 Solve the equation 9

2

)4(3 =+!

y

y .

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)

(b)

1

Question 4

7

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

For

Examiner's

Use

16 Make y the subject of the formula.

5

)2( +

=

yr

A

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,

[2]

(b) the perimeter of the sector.

2

Question 59

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

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.

[5]

3

Question 6

3

© UCLES 2011 0580/23/M/J/11 [Turn over

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.

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.

n = [2]

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

(b) Calculate 7.85 ÷ (2.366 ! 102

Question 74

For

Examiner's

Use

7 (a) Find the value of x when

x

27

24

18

= .

(b) Show that

3

2

÷ 1

6

1

=

7

4

.

Write down all the steps in your working.

[2]

8 Solve the simultaneous equations.

x + 2y = 3

2x – 3y = 13

y = [3]

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

Calculate the total amount Eva has after 2 years.

4

Question 8

4

For

Examiner's

Use

7 (a) Find the value of x when

x

27

24

18

= .

(b) Show that

3

2

÷ 1

6

1

=

7

4

.

Write down all the steps in your working.

[2]

8 Solve the simultaneous equations.

x + 2y = 3

2x – 3y = 13

y = [3]

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

Calculate the total amount Eva has after 2 years.

Question 95

© UCLES 2011 0580/23/M/J/11 [Turn over

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.

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.

[3]

5

Question 10

6

For

Examiner's

Use

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.

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

© UCLES 2011 0580/23/M/J/11 [Turn over

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.

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

.

Calculate the radius of the base of the cylinder.

6

Question 12 8

For

Examiner's

Use

16 Write

2

3

2

2

+

+

! xx

as a single fraction.

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.

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

Calculate the capacity of the smaller container.

Question 133

© UCLES 2012 0580/23/M/J/12 [Turn over

For

Examiner's

Use

4 Solve the inequality.

3y + 7 Y 2 – y

5

9 cm

12 cm

5 cm5 cm NOT TOSCALE

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

Calculate the upper bound of the perimeter of the quadrilateral.

6

C

AB

28°

9 cm

15 cm

NOT TOSCALE

Calculate the area of triangle ABC.

[2]

Question 14

4

For

Examiner's

Use

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.

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.

9 Make w the subject of the formula.

t = 2 – a

w3

7

Question 15

6

For

Examiner's

Use

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

(b)

4

1

÷

13

11

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

(b) Factorise completely.

3uv + 9vw

Question 16

7

© UCLES 2012 0580/23/M/J/12 [Turn over

For

Examiner's

Use

14 Simplify the following.

(a) ( )32

4pq

(b) ( )4

1

8

16

!

x

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

© UCLES 2012 0580/23/M/J/12 [Turn over

For

Examiner's

Use

20 Simplify fully.

xxx

xx

2510

20

23

2

+!

!!

Question 21 is printed on the next page.

9

Question 18 16

For

Examiner's

Use

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.

The total cost of 12 bottles of water and 18 bottles of juice is \$45.

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.

[4]

10

16

For

Examiner's

Use

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.

The total cost of 12 bottles of water and 18 bottles of juice is \$45.

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.

[4]

11

17

© UCLES 2010 0580/43/M/J/10 [Turn over

For

Examiner's

Use

(ii) Factorise y2

! 13y + 12.

(iii) Solve the equation y2

! 13y + 12 = 0.

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

(iv) Work out Roshni’s running speed.

(c) Solve the equation

u2

! u – 4 = 0.

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

Question 19

12

4

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 .

[3]

13

5

© UCLES 2011 0580/43/M/J/11 [Turn over

For

Examiner's

Use

(b) Find the value of x.

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

14

Question 20 18

For

Examiner's

Use

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.

(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 .

[3]

(ii) Factorise 25y2

+ 25y O 36 .