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Algebra Maths
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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.
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
© UCLES 2010 0580/23/M/J/10 [Turn over
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
Use
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
© 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
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
© 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.
Answer
[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.
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
For
Examiner's
Use
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
Use
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
© 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.
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
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.
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
© 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.
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
For
Examiner's
Use
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
© UCLES 2012 0580/23/M/J/12 [Turn over
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
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.
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
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
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
© UCLES 2012 0580/23/M/J/12 [Turn over
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
© UCLES 2012 0580/23/M/J/12 [Turn over
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
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.
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.
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.
Answer(b)(i)
[4]
11
17
© UCLES 2010 0580/43/M/J/10 [Turn over
For
Examiner's
Use
(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
© UCLES 2011 0580/43/M/J/11 [Turn over
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
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.
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