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1 Consecutive integers are set out in rows in a grid.
For
Examiner's
Use
(a) This grid has 5 columns.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
a
b
n
c
d
The shape drawn encloses five numbers 7, 9, 13, 17 and 19. This is the n = 13 shape.
In this shape, a = 7, b = 9, c = 17 and d = 19.
(i) Calculate bc O ad for the n = 13 shape.
Answer(a)(i)
[1]
(ii) For the 5 column grid, a = n O 6.
Write down b, c and d in terms of n for this grid.
Answer(a)(ii) b =
c=
d=
[2]
(iii) Write down bc O ad in terms of n.
Show clearly that it simplifies to 20.
Answer(a)(iii)
[2]
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(b) This grid has 6 columns. The shape is drawn for n = 10.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
For
Examiner's
Use
a
b
n
c
d
(i) Calculate the value of bc O ad for n = 10.
Answer(b)(i)
[1]
(ii) Without simplifying, write down bc O ad in terms of n for this grid.
Answer(b)(ii)
[2]
(c) This grid has 7 columns.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
a
b
n
c
d
Show clearly that bc O ad = 28 for n = 17.
Answer(c)
[1]
Question 1 continues on the next page.
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(d) Write down the value of bc O ad when there are t columns in the grid.
Answer(d)
[1]
(e) Find the values of c, d and bc O ad for this shape.
2
3
4
16
c
d
Answer (e) c =
d=
bc O ad =
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[2]
For
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(a) Marcos buys 2 bottles of water and 3 bottles of lemonade.
The total cost is $3.60.
The cost of one bottle of lemonade is $0.25 more than the cost of one bottle of water.
Find the cost of one bottle of water.
For
Examiner's
Use
[4]
Answer(a) $
(b)
5 cm2
y cm
x cm
6 cm2
Y cm
NOT TO
SCALE
(x + 2) cm
The diagram shows two rectangles.
The first rectangle measures x cm by y cm and has an area of 5 cm2.
The second rectangle measures (x + 2) cm by Y cm and has an area of 6 cm2.
(i) When y + Y = 1, show that x2 O 9x O 10 = 0 .
Answer (b)(i)
[4]
(ii) Factorise x2 O 9x O 10 .
Answer(b)(ii)
[2]
Answer(b)(iii)
cm [2]
(iii) Calculate the perimeter of the first rectangle.
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(c)
For
Examiner's
Use
(2x + 3) cm
NOT TO
SCALE
5 cm
(x + 3) cm
The diagram shows a right-angled triangle with sides of length 5 cm, (x + 3) cm and (2x + 3) cm.
(i) Show that 3x2 + 6x O 25 = 0 .
Answer (c)(i)
[4]
(ii) Solve the equation 3x2 + 6x O 25 = 0 .
Show all your working and give your answers correct to 2 decimal places.
or x =
Answer(c)(ii) x =
[4]
(iii) Calculate the area of the triangle.
Answer(c)(iii)
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cm2 [2]
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(a) Solve the equation 8x2 – 11x – 11 = 0.
Show all your working and give your answers correct to 2 decimal places.
For
Examiner′s
Use
Answer(a) x = ........................ or x = ........................ [4]
(b) y varies directly as the square root of x.
y = 18 when x = 9.
Find y when x = 484.
Answer(b) y = ........................... [3]
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For
Examiner′s
Use
(c) Sara spends $x on pens which cost $2.50 each.
She also spends $(x – 14.50) on pencils which cost $0.50 each.
The total of the number of pens and the number of pencils is 19.
Write down and solve an equation in x.
Answer(c) x = ............................................... [6]
_____________________________________________________________________________________
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(a) Calculate 20.7.
Answer(a) ................................................ [1]
(b) Find the value of x in each of the following.
(i) 2x = 128
Answer(b)(i) x = ................................................ [1]
(ii) 2x × 29 = 213
Answer(b)(ii) x = ................................................ [1]
(iii) 29 ÷ 2x = 4
Answer(b)(iii) x = ................................................ [1]
(iv) 2x = 3 2
Answer(b)(iv) x = ................................................ [1]
(c) (i) Complete this table of values for y = 2x.
x
–3
y
0.125
–2
–1
0
0.5
1
2
3
2
4
8
[2]
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(a) The area of shape ABCDEF is 24 cm2.
All lengths are in centimetres.
F
3x – 9
E
2x
3x + 13
A
(i)
C
D
NOT TO
SCALE
B
4x
Show that 5x 2 + 17x - 12 = 0 .
Answer(a)(i)
[3]
(ii)
Solve, by factorising, the equation 5x 2 + 17x - 12 = 0 .
You must show all your working.
Answer(a)(ii) x = .................... or x = .................... [3]
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(b) Solve the simultaneous equations.
You must show all your working.
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3x – 2y = 23
–4x – y = –5
Answer(b) x = ...............................................
y = ............................................... [3]
(c) Solve the equation.
2 ^t + 3h
t
=1
t
t+3
Answer(c) t = ............................................... [5]
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(a) The cost of a loaf of bread is x cents.
The cost of a cake is (x – 5) cents.
The total cost of 6 loaves of bread and 11 cakes is $13.56 .
Find the value of x.
Answer(a) x = ................................................ [4]
(b)
y+1
y
y+3
NOT TO
SCALE
2y + 1
The area of the rectangle and the area of the triangle are equal.
Find the value of y.
Answer(b) y = ................................................ [4]
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(c) The cost of a bottle of water is (w – 1) cents.
The cost of a bottle of milk is (2w – 11) cents.
A certain number of bottles of water costs $4.80 .
The same number of bottles of milk costs $7.80 .
Find the value of w.
Answer(c) w = ................................................ [4]
(d)
NOT TO
SCALE
u cm
t
(3u – 2) cm
The area of the triangle is 2.5 cm2.
(i) Show that 3u2 – 2u – 5 = 0.
Answer(d)(i)
[2]
(ii) Factorise 3u2 – 2u – 5.
Answer(d)(ii) ................................................ [2]
(iii) Find the size of angle t.
Answer(d)(iii) t = ................................................ [3]
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Apples cost x cents each and oranges cost (x + 2) cents each.
Dylan spends $3.23 on apples and $3.23 on oranges.
The total of the number of apples and the number of oranges Dylan buys is 36.
(a) Write an equation in x and show that it simplifies to 18x 2 - 287x - 323 = 0 .
[4]
(b) (i)
Find the two prime factors of 323.
....................... , .......................[1]
(ii)
Complete the statement.
18x 2 - 287x - 323 = (18x
(iii)
......................)(x
......................)
[2]
Solve the equation 18x 2 - 287x - 323 = 0 .
x = .......................... or x = ..........................[1]
(c) Find the largest number of apples Dylan can buy for $2.
................................................. [1]
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(a) $1= 3.67 dirhams
Calculate the value, in dollars, of 200 dirhams.
Give your answer correct to 2 decimal places.
$ ................................................. [2]
(b) (i)
Write as a single fraction, in its simplest form.
1000 1000
x
x+1
................................................. [3]
(ii)
One day in 2014, 1 euro was worth x rand.
One year later, 1 euro was worth (x + 1) rand.
Winston changed 1000 rand into euros in both years.
In 2014 he received 4.50 euros more than in 2015.
Write an equation in terms of x and show that it simplifies to
9x 2 + 9x - 2000 = 0 .
[3]
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(iii)
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Use the quadratic formula to solve the equation 9x 2 + 9x - 2000 = 0 .
Show all your working and give your answers correct to 2 decimal places.
x = ........................ or x = ........................ [4]
(iv)
Calculate the number of euros Winston received in 2014.
Give your answer correct to 2 decimal places.
........................................euros [2]
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The perimeter of each of the three shapes is 60 cm.
Find x in each part.
(a)
Rectangle
x
NOT TO
SCALE
3x
x = .......................................... cm [2]
(b)
Triangle
NOT TO
SCALE
3x
4x
x = .......................................... cm [3]
(c)
Sector
x
O
NOT TO
SCALE
x
x = .......................................... cm [3]
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(a) The cost of 1 apple is a cents.
The cost of 1 pear is p cents.
The total cost of 7 apples and 9 pears is 354 cents.
(i)
Write down an equation in terms of a and p.
................................................. [1]
(ii)
The cost of 1 pear is 2 cents more than the cost of 1 apple.
Find the value of a and the value of p.
a = ................................................
p = ................................................ [3]
(b) Rowena walks 2 km at an average speed of x km/h.
(i)
Write down an expression, in terms of x, for the time taken.
.............................................. h [1]
(ii)
Rowena then walks 3 km at an average speed of (x – 1) km/h.
The total time taken to walk the 5 km is 2 hours.
(a) Show that 2x 2 - 7x + 2 = 0 .
[3]
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(b) Find the value of x.
Show all your working and give your answer correct to 2 decimal places.
x = ................................................ [4]
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Luigi and Alfredo run in a 10 km race.
Luigi’s average speed was x km/h.
Alfredo’s average speed was 0.5 km/h slower than Luigi’s average speed.
(a) Luigi took
10
hours to run the race.
x
Write down an expression, in terms of x, for the time that Alfredo took to run the race.
................................................ h [1]
(b) Alfredo took 0.25 hours longer than Luigi to run the race.
(i)
Show that 2x 2 - x - 40 = 0 .
[4]
(ii)
Use the quadratic formula to solve 2x 2 - x - 40 = 0 .
Show all your working and give your answers correct to 2 decimal places.
x = ......................... or x = .......................... [4]
(iii)
Work out the time that Luigi took to run the 10 km race.
Give your answer in hours and minutes, correct to the nearest minute.
.................... h .................... min [3]
Question 10 is printed on the next page.
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Paulo and Jim each buy sacks of rice but from different shops.
Paulo pays $72 for sacks costing $m each.
Jim pays $72 for sacks costing $(m + 0.9) each.
(a) (i)
Find an expression, in terms of m, for the number of sacks Paulo buys.
................................................ [1]
(ii)
Find an expression, in terms of m, for the number of sacks Jim buys.
................................................ [1]
(b) Paulo buys 4 more sacks than Jim.
Write down an equation, in terms of m, and show that it simplifies to 10m 2 + 9m - 162 = 0 .
[4]
(c)
(i)
Solve 10m 2 + 9m - 162 = 0 .
m = .................... or m = .................... [3]
(ii)
Find the number of sacks of rice that Paulo buys.
................................................ [1]
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The diagram shows a right-angled triangle ABC.
B
NOT TO
SCALE
4(x – 1) cm
A
(2x + 5) cm
C
The area of this triangle is 30 cm2.
(a) Show that 2x2 + 3x – 20 = 0.
[3]
(b) Use factorisation to solve the equation 2x2 + 3x – 20 = 0.
x = ...................... or x = ...................... [3]
(c) Calculate BC.
BC = .......................................... cm [3]
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