2
1
Dhanu has a model railway.
(a) He has a train that consists of a locomotive and 4 coaches.
The mass of the locomotive is 87 g and the mass of each coach is 52 g.
(i) Work out the total mass of the train.
.............................................. g [2]
(ii) Work out the mass of the locomotive as a percentage of the total mass of the train.
............................................. % [1]
(b) The train is 61 cm long and travels at a speed of 18 cm/s.
It takes 4 seconds for the whole of the train to cross a bridge.
Calculate the length of the bridge.
........................................... cm [2]
(c) A new locomotive costs $64.
Calculate the cost of the locomotive in rupees when the exchange rate is 1 rupee = $0.0154 .
Give your answer correct to the nearest 10 rupees.
...................................... rupees [2]
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(d) The cost of a railway magazine increases by 12.5% to $2.70 .
Calculate the cost of the magazine before this increase.
$ ................................................. [2]
(e) Dhanu plays with his model railway from 06 50 to 11 15.
He then rides his bicycle for 3 hours.
Find the ratio time playing with model railway : time riding bicycle.
Give your answer in its simplest form.
......................... : ......................... [3]
(f) The value of Dhanu’s model railway is $550.
This value increases exponentially at a rate of r % per year.
At the end of 5 years the value will be $736.
Calculate the value of r.
r = ................................................. [3]
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4
2
(a) The table shows some values for y = 2x 3 - 4x 2 + 3.
x
-1
- 0.5
y
-3
1.75
0
0.5
1
1.5
2
0.75
3
(i) Complete the table.
[3]
(ii) On the grid, draw the graph of y = 2x 3 - 4x 2 + 3 for - 1 G x G 2 .
y
3
2
1
–1
0
2 x
1
–1
–2
–3
[4]
(iii) Use your graph to solve the equation 2x 3 - 4x 2 + 3 = 1.5 .
x = ........................ or x = ........................ or x = ........................ [3]
(iv) The equation 2x 3 - 4x 2 + 3 = k has only one solution for - 1 G x G 2 .
Write down a possible integer value of k.
................................................. [1]
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5
(b)
y
6
5
4
3
2
1
–1
0
1
x
–1
–2
–3
–4
(i) On the grid, draw the tangent to the curve at x = 1.
[1]
(ii) Use your tangent to estimate the gradient of the curve at x = 1.
................................................. [2]
(iii) Write down the equation of your tangent in the form y = mx + c .
y = ................................................. [2]
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6
3
(a) Manjeet uses 220 litres of water each day.
She reduces the amount of water she uses by 15%.
Calculate the number of litres of water she now uses each day.
......................................... litres [2]
(b) Manjeet has two mathematically similar bottles in her bathroom.
The large bottle holds 1.35 litres and is 29.7 cm high.
The small bottle holds 0.4 litres.
Calculate the height of the small bottle.
............................................ cm [3]
(c) Water from Manjeet’s shower flows at a rate of 12 litres per minute.
The water from the shower flows into a tank that is a cuboid of length 90 cm and width 75 cm.
Calculate the increase in the level of water in the tank when the shower is used for 7 minutes.
............................................ cm [3]
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7
4
A solid metal cone has radius 1.65 cm and slant height 4.70 cm.
(a) Calculate the total surface area of the cone.
[The curved surface area, A, of a cone with radius r and slant height l is A = rrl .]
.......................................... cm2 [2]
(b) Find the angle the slant height makes with the base of the cone.
................................................. [2]
(c) (i) Calculate the volume of the cone.
1
[The volume, V, of a cone with radius r and height h is V = rr 2 h .]
3
.......................................... cm3 [4]
(ii) A metal sphere with radius 5 cm is melted down to make cones identical to this one.
Calculate the number of complete identical cones that are made.
4
[The volume, V, of a sphere with radius r is V = rr 3 .]
3
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................................................. [4]
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8
5
(a) Write as a single fraction in its simplest form.
x+3 x-2
x-3 x+2
................................................. [4]
k
2 12 ' 2 2 = 32
(b)
Find the value of k.
k = ................................................. [2]
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(c) Expand and simplify.
( y + 3) ( y - 4) (2y - 1)
................................................. [3]
(d) Make x the subject of the formula.
x=
3+x
y
x = ................................................. [3]
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11
7
(a) Naga has n marbles.
Panav has three times as many marbles as Naga.
Naga loses 5 marbles and Panav buys 10 marbles.
Together they now have more than 105 marbles.
Write down and solve an inequality in n.
................................................. [3]
(b) y is inversely proportional to x2.
When x = 4 , y = 7.5 .
Find y when x = 5.
y = ................................................. [3]
(c) Find the nth term of each sequence.
(i)
4
2
0
-2
-4
…
................................................. [2]
(ii)
1
7
17
31
49
…
................................................. [2]
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12
8
(a)
S
25°
72°
NOT TO
SCALE
6 cm
34°
7.4 cm
P
R
Q
The diagram shows a quadrilateral PQRS formed from two triangles, PQS and QRS.
Calculate
(i) QR,
QR = ............................................ cm [3]
(ii) PS,
PS = ............................................ cm [3]
(iii) the area of quadrilateral PQRS.
........................................... cm2 [4]
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13
(b)
X
H
G
E
16 cm
F
D
NOT TO
SCALE
C
18 cm
A
20 cm
B
The diagram shows an open box ABCDEFGH in the shape of a cuboid.
AB = 20 cm, BC = 18 cm and AE = 16 cm .
A thin rod AGX rests partly in the box as shown.
The rod is 40 cm long.
(i) Calculate GX, the length of the rod which is outside the box.
GX = ............................................ cm [4]
(ii) Calculate the angle the rod makes with the base of the box.
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................................................. [3]
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14
9
This year, 40 students have each travelled by one or more of plane (P), train (T) or boat (B).
7 have travelled only by plane.
11 have travelled only by train.
9 have travelled only by boat.
n (P + T ) = 8
n (B + T ) = 3
n (B + P ) = 6
P
T
B
(a) Complete the Venn diagram.
[3]
(b) Find n `(P , B)lj .
................................................. [1]
(c) Use set notation to complete the statement.
(P , T , B)l = ...........................
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[1]
16
10
f (x) = 4x - 1
g (x) = x 2
h (x) = 3 -x
(a) Find in its simplest form
(i)
f (x - 3) ,
................................................. [1]
(ii)
g (5x) .
................................................. [1]
(b) Find f -1 (x) .
f -1 (x) = ................................................. [2]
(c) Find the value of hh (l) , correct to 4 significant figures.
................................................. [3]
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17
(d) (i) Show that g (3x - 2) - h (- 3) can be written as 9x 2 - 12x - 23.
[2]
(ii) Use the quadratic formula to solve 9x 2 - 12x - 23 = 0 .
Give your answers correct to 2 decimal places.
x = ............................ or x = ............................ [4]
(e) Find x when f (61) = h (x) .
x = ................................................. [2]
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18
11
A curve has equation y = x 3 - 3x + 4 .
(a) Work out the coordinates of the two stationary points.
( .................... , ....................)
( .................... , ....................) [5]
(b) Determine whether each stationary point is a maximum or a minimum.
Give reasons for your answers.
[3]
© UCLES 2020
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2
1
Painter
Plumber
Electrician
$35 per hour
Fixed charge $40
$48 per hour
for the first 2 hours
plus
$26.50 per hour
then
$32 per hour
These are the rates charged by a painter, a plumber and an electrician who do some work for Mr Sharma.
(a) The painter works for 7 hours.
Calculate the amount Mr Sharma pays the painter.
$ ................................................. [1]
(b) Mr Sharma pays the plumber $252.
Calculate how many hours the plumber works.
........................................ hours [2]
(c) Mr Sharma pays the electrician $224.
Calculate how many hours the electrician works.
........................................ hours [2]
(d) Write down the ratio of the amount Mr Sharma pays to the painter, the plumber and the electrician.
Give your answer in its lowest terms.
painter : plumber : electrician = .................. : .................. : .................. [2]
© UCLES 2021
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3
2
y
14
12
10
B
8
A
6
4
2
– 14
– 12
– 10
–8
–6
–4
–2
0
2
4
6
8
10 x
–2
–4
C
–6
–8
– 10
(a) Describe fully the single transformation that maps
(i) triangle A onto triangle B,
.............................................................................................................................................
............................................................................................................................................. [3]
(ii) triangle A onto triangle C.
.............................................................................................................................................
............................................................................................................................................. [3]
-5
o.
(b) Draw the image of triangle A after a translation by the vector e
- 10
[2]
(c) Draw the image of triangle A after a reflection in the line y = 4 .
[2]
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4
3
(a)
a°
126°
b°
NOT TO
SCALE
c°
63°
The diagram shows two straight lines intersecting two parallel lines.
Find the values of a, b and c.
a = ................................................
b = ................................................
c = ................................................ [3]
(b)
Q
NOT TO
SCALE
S
R
58° x°
P
Points R and S lie on a circle with diameter PQ.
RQ is parallel to PS.
Angle RPQ = 58° .
Find the value of x, giving a geometrical reason for each stage of your working.
.....................................................................................................................................................
.....................................................................................................................................................
.....................................................................................................................................................
x = ................................................ [3]
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5
(c)
O
142°
A
C
NOT TO
SCALE
y°
B
Points A, B and C lie on a circle, centre O.
Angle AOC = 142° .
Find the value of y.
y = ................................................ [2]
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6
4
(a) A shop gives each of 1000 people a voucher.
28 people use their voucher.
The shop now gives each of 16 500 people a voucher.
Calculate how many of these 16 500 people are expected to use their voucher.
................................................. [1]
(b) In a class activity, all the 15 students wear hats.
7 students wear red hats, 6 students wear green hats and 2 students wear white hats.
(i) One of these students is picked at random.
Find the probability that this student wears a red hat.
................................................. [1]
(ii) Two of the 15 students are picked at random.
Show that the probability that these two students wear hats of the same colour is
37
.
105
[3]
(iii) Three of the 15 students are picked at random.
Find the probability that at least two of these three students wear red hats.
................................................. [4]
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7
5
C
B
65°
NOT TO
SCALE
4.4 cm
9.7 cm
A
8.6 cm
42°
D
(a) Calculate angle ADB.
Angle ADB = ................................................ [3]
(b) Calculate DC.
DC = ........................................... cm [4]
(c) Calculate the shortest distance from C to BD.
© UCLES 2021
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............................................ cm [3]
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8
6
y
5
4
3
2
1
–2
–1
0
1
2
x
–1
–2
(a) The grid shows the graph of y = a + bx 2 .
The graph passes through the points with coordinates (0, 4) and (1, 1).
(i) Find the value of a and the value of b.
a = ................................................
b = ................................................ [2]
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9
(ii) Write down the equation of the tangent to the graph at (0, 4).
................................................. [1]
(iii) The equation of the tangent to the graph at x =- 1 is y = 6x + 7 .
Find the equation of the tangent to the graph at x = 1.
................................................. [2]
(b) The table shows some values for y = 1 +
x
–2
– 1.5
y
2
2.11
–1
5
for - 2 G x G 1.5 .
3-x
– 0.5
0
2.43
0.5
1
3
1.5
4.33
(i) Complete the table.
[3]
(ii) On the grid, draw the graph of y = 1 +
5
for - 2 G x G 1.5 .
3-x
[4]
(c) (i) Write down the values of x where the two graphs intersect.
x = ................... or x = .................. [2]
(ii) The answers to part(c)(i) are two solutions of a cubic equation in terms of x.
Find this equation in the form ax 3 + bx 2 + cx + d = 0 , where a, b, c and d are integers.
............................................................................... [4]
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12
8
(a)
O
53°
A
X
B
NOT TO
SCALE
9.5 cm
Y
The diagram shows a sector OXY of a circle with centre O and radius 9.5 cm.
The sector angle is 53°.
A lies on OX, B lies on OY and OA = OB .
(i) Show that the area of the sector is 41.7 cm 2 , correct to 1 decimal place.
[2]
(ii) The area of triangle OAB is
1
of the area of sector OXY.
3
Calculate OA.
OA = ........................................... cm [4]
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13
(b)
O
60°
NOT TO
SCALE
24 cm
P
Q
The diagram shows a sector OPQ of a circle with centre O and radius 24 cm.
The sector angle is 60°.
A cone is made from this sector by joining OP to OQ.
O
NOT TO
SCALE
P
Q
Calculate the volume of the cone.
1
[The volume, V, of a cone with radius r and height h is V = rr 2 h .]
3
......................................... cm 3 [6]
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14
9
(a) Factorise.
(i)
5am + 10ap - bm - 2bp
................................................. [2]
(ii) 15 (k + g) 2 - 20 (k + g)
................................................. [2]
(iii)
4x 2 - y 4
................................................. [2]
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15
(b) Expand and simplify.
(x - 3) (x + 1) (3x - 4)
................................................. [3]
(x + a) 2 = x 2 + 22x + b
(c)
Find the value of a and the value of b.
a = ................................................
b = ................................................ [2]
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16
10 (a) A box is a cuboid with length 45 cm, width 30 cm and height 42 cm.
The box is completely filled with 90.72 kg of sand.
Calculate the density of this sand in kg/m 3 .
[Density = mass ÷ volume]
....................................... kg/m 3 [3]
(b) A bag contains 15000 cm 3 of sand.
Some of this sand is used to completely fill a hole in the shape of a cylinder.
The hole is 30 cm deep and has radius 10 cm.
Calculate the percentage of the sand from the bag that is used.
.............................................. % [3]
(c) Sand costs $98.90 per tonne.
This cost includes a tax of 15%.
Calculate the amount of tax paid per tonne of sand.
$ ................................................. [3]
(d) Raj buys some sand for 3540 rupees.
Calculate the cost in dollars when the exchange rate is $1 = 70.8 rupees.
$ ................................................. [2]
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17
11
Gaya spends $48 to buy books that cost $x each.
(a) Write down an expression, in terms of x, for the number of books Gaya buys.
................................................. [1]
(b) Myra spends $60 to buy books that cost $ (x + 2) each.
Gaya buys 4 more books than Myra.
Show that x 2 + 5x - 24 = 0 .
(c) Solve by factorisation.
[4]
x 2 + 5x - 24 = 0
x = .................. or x = ................... [3]
(d) Find the number of books Myra buys.
................................................. [1]
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18
12 (a) Find the gradient of the curve y = 2x 3 - 7x + 4 when x =- 2 .
................................................. [3]
(b) A is the point (7, 2) and B is the point (−5, 8).
(i) Calculate the length of AB.
................................................. [3]
(ii) Find the equation of the line that is perpendicular to AB and that passes through
the point (−1, 3).
Give your answer in the form y = mx + c .
y = ................................................ [4]
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19
(iii) AB is one side of the parallelogram ABCD and
•
-a
BC = e o where a 2 0 and b 2 0
-b
•
the gradient of BC is 1
•
BC = 8 .
Find the coordinates of D.
( ....................... , ......................) [4]
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