January 2021 · Paper 2R (4MB1/02R) (Edexcel IGCSE Maths): Exam Questions

E is the universal set and A, B and C are three sets such that

E = {even numbers between 5 and 31}

A = {factors of 24}

B = {8, 16}

C = {multiples of 6}

The Venn diagram on the opposite page can be used to show these sets.

Complete the Venn diagram for the sets ℰ, A, B and C

List the elements of the set

E is the universal set and A, B and C are three sets such that

E = {even numbers between 5 and 31}

A = {factors of 24}

B = {8, 16}

C = {multiples of 6}

The Venn diagram on the opposite page can be used to show these sets.

A ∩ C

E is the universal set and A, B and C are three sets such that

E = {even numbers between 5 and 31}

A = {factors of 24}

B = {8, 16}

C = {multiples of 6}

The Venn diagram on the opposite page can be used to show these sets.

(A ∪ B ∪ C)′

Find

E is the universal set and A, B and C are three sets such that

E = {even numbers between 5 and 31}

A = {factors of 24}

B = {8, 16}

C = {multiples of 6}

The Venn diagram on the opposite page can be used to show these sets.

n ([A ∪ B]′)

E is the universal set and A, B and C are three sets such that

E = {even numbers between 5 and 31}

A = {factors of 24}

B = {8, 16}

C = {multiples of 6}

The Venn diagram on the opposite page can be used to show these sets.

n ([A ∩ B] ∪ C)

A number is selected at random from ℰ

E is the universal set and A, B and C are three sets such that

E = {even numbers between 5 and 31}

A = {factors of 24}

B = {8, 16}

C = {multiples of 6}

The Venn diagram on the opposite page can be used to show these sets.

Find the probability that the number is in set B

Question 1 continued

E

A

B

C

Turn over for a spare copy of the Venn diagram.

Question 1 continued

Question 1 continued

Only use this Venn diagram if you wish to replace your answer to part (a)

E

A

B

C

Each year the students at a college organise a music concert.

In 2017, the total cost of organising the concert was $675

In 2018, the total cost of organising the concert was 20% more than the total cost in 2017

Calculate the total cost of organising the concert in 2018

The tickets sold each year were either adult tickets or student tickets.

In 2019, the total number of tickets sold was 385

In 2019, the number of adult tickets sold and the number of student tickets sold were in

the ratio

number of adult tickets : number of student tickets = 19 : 16

Each year the students at a college organise a music concert.

In 2017, the total cost of organising the concert was $675

In 2018, the total cost of organising the concert was 20% more than the total cost in 2017

Calculate the number of adult tickets sold in 2019

In 2019, the price of each adult ticket sold was $8.50 and the price of each student ticket

sold was $4.50

Each year the students at a college organise a music concert.

In 2017, the total cost of organising the concert was $675

In 2018, the total cost of organising the concert was 20% more than the total cost in 2017

Calculate the total amount of money, in $, received for all the tickets sold in 2019

In 2019, the total cost of organising the concert was double the total cost in 2017

Each year the students at a college organise a music concert.

In 2017, the total cost of organising the concert was $675

In 2018, the total cost of organising the concert was 20% more than the total cost in 2017

Calculate the percentage profit made in 2019

Give your answer to 1 decimal place.

Question 2 continued

The equation of a curve is y = −2x3 + 6x + 5

2x

Complete the table of values for y = −2x3 + 6x + 5

2x

Give your values of y to 2 decimal places where necessary.

−2

−1.8

−1.7

−1.6

−1.4

−1.2

−1

−0.9

−0.8

x

The equation of a curve is y = −2x3 + 6x + 5

2x

On the grid opposite, plot the points from your completed table and join them to

form a smooth curve.

The equation of a curve is y = −2x3 + 6x + 5

2x

Use your graph to find an estimate, to 2 decimal places, for the minimum value of

−2x3 + 6x + 5

2x for values of x in −2 ≤ x ≤ −0.8

Question 3 continued

x

Turn over for a spare grid if you need to redraw your curve.

Question 3 continued

Question 3 continued

Only use this grid if you need to redraw your curve.

y

6

x

Ramesh, Maya, Kalil, Chen and Andreia each have a bag containing an identical set of

six cards.

There is a number on each of the six cards.

Here are the cards in each of the bags.

2

4

4

7

9

10

Ramesh takes at random one of the six cards in his bag.

Write down the probability that the number on the card Ramesh takes is a

prime number.

Maya takes at random from her bag two of the six cards in her bag.

Ramesh, Maya, Kalil, Chen and Andreia each have a bag containing an identical set of

six cards.

There is a number on each of the six cards.

Here are the cards in each of the bags.

2

4

4

7

9

10

Ramesh takes at random one of the six cards in his bag.

Find the probability that neither of the two cards has a number 4 on it.

Kalil takes at random from his bag two of the six cards in his bag.

Ramesh, Maya, Kalil, Chen and Andreia each have a bag containing an identical set of

six cards.

There is a number on each of the six cards.

Here are the cards in each of the bags.

2

4

4

7

9

10

Ramesh takes at random one of the six cards in his bag.

Find the probability that the total of the two numbers on the cards is 11

Chen takes at random one card at a time, without replacement, from her bag until she

gets a card with a number 4 on it. She then stops taking cards from her bag.

Ramesh, Maya, Kalil, Chen and Andreia each have a bag containing an identical set of

six cards.

There is a number on each of the six cards.

Here are the cards in each of the bags.

2

4

4

7

9

10

Ramesh takes at random one of the six cards in his bag.

Find the probability that Chen stops taking cards from her bag before she takes the

fourth card.

Andreia puts another card with a number on it into her bag so that she has seven cards

in her bag.

The mean of the numbers on the seven cards in Andreia’s bag is 8

Ramesh, Maya, Kalil, Chen and Andreia each have a bag containing an identical set of

six cards.

There is a number on each of the six cards.

Here are the cards in each of the bags.

2

4

4

7

9

10

Ramesh takes at random one of the six cards in his bag.

Find the value of the number on the card that Andreia put into her bag.

Question 4 continued

Question 4 continued

Question 4 continued

D

Diagram NOT

accurately drawn

10.6 cm

C

E

34°

65°

B

9.2 cm

A

Figure 1

In Figure 1, ABCDE is a circle.

AB = 9.2 cm AD = 10.6 cm ∠BAD = 65° ∠CBD = 34°

Calculate the length, in cm to 3 significant figures, of BD.

D

Diagram NOT

accurately drawn

10.6 cm

C

E

34°

65°

B

9.2 cm

A

Figure 1

In Figure 1, ABCDE is a circle.

AB = 9.2 cm AD = 10.6 cm ∠BAD = 65° ∠CBD = 34°

Explain why ∠BCD = 115°

D

Diagram NOT

accurately drawn

10.6 cm

C

E

34°

65°

B

9.2 cm

A

Figure 1

In Figure 1, ABCDE is a circle.

AB = 9.2 cm AD = 10.6 cm ∠BAD = 65° ∠CBD = 34°

Calculate the length, in cm to 3 significant figures, of BC.

The point E is such that ΔBDE is isosceles, with DE = BE.

D

10.6 cm

C

E

34°

65°

B

9.2 cm

A

Figure 1

In Figure 1, ABCDE is a circle.

AB = 9.2 cm AD = 10.6 cm ∠BAD = 65° ∠CBD = 34°

Calculate the area, in cm² to 3 significant figures, of the quadrilateral BCDE.

Cosine rule

bc

A

a

b

c

=

−

2

cos

2

2

2

a

b

c

Sine rule

=

=

A

B

C

sin

sin

sin

1

a of triangle =

Are

2 ab

C

sin

Question 5 continued

Question 5 continued

Question 5 continued

The two functions, f and g, are defined for all values of x as

f : x ↦ 3x + 1

g : x ↦ x2 − 2

Find (i) g (−3) (ii) fg

The two functions, f and g, are defined for all values of x as

f : x ↦ 3x + 1

g : x ↦ x2 − 2

Write down the range of g

The two functions, f and g, are defined for all values of x as

f : x ↦ 3x + 1

g : x ↦ x2 − 2

Express the composite function gf in the form gf : x ↦ …

The function h is defined as

x

−

h : x ↦ 2

1

where x ≠ −3

x

3

The two functions, f and g, are defined for all values of x as

f : x ↦ 3x + 1

g : x ↦ x2 − 2

Solve the equation h (x) = 1

The two functions, f and g, are defined for all values of x as

f : x ↦ 3x + 1

g : x ↦ x2 − 2

Express the inverse function h−1 in the form h−1 : x ↦ …

The two functions, f and g, are defined for all values of x as

f : x ↦ 3x + 1

g : x ↦ x2 − 2

Write down the value of x that must be excluded from the domain of h−1

The two functions, f and g, are defined for all values of x as

f : x ↦ 3x + 1

g : x ↦ x2 − 2

Find the exact values of p for which

4gf (p) = fh−1

Show your working clearly.

Give your values in the form a

b

±

where a, b and c are integers.

c

b

b

ac

= −±

−

4

2

ax

bx

c

x

=

Solutions of

0

are

2

a

2

Question 6 continued

Question 6 continued

Question 6 continued

The points with coordinates (2, 2), (5, 2) and (4, −1) are the vertices of triangle A.

On the grid, draw and label triangle A.

Triangle A is transformed to triangle B under the transformation with matrix M where

1

3

2

M =

1

−

−

1

2

The points with coordinates (2, 2), (5, 2) and (4, −1) are the vertices of triangle A.

On the grid, draw and label triangle B.

Triangle B is transformed to triangle C under the transformation with matrix N where

1

1

−

2

2

N =

1

1

2

2

The points with coordinates (2, 2), (5, 2) and (4, −1) are the vertices of triangle A.

On the grid, draw and label triangle C.

Triangle B is the image of triangle C under the transformation with matrix P

The points with coordinates (2, 2), (5, 2) and (4, −1) are the vertices of triangle A.

Find the matrix P

Triangle A is transformed to triangle C under the transformation with matrix Q

The points with coordinates (2, 2), (5, 2) and (4, −1) are the vertices of triangle A.

Find the matrix Q

a

b

d

b

−

1

The inverse of matrix

is

c

d

c

a

−

ad

bc

−

Question 7 continued

y

−8

Turn over for a spare grid if you need to redraw your triangles.

Question 7 continued

Question 7 continued

Only use this grid if you need to redraw your triangles.

y

E

B

F

D

x cm

A

3x cm

K

H

J

(x − 2) cm

2x cm

Figure 2

Figure 2 shows a solid right pentagonal prism ABCDEFGHJK which is made by fixing

a solid right triangular prism ABCDEF onto a solid cuboid ACDFGHJK.

The triangle ABC is isosceles with BA = BC and the height of the triangle is x cm.

AH = FG = CJ = DK = 3x cm

AC = HJ = FD = GK = 2x cm

HG = JK = AF = CD = (x − 2) cm

The volume of the pentagonal prism is 1008 cm³

Show that x3 − 2x2 − 144 = 0

Given that f (x) = x3 − 2x2 − 144

E

B

F

D

x cm

A

3x cm

K

H

J

(x − 2) cm

2x cm

Figure 2

Figure 2 shows a solid right pentagonal prism ABCDEFGHJK which is made by fixing

a solid right triangular prism ABCDEF onto a solid cuboid ACDFGHJK.

The triangle ABC is isosceles with BA = BC and the height of the triangle is x cm.

AH = FG = CJ = DK = 3x cm

AC = HJ = FD = GK = 2x cm

HG = JK = AF = CD = (x − 2) cm

The volume of the pentagonal prism is 1008 cm³

use the factor theorem to show that (x − 6) is a factor of f (x)

E

B

F

D

x cm

A

3x cm

K

H

J

(x − 2) cm

2x cm

Figure 2

Figure 2 shows a solid right pentagonal prism ABCDEFGHJK which is made by fixing

a solid right triangular prism ABCDEF onto a solid cuboid ACDFGHJK.

The triangle ABC is isosceles with BA = BC and the height of the triangle is x cm.

AH = FG = CJ = DK = 3x cm

AC = HJ = FD = GK = 2x cm

HG = JK = AF = CD = (x − 2) cm

The volume of the pentagonal prism is 1008 cm³

Find the value of p, the value of q and the value of r so that

f (x) = (x − 6)(px2 + qx + r)

E

B

F

D

x cm

A

3x cm

K

H

J

(x − 2) cm

2x cm

Figure 2

Figure 2 shows a solid right pentagonal prism ABCDEFGHJK which is made by fixing

a solid right triangular prism ABCDEF onto a solid cuboid ACDFGHJK.

The triangle ABC is isosceles with BA = BC and the height of the triangle is x cm.

AH = FG = CJ = DK = 3x cm

AC = HJ = FD = GK = 2x cm

HG = JK = AF = CD = (x − 2) cm

The volume of the pentagonal prism is 1008 cm³

Hence explain why the equation f (x) = 0 has only one solution.

b

b

ac

= −±

−

4

2

ax

bx

c

x

=

Solutions of

0

are

2

a

2

Question 8 continued

Question 8 continued

Question 8 continued

The equation of a curve C is y = x (2x − 3)(x + 2)

The point P lies on C

The x coordinate of P is a

Find the range of values of a for which the gradient of C at P is at least −2

Show clear algebraic working.

Question 9 continued

F

C

B

Diagram NOT

accurately drawn

c

D

X

O

A

a

Figure 3

→

→

= a and OC

= c

Figure 3 shows a rectangle OABC in which OA

F is the midpoint of CB and D is the point on AB such that AD : DB = 2 : 3

Find

→

→

in terms of a (ii) AD

in terms of c

F

C

B

Diagram NOT

accurately drawn

D

X

O

A

Figure 3

→

= a and OC

= c

Figure 3 shows a rectangle OABC in which OA

F is the midpoint of CB and D is the point on AB such that AD : DB = 2 : 3

CF

The lines OD and AF intersect at the point X

= λOD

and AX

= μAF

, where λ and μ are scalars,

Given that OX

find the value of λ and the value of μ

Given that OX : XD = n : 1

F

C

B

Diagram NOT

accurately drawn

c

D

X

O

A

a

Figure 3

→

→

= a and OC

= c

Figure 3 shows a rectangle OABC in which OA

F is the midpoint of CB and D is the point on AB such that AD : DB = 2 : 3

find the value of n

Given also that | a | = 12 cm and | c | = 12.5 cm,

F

C

B

c

D

X

O

A

a

Figure 3

→

→

= a and OC

= c

Figure 3 shows a rectangle OABC in which OA

F is the midpoint of CB and D is the point on AB such that AD : DB = 2 : 3

find the area, in cm², of quadrilateral XDBF

Question 10 continued

Question 10 continued

Question 10 continued

Diagram NOT

accurately drawn

Figure 4

Figure 4 shows six identical circles inside a rectangle.

The radius of each circle is 12 cm.

The radius of the circles is the greatest possible radius so that the circles fit inside

the rectangle.

The six circles form the pattern shown in Figure 4 so that

•

each circle touches at least two other circles

•

the circle in the top row of the pattern and the circles in the bottom row of the

pattern touch at least one side of the rectangle

•

the centres of the circles all lie on the perimeter of a single triangle

Show that the total area of the six circles is almost 57.5% of the area of the rectangle.

Question 11 continued

Question 11 continued