June 2018 · Paper 2 (4MB1/02) (Edexcel IGCSE Maths): Exam Questions

For concerts in a theatre, people can buy tickets in the Stalls, in the Dress Circle and in

the Balcony.

The number of seats in the Stalls is S, the number of seats in the Dress Circle is D and

the number of seats in the Balcony is B where S : D : B = 12 : 9 : 4

Given that B = 600

find the total number of seats in the theatre.

The following table gives information about the cost of tickets and the number of tickets

sold for a concert held in the theatre in 2017

For concerts in a theatre, people can buy tickets in the Stalls, in the Dress Circle and in

the Balcony.

The number of seats in the Stalls is S, the number of seats in the Dress Circle is D and

the number of seats in the Balcony is B where S : D : B = 12 : 9 : 4

Given that B = 600

Calculate the total amount of money, in $, paid for all the tickets sold for this concert.

For a concert held in 2018, the total amount of money paid for all the tickets sold

was $175 000

For this concert in 2018, the costs of each ticket for a seat in the Stalls, in the

Dress Circle and in the Balcony were the same as the costs in 2017

Also for this concert, the number of tickets sold for seats in the Stalls was the same as in

2017 and the number of tickets sold for seats in the Dress Circle was the same as in 2017

For concerts in a theatre, people can buy tickets in the Stalls, in the Dress Circle and in

the Balcony.

The number of seats in the Stalls is S, the number of seats in the Dress Circle is D and

the number of seats in the Balcony is B where S : D : B = 12 : 9 : 4

Given that B = 600

Calculate the number of tickets sold for seats in the Balcony for the concert in 2018

Question 1 continued

Question 1 continued

Question 1 continued

E

11 cm

D

C

O

5 cm

A

B

12 cm

Figure 1

Figure 1 shows a rectangular based right pyramid, ABCDE. The point O is the centre of

the rectangular base so that OE is the perpendicular height of the pyramid.

AB = 12 cm and BC = 5 cm

AE = BE = CE = DE = 11 cm

Calculate the exact volume of the pyramid.

Give your answer in the form $p\sqrt{q}$ cm³ where $p$ and $q$ are integers.

$\left[\text{Volume of pyramid} = \tfrac{1}{3} \times \text{base area} \times \text{height}\right]$

$A = \begin{pmatrix} a & 3 \\ 4 & -1 \end{pmatrix} \quad B = \begin{pmatrix} 13 & 3 \\ 4 & 10 \end{pmatrix}$

Given that $A^2 - B = \lambda I$ where $\lambda$ is an integer and $I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$

find the value of $a$ and the value of $\lambda$

The Venn diagram shows information about the elements in a universal set, E, and the

three sets X, Y and Z.

E

X

Y

a

h

f

b

g

c

l

i

k

j

d

e

m

n

Z

Using the information in the Venn diagram, write down the elements in

X ˆ
Y

The Venn diagram shows information about the elements in a universal set, E, and the

three sets X, Y and Z.

E

X

Y

a

h

f

b

g

c

l

i

k

j

d

e

m

n

Z

Using the information in the Venn diagram, write down the elements in

(X ‰ Y ) ˆ
=ү

Find

The Venn diagram shows information about the elements in a universal set, E, and the

three sets X, Y and Z.

E

X

Y

a

h

f

b

g

c

l

i

k

j

d

e

m

n

Z

Using the information in the Venn diagram, write down the elements in

n([X ‰ Y ]ү)

The Venn diagram shows information about the elements in a universal set, E, and the

three sets X, Y and Z.

E

X

Y

a

h

f

b

g

c

l

i

k

j

d

e

m

n

Z

Using the information in the Venn diagram, write down the elements in

n([X ˆ
Z] ‰<ү)

Here are five symbols used in connection with sets.

   ‡         ˆ

The Venn diagram shows information about the elements in a universal set, E, and the

three sets X, Y and Z.

E

X

Y

a

h

f

b

g

c

l

i

k

j

d

e

m

n

Z

Using the information in the Venn diagram, write down the elements in

Use one of these symbols to complete the following statement correctly

{d, e} .................................. Z

A letter is chosen at random from E

The Venn diagram shows information about the elements in a universal set, E, and the

three sets X, Y and Z.

E

X

Y

a

h

f

b

g

c

l

i

k

j

d

e

m

n

Z

Using the information in the Venn diagram, write down the elements in

Find the probability that this letter is in set Y given that it is in set X.

Question 4 continued

f and g are two functions such that

f : x 6 3 + 5x where x < 0

g : x 6 5

4

x

−

3

x

2

−

Find the value of a for which f(a) = 0

f and g are two functions such that

f : x 6 3 + 5x where x < 0

g : x 6 5

4

x

−

3

x

2

−

Write down the range of f

f and g are two functions such that

f : x 6 3 + 5x where x < 0

g : x 6 5

4

x

−

3

x

2

−

State the value of x that must be excluded from any domain of g

f and g are two functions such that

f : x 6 3 + 5x where x < 0

g : x 6 5

4

x

−

3

x

2

−

Find the value of x for which f(x) = g(x)

Show your working clearly.

Give your answer to 3 significant figures.

$f$ and $g$ are two functions such that

$f : x \mapsto 3 + 5x$ where $x < 0$

$g : x \mapsto \dfrac{5 - 4x}{3x - 2}$

Express the inverse function $g^{-1}$ in the form $g^{-1} : x \mapsto \ldots$

$\left[\text{Solutions of } ax^2 + bx + c = 0 \text{ are } x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}\right]$

The six numbers

x y y 12 (x + y) 12

are in ascending order of size.

The mode of the six numbers is 12 and the mean of the six numbers is 9.5

Find the value of x and the value of y.

The six numbers

x y y 12 (x + y) 12

are in ascending order of size.

The mode of the six numbers is 12 and the mean of the six numbers is 9.5

Hence find the median of the six numbers.

Question 6 continued

The points (1, 1), (1, 3) and (0, 3) are the vertices of triangle A.

On the grid, draw and label triangle A.

Triangle A is transformed to triangle B by a reflection in the line with equation x = –2

The points (1, 1), (1, 3) and (0, 3) are the vertices of triangle A.

On the grid, draw and label triangle B.

Triangle B is transformed to triangle C under the translation $\binom{1}{-3}$

The points (1, 1), (1, 3) and (0, 3) are the vertices of triangle A.

On the grid, draw and label triangle C.

Triangle C is transformed to triangle D under the transformation with matrix M where

Triangle C is transformed to triangle D under the transformation with matrix $\mathbf{M}$ where

$\mathbf{M} = \begin{pmatrix} -2 & 0 \\ 0 & -2 \end{pmatrix}$

The points (1, 1), (1, 3) and (0, 3) are the vertices of triangle A.

On the grid, draw and label triangle D.

The points (1, 1), (1, 3) and (0, 3) are the vertices of triangle A.

Find the inverse matrix, M–1, of the matrix M

The points (1, 1), (1, 3) and (0, 3) are the vertices of triangle A.

Describe fully the single transformation represented by the matrix $\mathbf{M}^{-1}$

$\left[\text{The inverse of matrix } \begin{pmatrix} a & b \\ c & d \end{pmatrix} \text{ is } \tfrac{1}{ad-bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}\right]$

y

x

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

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

y

x

The equation of a curve is y

2

2

3

x

(

x

1

)

x

11

x

=

−

−

2

3

3

The point A on the curve has coordinates (p, q)

The gradient of the curve at A is 5

Show that p is a root of the equation

2p3 - 5p2 - p + 6 = 0

The equation of a curve is y

2

2

3

x

(

x

1

)

x

11

x

=

−

−

2

3

3

The point A on the curve has coordinates (p, q)

The gradient of the curve at A is 5

Show that (2p - 3) is a factor of 2p3 - 5p2 - p + 6

The equation of a curve is y

2

2

3

x

(

x

1

)

x

11

x

=

−

−

2

3

3

The point A on the curve has coordinates (p, q)

The gradient of the curve at A is 5

Hence factorise completely 2p3 - 5p2 - p + 6

Given that p > 3

2

The equation of a curve is y

2

2

3

x

(

x

1

)

x

11

x

=

−

−

2

3

3

The point A on the curve has coordinates (p, q)

The gradient of the curve at A is 5

find the value of q.

Question 8 continued

Question 8 continued

Question 8 continued

In Figure 2, $\overrightarrow{OA} = \mathbf{a}$, $\overrightarrow{OB} = \mathbf{b}$, $\overrightarrow{FE} = \lambda\mathbf{b}$ and $\overrightarrow{ED} = \mu\mathbf{a}$, where $\lambda$ and $\mu$ are positive constants.

$B$ is the point of intersection of $OC$ and $AF$ such that $OB : OC = AB : AF = 1 : 3$

Find, in terms of $\mathbf{a}$ or $\mathbf{b}$ or $\mathbf{a}$ and $\mathbf{b}$, simplifying your answers where possible,

(i) $\overrightarrow{AB}$

In Figure 2, $\overrightarrow{OA} = \mathbf{a}$, $\overrightarrow{OB} = \mathbf{b}$, $\overrightarrow{FE} = \lambda\mathbf{b}$ and $\overrightarrow{ED} = \mu\mathbf{a}$, where $\lambda$ and $\mu$ are positive constants.

$B$ is the point of intersection of $OC$ and $AF$ such that $OB : OC = AB : AF = 1 : 3$

Find, in terms of $\mathbf{a}$ or $\mathbf{b}$ or $\mathbf{a}$ and $\mathbf{b}$, simplifying your answers where possible,

(ii) $\overrightarrow{CF}$

In Figure 2, $\overrightarrow{OA} = \mathbf{a}$, $\overrightarrow{OB} = \mathbf{b}$, $\overrightarrow{FE} = \lambda\mathbf{b}$ and $\overrightarrow{ED} = \mu\mathbf{a}$, where $\lambda$ and $\mu$ are positive constants.

$B$ is the point of intersection of $OC$ and $AF$ such that $OB : OC = AB : AF = 1 : 3$

Find, in terms of $\mathbf{a}$, $\mathbf{b}$, $\lambda$ and where necessary $\mu$, simplifying your answers where possible,

(ii) $\overrightarrow{AE}$

Given that $\overrightarrow{AE} = 4\overrightarrow{CD}$

In Figure 2, $\overrightarrow{OA} = \mathbf{a}$, $\overrightarrow{OB} = \mathbf{b}$, $\overrightarrow{FE} = \lambda\mathbf{b}$ and $\overrightarrow{ED} = \mu\mathbf{a}$, where $\lambda$ and $\mu$ are positive constants.

$B$ is the point of intersection of $OC$ and $AF$ such that $OB : OC = AB : AF = 1 : 3$

find the value of $\mu$ and the value of $\lambda$.

Given also that $|\mathbf{a}| = 4$ cm, $|\mathbf{b}| = 1$ cm and that the area of the trapezium $CDEF$ is 5 cm²

$\mu$a

E

D

$\lambda$b

F

C

B

b

O

A

a

Figure 2

→ = a, OB

→ = b, FE

→ = $\lambda$b and ED

→ = $\mu$a, where $\lambda$ and $\mu$ are positive constants.

In Figure 2, OA

B is the point of intersection of OC and AF such that OB : OC = AB : AF = 1 : 3

calculate the size, in degrees to 3 significant figures, of ∠CFE.

$\left[\text{Area of trapezium} = \tfrac{1}{2}(a+b)h\right]$

The equation of a curve is given by

8

2

y

x

5

x

=

−

x

Complete the table of values for y, giving your values of y to 2 decimal places where

necessary.

The equation of a curve is given by

8

2

y

x

5

x

=

−

x

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

smooth curve.

The equation of a curve is given by

8

2

y

x

5

x

=

−

x

By drawing a suitable straight line on the grid, find an estimate, to 1 decimal

place, for the range of values of x in 0.8 - x - 6 for which 4x3 - 21x2 - 12x + 32 0

Question 10 continued

y

8 –

x

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

Question 10 continued

Question 10 continued

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

y

8 –

x

A bag contains 6 five cent coins and 9 ten cent coins.

Coins are taken at random from the bag, one at a time without replacement, until the total

value of the coins taken from the bag is at least 15 cents.

No more coins are then taken from the bag.

Complete the probability tree diagram.

First coin

Second coin

Third coin

5 cent

6

15

9

15

10 cent

A bag contains 6 five cent coins and 9 ten cent coins.

Coins are taken at random from the bag, one at a time without replacement, until the total

value of the coins taken from the bag is at least 15 cents.

No more coins are then taken from the bag.

Calculate the probability that exactly two coins are taken from the bag.

A is the event that the total value of the coins taken from the bag is 15 cents.

B is the event that the total value of the coins taken from the bag is 20 cents.

A bag contains 6 five cent coins and 9 ten cent coins.

Coins are taken at random from the bag, one at a time without replacement, until the total

value of the coins taken from the bag is at least 15 cents.

No more coins are then taken from the bag.

Determine which of the events, A or B, is the more likely.

Show your working clearly.

Question 11 continued

Question 11 continued