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

Diagram NOT

accurately drawn

(2x + 3) cm

x cm

(2x - 2) cm

(x + 3) cm

(x + 1) cm

Figure 1

Figure 1 shows a triangle and a rectangle.

The triangle has sides of length x cm, (2x - 2) cm and (x + 1) cm.

The rectangle has length (2x + 3) cm and width (x + 3) cm.

The perimeter of the rectangle is 3 times the perimeter of the triangle.

Write down an equation in x to represent the given information.

Diagram NOT

accurately drawn

(2x + 3) cm

x cm

(2x - 2) cm

(x + 3) cm

(x + 1) cm

Figure 1

Figure 1 shows a triangle and a rectangle.

The triangle has sides of length x cm, (2x - 2) cm and (x + 1) cm.

The rectangle has length (2x + 3) cm and width (x + 3) cm.

The perimeter of the rectangle is 3 times the perimeter of the triangle.

Solve your equation to find the value of x.

Show clear algebraic working.

Question 1 continued

Part of the curve with equation y = x2 - 3x - 1 is drawn on the grid on the opposite page.

The equation of another curve is y = –x2 - 2x + 4

Complete the table of values for y = –x2 - 2x + 4

Part of the curve with equation y = x2 - 3x - 1 is drawn on the grid on the opposite page.

The equation of another curve is y = –x2 - 2x + 4

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

smooth curve.

Part of the curve with equation y = x2 - 3x - 1 is drawn on the grid on the opposite page.

The equation of another curve is y = –x2 - 2x + 4

Use the two curves on the grid to find estimates, to 1 decimal place, for the solutions

of the equation 2x2 - x - 5 = 0

Question 2 continued

y

x

Figure 2

Figure 2 shows an empty tank in the shape of a right circular cylinder.

The axis of the cylinder is vertical.

Adelih is going to put water into the tank so that the surface of the water is level with the

top of the tank.

The water flows into the tank at a constant rate of R litres per minute.

Given that R = 109

calculate, in hours and minutes to the nearest minute, the time taken by the water to

flow into the tank.

Adelih wants the time taken by the water to flow into the tank to be 3 hours.

Figure 2

Figure 2 shows an empty tank in the shape of a right circular cylinder.

The axis of the cylinder is vertical.

Adelih is going to put water into the tank so that the surface of the water is level with the

top of the tank.

The water flows into the tank at a constant rate of R litres per minute.

Given that R = 109

Calculate the value, to the nearest whole number, of R.

$\left[1\text{ m}^3 = 1000 \text{ litres} \quad \text{Volume of cylinder} = \pi r^2 h\right]$

The functions f and g are defined as

f : x 6 3x + 4

5

g : x 6

x −

3

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

The functions f and g are defined as

f : x 6 3x + 4

5

g : x 6

x −

3

Find f

The functions f and g are defined as

f : x 6 3x + 4

5

g : x 6

x −

3

Find fg

The function h is defined as

h : x 6 x2 - 6x - 8 where x . 3

The functions $f$ and $g$ are defined as

$f : x \mapsto 3x + 4$

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

The function $h$ is defined as $h : x \mapsto x^2 - 6x - 8$ where $x \ge 3$

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

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

Solve the inequality 5(x + 1) < x

Show clear algebraic working.

Solve the simultaneous equations

3x2 + y2 - 7 = 0

y - 3x - 5 = 0

Show clear algebraic working.

Hence find the value of x for which

5(x + 1) < x and 3x2 + y2 - 7 = 0 and y - 3x - 5 = 0

Question 5 continued

The curve C has equation y = 2x3 + 3x2 - 12x + 1

The point P and the point Q are the turning points on C.

Find an equation of the straight line that passes through P and Q.

Question 6 continued

Mr Ng is the head of mathematics at a school.

He asked the students in Year 9 which of a ruler (R), a protractor (P) and a calculator (C)

they each had.

Of these students

10 students had a ruler, a protractor and a calculator

25 students had a ruler and a protractor

18 students had a protractor and a calculator

17 students had a ruler and a calculator

54 students had a ruler

49 students had a protractor

29 students had a calculator

8 students did not have a ruler or a protractor or a calculator.

Show all this information in the Venn diagram.

E

R

P

C

Mr Ng is the head of mathematics at a school.

He asked the students in Year 9 which of a ruler (R), a protractor (P) and a calculator (C)

they each had.

Of these students

10 students had a ruler, a protractor and a calculator

25 students had a ruler and a protractor

18 students had a protractor and a calculator

17 students had a ruler and a calculator

54 students had a ruler

49 students had a protractor

29 students had a calculator

8 students did not have a ruler or a protractor or a calculator.

Find the number of students in Year 9

Mr Ng is the head of mathematics at a school.

He asked the students in Year 9 which of a ruler (R), a protractor (P) and a calculator (C)

they each had.

Of these students

10 students had a ruler, a protractor and a calculator

25 students had a ruler and a protractor

18 students had a protractor and a calculator

17 students had a ruler and a calculator

54 students had a ruler

49 students had a protractor

29 students had a calculator

8 students did not have a ruler or a protractor or a calculator.

Find n([R ∪ P]a)

One of the students in Year 9 is chosen at random.

Mr Ng is the head of mathematics at a school.

He asked the students in Year 9 which of a ruler (R), a protractor (P) and a calculator (C)

they each had.

Of these students

10 students had a ruler, a protractor and a calculator

25 students had a ruler and a protractor

18 students had a protractor and a calculator

17 students had a ruler and a calculator

54 students had a ruler

49 students had a protractor

29 students had a calculator

8 students did not have a ruler or a protractor or a calculator.

Given that this student had a calculator, find the probability that this student had a

ruler but not a protractor.

Question 7 continued

y

x

Figure 3

Figure 3 shows a triangle, A, drawn on a grid.

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

$\mathbf{P} = \begin{pmatrix} 2 & -1 \\ \tfrac{3}{2} & \tfrac{1}{2} \end{pmatrix}$

On the grid in Figure 3, draw and label triangle B.

y

x

Figure 4

Figure 4 shows triangle A drawn on a grid.

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

$\mathbf{Q} = \begin{pmatrix} 3 & -4 \\ 1 & -2 \end{pmatrix}$

y

x

Figure 3

Figure 3 shows a triangle, A, drawn on a grid.

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

$\mathbf{P} = \begin{pmatrix} 2 & -1 \\ \tfrac{3}{2} & \tfrac{1}{2} \end{pmatrix}$

On the grid in Figure 4, draw and label triangle C.

Triangle C is the image of triangle A under a single transformation.

y

x

Figure 3

Figure 3 shows a triangle, A, drawn on a grid.

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

$\mathbf{P} = \begin{pmatrix} 2 & -1 \\ \tfrac{3}{2} & \tfrac{1}{2} \end{pmatrix}$

Describe fully this transformation.

Turn over for a spare copy of Figure 3 and Figure 4 if you need to redraw your triangles.

Only use these grids if you need to redraw your triangles.

y

x

Figure 3

y

x

Figure 4

One Saturday, each of the 100 people who visited a library was asked how long they

were in the library.

The table below shows information about the results.

Calculate an estimate for the mean length of time, in minutes to 3 significant figures,

these people were in the library.

Two of the 100 people who visited the library that Saturday are picked at random.

One Saturday, each of the 100 people who visited a library was asked how long they

were in the library.

The table below shows information about the results.

Find, to 3 decimal places, the probability that

both people were in the library for more than 30 minutes,

One Saturday, each of the 100 people who visited a library was asked how long they

were in the library.

The table below shows information about the results.

one of the two people was in the library for more than 30 minutes and one was in

the library for 30 minutes or less.

Question 9 continued

Zahur made 250 cakes to sell at a cake sale.

Of the cakes made by Zahur, 28% were chocolate cakes.

Calculate the number of chocolate cakes made by Zahur.

All the other cakes made by Zahur were either lemon cakes or vanilla cakes.

The ratio of the number of lemon cakes to the number of vanilla cakes was 4 : 5

Zahur made 250 cakes to sell at a cake sale.

Of the cakes made by Zahur, 28% were chocolate cakes.

Calculate the number of lemon cakes made by Zahur.

Zahur put icing on each of the vanilla cakes he made.

The icing for each vanilla cake needed 75 g of icing sugar.

Zahur made 250 cakes to sell at a cake sale.

Of the cakes made by Zahur, 28% were chocolate cakes.

Calculate the total amount, in kg, of icing sugar needed for all the vanilla cakes made

by Zahur.

At the start of the cake sale, the selling price of each of the cakes made by Zahur was $4

and he sold 204 cakes at this price.

Zahur then reduced the selling price of each cake by 30% and he sold all the remaining

cakes.

Zahur made 250 cakes to sell at a cake sale.

Of the cakes made by Zahur, 28% were chocolate cakes.

Calculate the total amount of money, in $, that Zahur received by selling all 250 cakes.

When Zahur had subtracted the cost of all the ingredients he needed to make his cakes

from the total amount of money he received by selling all the cakes, he found that he had

made a profit of 60%

Zahur made 250 cakes to sell at a cake sale.

Of the cakes made by Zahur, 28% were chocolate cakes.

Calculate, in $, the cost of all the ingredients Zahur needed.

Question 10 continued

Question 10 continued

Question 10 continued

K

Diagram NOT

F

accurately drawn

J

O

D

C

E

N

G

H

6 cm

8 cm

A

B

M

Figure 5

Figure 5 shows a right prism ABCDEFGHJK.

A cross section of the prism is a regular pentagon with sides of length 6 cm.

BH = 8 cm.

M is the midpoint of AB.

N is the midpoint of GH.

O is the centre of pentagon FGHJK.

Find, in cm to 3 significant figures, the length of AO.

K

Diagram NOT

F

accurately drawn

J

O

D

C

E

N

G

H

6 cm

8 cm

A

B

M

Figure 5

Figure 5 shows a right prism ABCDEFGHJK.

A cross section of the prism is a regular pentagon with sides of length 6 cm.

BH = 8 cm.

M is the midpoint of AB.

N is the midpoint of GH.

O is the centre of pentagon FGHJK.

Calculate the size, in degrees to 1 decimal place, of the angle between MK and MN.

Question 11 continued

Question 11 continued

Question 11 continued

A

Diagram NOT

P

accurately drawn

14 cm

8 cm

R

O

C

B

Q

12 cm

Figure 6

Figure 6 shows a triangle ABC and a circle PQR, centre O. The triangle is such that side

AB is the tangent to the circle at P, side BC is the tangent to the circle at Q and side AC is

the tangent to the circle at R. The region inside the circle is shaded, as shown in Figure 6.

AB = 14 cm, BC = 12 cm and AC = 8 cm.

Let BP = x cm and by considering the lengths of the tangents to the circle,

obtain an equation in x only and solve it to find the length, in cm, of BP.

A

Diagram NOT

P

accurately drawn

14 cm

8 cm

R

O

C

B

Q

12 cm

Figure 6

Figure 6 shows a triangle ABC and a circle PQR, centre O. The triangle is such that side

AB is the tangent to the circle at P, side BC is the tangent to the circle at Q and side AC is

the tangent to the circle at R. The region inside the circle is shaded, as shown in Figure 6.

AB = 14 cm, BC = 12 cm and AC = 8 cm.

Let BP = x cm and by considering the lengths of the tangents to the circle,

Find, to 3 significant figures, the area of the circle as a percentage of the total area of

triangle ABC.

$\left[\text{Cosine rule: } a^2 = b^2 + c^2 - 2bc\cos A \quad \text{Area of triangle} = \tfrac{1}{2}ab\sin C\right]$