4MB1/02 — June 2019 (Edexcel IGCSE Maths): Past Paper

Malcolm is investigating the ages of the 200 members of a theatre company. The table below gives information about the age group of each member of the company.

Malcolm is going to draw a pie chart for this information.

Calculate the size, in degrees, of the angle of the sector in the pie chart for those members of the company whose age group is $40 < t \le 50$

Calculate an estimate for the mean age, to the nearest year, of the members of the theatre company.

In 2017, country A had a population of 2.35 × 107 people. Of these people, 48% were male.

Calculate the number of males in country A in 2017 Country A is divided into three regions. These three regions are called East Region, Central Region and West Region. In 2017, the ratio of the number of males in the East Region to the number of males in the Central Region to the number of males in the West Region was 5 : 3 : 2

Calculate the number of males in the Central Region in 2017 Give your answer in standard form. In 2017, the number of females in the Central Region was 12.5% greater than the number of males in the Central Region.

Calculate the number of females in the Central Region in 2017 In 2010, country B had a population of 2.5 × 107 people. From 2010 to 2014, the population of country B increased by 2.4% From 2014 to 2018, the population of country B decreased by 2.4%

Calculate the population of country B in 2018

Diagram NOT accurately drawn C 8 cm D T O B 10 cm A Figure 1 Figure 1 shows the circle ABCD with centre O and diameter DC. The point T is such that TCOD is a straight line and TA is the tangent to the circle at A. AT = 10 cm TC = 8 cm

Calculate the radius, in cm, of the circle.

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

Triangle A is drawn on the grid opposite. The points (1, 2), (3, −2) and (1, 4) are the vertices of triangle B.

On the grid, draw and label triangle B.

Describe fully the single transformation that maps triangle A onto triangle B. Triangle C is the image of triangle B under an enlargement with scale factor 2 and centre of enlargement the origin.

On the grid, draw and label triangle C. Triangle D is the image of triangle C under the transformation with matrix M where ⎛ 1 ⎞ − 0 ⎜ 2 ⎟ M= 1 −1 2 ⎠

On the grid, draw and label triangle D. Triangle C is the image of triangle D under the transformation with matrix N.

Triangle C is the image of triangle D under the transformation with matrix N. Find the matrix N.

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

cm E D Figure 2 Figure 2 shows the triangles ABE and ACD where B lies on AC and E lies on AD. The triangles are similar with ·ABE = ·ACD. AB = 8 cm AE = 5 cm ·BAE = 30°

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

Calculate the size, in degrees to 3 significant figures, of ·ABE. The area of quadrilateral BCDE is 18.9

…………… cm²

Calculate the length, in cm to 3 significant figures, of CD. ⎡ ⎤ ⎢Cosine rule : a = b + c − 2bc cos A⎥ 2 2 2 ⎢ a b c ⎥ ⎢ Sine rule : = = ⎥ ⎢ sin A sin B sin C ⎥ ⎢ 1 ⎥ ⎢ Area of triangle = ab sin C ⎥ ⎢⎣ 2 ⎥⎦

A = {a, b, c, d} Write down all the subsets of A that contain the element b.

To pass a course, Preety has to pass two tests, test A and test B. Each test only has to be passed once. Passing test A and passing test B are independent events. Preety has three attempts to pass test A. The probability that she passes test A on her first attempt is If she fails on her first attempt, the probability that she passes on her second attempt is If she fails on her second attempt, the probability that she passes on her third attempt is

Calculate the probability she passes test A. Preety has two attempts to pass test B. The probability that she passes test B on her first attempt is The probability that she passes test B is

Calculate the probability that Preety passes test B on her second attempt, given that she fails test B on her first attempt. Preety decides to take the two tests only if the probability that she passes the course is greater than 0.5

State whether or not Preety should take the two tests. Give a reason for your answer.

C 5x m D Diagram NOT accurately drawn (5x + 3) m (2x + 3) m A 4x m B Figure 3 Figure 3 shows the plan for a lawn that is in the shape of a trapezium ABCD in which AB = 4x metres BC = (5x + 3) metres CD = 5x metres DA = (2x + 3) metres The perimeter of the lawn is P metres.

Find and simplify an expression for P in terms of x. The area of the lawn is A

…………… m²

Show that A = 14x2 + 12x The owner of the lawn wants the perimeter of the lawn to be greater than 52 m. He also wants the area of the lawn to be at most 162

…………… m²

Find the range of possible values of x. Show clear algebraic working. ⎡ 1 ⎤ ⎢⎣ Area of trapezium = 2 (a + b)h ⎥⎦

⎛ 2 x − 1⎞ ⎛ y + 3⎞ p=⎜ q= y − y 7⎞ The vectors p and q are such that |p| = 98 and p + q = 0⎠

Show that x² − 3x − 9 = 0 Given that x > 0

find the exact value of x …………… , ……………

show that y = 2 − 3√5

Find the exact value of |q|2 Show your working clearly. ⎡ 2 −b ± b 2 − 4ac ⎤ ⎢ Solutions of ax + bx + c = 0 are x = ⎥ ⎢⎣ 2a ⎥⎦

The curve C has equation y = 2x3 + ax2 + bx + 10 where a and b are constants. The point A, with coordinates (1, 3), lies on C.

Write down an equation in a and b. The tangent to C at the point A has gradient −8

Show that 2a + b = −14

Hence find the value of a and the value of b. Show clear algebraic working. Using the value of a and the value of b found in part (c) …………… , ……………

find the x coordinate of each of the points on C where the tangent to C is parallel to the line with equation y = 7 − 4x. Show clear algebraic working.

Diagram NOT accurately drawn h cm r cm Figure 4 A child’s toy is made by fixing a solid right circular cone, with base radius r cm and height h cm, on the flat circular face of a solid hemisphere of radius r cm. The centre of the base of the cone coincides with the centre of the hemisphere, as shown in Figure 4 Given that h + 6r = 15

find the upper bound for the value of r. Give a reason for your answer. The volume of the toy is V

…………… cm³

Show that V = (1/3)πr2(15 − 4r) ⎡ 1 2 ⎤ ⎢ Volume of cone = 3 πr h ⎥ ⎢ ⎥ ⎢ Volume of sphere = 4 πr 3 ⎥ ⎢⎣ 3 ⎥⎦ 1 2

Complete the following table of values for $V = \frac{1}{3}\pi r^2 (15 - 4r)$ Give your values of $V$ to 1 decimal place.

On the grid opposite, plot the points from your completed table and join them to form a smooth curve. The volume of one particular toy is 26 cm³

Use your curve to find, to one decimal place, the value of r. The manufacturer of the toy decides that the value of V should be twice the value of h.

By drawing a suitable straight line on the grid, find an estimate, to the nearest integer, for the value of V.