4MB1/02R — January 2021 (Edexcel IGCSE Maths): Past Paper

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

Figure not shown in this reconstruction — the printed paper includes a diagram (not drawn accurately). Use the text description to work through it.

A ∩ C

(A ∪ B ∪ C)′ Find

n ([A ∪ B]′)

n ([A ∩ B] ∪ C) A number is selected at random from ℰ

Find the probability that the number is in set B E A B C for a spare copy of the Venn diagram. Only use this Venn diagram if you wish to replace your answer to part (a) E A B C

Figure not shown in this reconstruction — the printed paper includes a diagram (not drawn accurately). Use the text description to work through it.

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

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

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

Calculate the percentage profit made in 2019 Give your answer to 1 decimal place.

The equation of a curve is $y = -2x^3 + 6x + \frac{5}{x^2}$

Complete the table of values for $y = -2x^3 + 6x + \frac{5}{x^2}$ Give your values of $y$ to 2 decimal places where necessary.

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

Use your graph to find an estimate, to 2 decimal places, for the minimum value of −2x3 + 6x + 2 for values of x in −2 ≤ x ≤ −0.8

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.

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.

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.

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.

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

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

D Diagram NOT accurately drawn 10.6 cm C E 34° 65° B A 9.2 cm 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.

Explain why ∠BCD = 115°

Calculate the length, in cm to 3 significant figures, of BC. The point E is such that ΔBDE is isosceles, with DE = BE.

Calculate the area, in cm² to 3 significant figures, of the quadrilateral BCDE. ⎡ ⎤ ⎢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 ⎦⎥

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 (1)

Write down the range of g

Express the composite function gf in the form gf : x ↦ … The function h is defined as 2x − 1 h:xα where x ≠ −3 x+3

Solve the equation h (x) = 1

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

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

Find the exact values of p for which 4gf ( p) = fh−1 (1) Show your working clearly. a± b Give your values in the form where a, b and c are integers. c ⎡ −b ± b 2 − 4ac ⎤ ⎢Solutions of ax 2 + bx + c = 0 are x = ⎥ ⎣ 2a ⎦

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 2⎟ M= −1 − 1 2⎠

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⎠

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

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

Find the matrix Q.

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

E B x cm F D A C 3x cm G 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

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

Find the value of p, the value of q and the value of r so that f (x) = (x − 6)( px2 + qx + r)

Hence explain why the equation f (x) = 0 has only one solution. ⎡ −b ± b 2 − 4ac ⎤ ⎢Solutions of ax + bx + c = 0 are x = ⎥ ⎣ 2a ⎦

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.

C F B Diagram NOT accurately drawn c X D O A a Figure 3 → → Figure 3 shows a rectangle OABC in which OA = a and OC = c F is the midpoint of CB and D is the point on AB such that AD : DB = 2 : 3

CF in terms of a (ii) AD in terms of c The lines OD and AF intersect at the point X → → → → Given that OX = λOD and AX = μAF , where λ and μ are scalars …………… , ……………

find the value of λ and the value of μ Given that OX : XD = n : 1

find the value of n Given also that | a | = 12 cm and | c | = 12.5 cm …………… , ……………

find the area, in cm², of quadrilateral XDBF

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.