4MA1 Mathematics A: Oct/Nov 2020 Paper 1HRHigher Tier Paper 1

Show that 3 34 × 79 = 2 1112[3]

Using ruler and compasses only, construct the bisector of angle ABC. You must show all your construction lines.

[2]

(a)Simplify h⁷× h²

[1]

G = c²– 4c

(b)Find the value of G when c = –5

G = [2]

(c)Solve 5x − 34 = 2x + 3

Show clear algebraic working.

x = [3]

The table gives information about the length of time, in minutes, that each of 60 students took to travel to school on Monday.

(a)Write down the modal class interval.

[1]

(b)Work out an estimate for the mean length of time taken by these 60 students to travel to school on Monday. Give your answer correct to one decimal place.

_minutes[4]

In 2017, the population of a village was 7500 In 2019, the population of the village was 8265

(a)Work out the percentage increase in the population of the village from 2017 to 2019

%[3]

In a sale, normal prices are reduced by 30% The sale price of a T‑shirt was 31.50 euros.

(b)Work out the normal price of the T‑shirt.

_euros[3]

Point A has coordinates (–3, 11) Point B has coordinates (47, b) The midpoint of AB has coordinates (a, –19)

Find the value of a and the value of b.

a =

b = [2]

Pedro drove from Toulouse to Montpellier in 2 hours 42 minutes. He drove at an average speed of 90 km/hour.

Janine drove from Toulouse to Montpellier along the same route as Pedro. The journey took her 3 hours.

Work out Janine’s average speed for the journey.

_km/hour[4]

Harold bought an antique clock for £1200 The clock increased in value by 8% per year.

Find the value of the clock exactly 3 years after Harold bought the clock. Give your answer correct to the nearest £.

£ [3]

A box is put on a horizontal table.

The face of the box in contact with the table is a square of side 1.5 metres. The pressure on the table due to the box is 34.8 newtons/m²

Work out the force exerted by the box on the table. pressure =

_newtons[3]

Alex makes 80 cakes to sell. He makes chocolate cakes, lemon cakes and fruit cakes where

all of the chocolate cakes

34 of the lemon cakes

78 of the fruit cakes

The profit he makes on each cake he sells is shown in the table.

Work out the total profit that Alex makes from the cakes he sells.

£ [5]

The frequency table gives information about the ages of the 80 people in a train carriage.

(a)Complete the cumulative frequency table.

[1]

(b)On the grid, draw a cumulative frequency graph for your table.

80

70

60

50

[2]

(c)Use your graph to find an estimate for the median age of the people in the train carriage.

_years[2]

Solve the simultaneous equations

Show clear algebraic working.

x =

y = [4]

The curve C has equation y = 5x³– x²– 6x + 4

(a)Find dydx

dydx = [2]

There are two points on the curve C at which the gradient of the curve is 2

(b)Find the x coordinate of each of these two points. Show clear algebraic working.

[4]

Expand and simplify (4x + 1)(x – 3)(5x + 6)

[3]

B, D, E and F are points on a circle. ABC is the tangent to the circle at B.

Angle EDF = 40° Angle FBC = 70°

Prove that the tangent ABC is parallel to EF. Give a reason for each stage of your working.[4]

The functions f and g are defined as

(a)Write down the value of x that must be excluded from any domain of g

[1]

(b)Find gf(2.6)

[2]

(c)Solve fg(x) = 2

x = [3]

(d)Express the inverse function g^–1 in the form g^–1 : x  ...

g^–1 : x  [3]

The diagram shows a prism ABCDEFGH with a horizontal base.

The base of the prism, EFGH, is a square of side 12 cm. Trapezium ADEF is a cross section of the prism where AF and DE are vertical edges.

DE = CH = 10 cm AD = BC = 15 cm

(a)Work out the size of the angle between CF and the base EFGH. Give your answer correct to one decimal place.

[3]

(b)Work out the length of BE. Give your answer correct to one decimal place.

cm[3]

In an arithmetic series, the 6th term is 39 In the same arithmetic series, the 19th term is 7.8

Work out the sum of the first 25 terms of the arithmetic series.

[4]

The diagram shows rectangle ABCD with rectangle EFGH cut out to form the shaded region.

AD = 8.3 cm correct to one decimal place DC = 7.2 cm correct to one decimal place EH = 6.2 cm correct to one decimal place HG = 5.3 cm correct to one decimal place

Work out the upper bound of the area of the shaded region. Show your working clearly.

_cm²[3]

A curve has equation y = f(x) There is only one maximum point on the curve. The coordinates of this maximum point are (–3, 4)

Write down the coordinates of the maximum point on the curve with equation

(i)y = f(x) – 6

( , )

(ii)y = f(2x)

( , )

[2]

Given that M =₂

find the values of n for which M = 2

[5]

The diagram shows a regular octagon ABCDEFGH.

Each side of the octagon has length 10 cm.

Find the area of the shaded region ACDEH. Give your answer correct to the nearest cm²

cm²[6]

In a bag, there are only

Jean is going to take at random two beads from the bag.

The probability that Jean will take two beads of the same colour is 38

Find the total number of beads in the bag. Show clear algebraic working.

[4]