4MA1 Mathematics A: January 2021 Paper 1HHigher Tier Paper 1

Pieter owns a currency conversion shop.

Last Monday, Pieter changed a total of 20 160 rand into a number of different currencies.

He changed 310 of the 20 160 rand into euros.

He changed the rest of the rands into dollars, rupees and francs in the ratios 9 : 5 : 2

Pieter changed more rands into dollars than he changed into francs.

Work out how many more.

_rand[4]

The table gives information about the speeds, in kilometres per hour, of 80 motorbikes as each pass under a bridge.

Speed

(a)Write down the modal class.

[1]

(b)Work out an estimate for the mean speed of the motorbikes as they pass under the bridge. Give your answer correct to 3 significant figures.

_{kilometresperhour}[4]

The diagram shows a container for water in the shape of a prism.

The rectangular base of the prism, shown shaded in the diagram, is horizontal. The container is completely full of water.

Tuah is going to use a pump to empty the water from the container so that the volume of water in the container decreases at a constant rate.

The pump starts to empty water from the container at 10 30 and at 12 00 the water level in the container has dropped by 20 cm.

Find the time at which all the water has been pumped out of the container.

[4]

E = {20, 21, 22, 23, 24, 25, 26, 27, 28, 29}

A = {odd numbers} B = {multiples of 3}

List the members of the set

(i)A ∩ B

[1]

(ii)A ∪ B

[1]

(a)Factorise fully 15y⁴+ 20uy³

[2]

(b)Solve 4 – 3x = 5 − 8x4

Show clear algebraic working.

x = [3]

(a)Write 2 840 000 000 in standard form.

[1]

(b)Write 2.5 × 10^–4 as an ordinary number.

[1]

Chen invests 40 000 yuan in a fixed-term bond for 3 years.

The fixed-term bond pays compound interest at a rate of 3.5% each year.

(a)Work out the value of Chen’s investment at the end of 3 years. Give your answer to the nearest yuan.

_yuan[3]

Wang invested P yuan. The value of his investment decreased by 6.5% each year.

At the end of the first year, the value of Wang’s investment was 30 481 yuan.

(b)Work out the value of P.

P = [3]

The region, shown shaded in the diagram, is a path.

2 m accurately drawn

The boundary of the path is formed by two semicircles, with the same centre O, and two straight lines.

The inner semicircle has a radius of 7 metres. The path has a width of 2 metres.

Work out the perimeter of the path. Give your answer correct to one decimal place.

_m[3]

(a)Simplify (2x³ y⁵)⁴

[2]

(b)(i)Factorise x²+ 5x – 36

[2]

(ii)Hence, solve x²+ 5x – 36 = 0

[1]

Here is isosceles triangle ABC.

D is the midpoint of AC and DB = 16 cm.

Angle DAB = 65°

Work out the perimeter of triangle ABC. Give your answer correct to one decimal place.

cm[4]

The cumulative frequency graph gives information about the weights, in grams, of 90 bags of sweets.

90

80

70

60

(a)Find an estimate for the median of the weights of these bags of sweets.

_grams[2]

Roberto sells the bags of sweets to raise money for charity. Bags with a weight greater than d grams are labelled large bags and sold for 3.75 euros each bag.

The total amount of money he receives by selling all the large bags is 93.75 euros.

(b)Find the value of d.

d = [3]

(a)Express 4x − 2 − 3x + 1 as a single fraction.

Give your answer in its simplest form.

[3]

(b)Expand and simplify 2x(x – 5)(x – 3)

[3]

Point A has coordinates (5, 8) Point B has coordinates (9, –4)

(a)Work out the gradient of AB.

[2]

The straight line L has equation y = –4x + 5

(b)Write down the gradient of a straight line that is perpendicular to L.

[1]

Ding is going to play one game of snooker against each of two of his friends, Marco and Judd.

The probability tree diagram gives information about the probabilities that Ding will win or lose each of these two games.

(a)Work out the probability that Ding will win both games.

[2]

(b)Work out the probability that Ding will win exactly one of the games.

[3]

v = 9.6 correct to 1 decimal place u = 3.8 correct to 1 decimal place t = 1.84 correct to 2 decimal places

Calculate the upper bound for the value of a. Give your answer as a decimal correct to 2 decimal places. Show your working clearly.

[3]

The diagram shows the positions of three ships, A, B and C.

Ship B is due north of ship A.

The bearing of ship C from ship A is 120°

Calculate the bearing of ship C from ship B. Give your answer correct to the nearest degree.

[5]

A solid, S, is made from a hemisphere and a cylinder.

The centre of the circular face of the hemisphere and the centre of the top face of the cylinder are at the same point.

(20 – 4x) cm

The radius of the cylinder and the radius of the hemisphere are both x cm. The height of the cylinder is (20 – 4x) cm.

The volume of S is V cm³ where V = 13 πy

Find the maximum value of y. Show clear algebraic working.

[5]

Given that (8 – x )(5 + x ) = y x + 21 where x is a prime number and y is an integer,

find the value of x and the value of y. Show each stage of your working clearly.

x =

y = [3]

Solve the simultaneous equations

Show clear algebraic working.

[5]

A and B are two similar solids.

A has a volume of 1836 cm³ B has a volume of 4352 cm³

B has a total surface area of 1120 cm²

Work out the total surface area of A.

cm²[3]

A curve has equation y = f(x)

The coordinates of the minimum point on this curve are (–9, 15)

(a)Write down the coordinates of the minimum point on the curve with equation

(i)y = f(x + 3)

( , )

(ii)y = 13 f(x)

( , )[2]

The graph of y = a cos (x + b)° for 0  x  360 is drawn on the grid below.

Given that a > 0 and that 0 < b < 360

(b)find the value of a and the value of b.

a =

b = [2]

The function f is such that f(x) = x²– 8x + 5 where x  4

Express the inverse function f^–1 in the form f^–1(x) = ...

f ^–1(x) = [3]

OAB is a triangle.

OA = 2a and OB = 2b

M is the midpoint of AB. N is the point on OB such that ON : NB = 2 : 1

P is the point on AN such that OPM is a straight line.

Use a vector method to find OP : PM Show your working clearly.

[6]

An arithmetic series has first term a and common difference d.

The sum of the first 2n terms of the series is four times the sum of the first n terms of the series.

Find an expression for a in terms of d. Show your working clearly.

a = [4]