4MA1 Mathematics A: Oct/Nov 2021 Paper 1HHigher Tier Paper 1

(a)Simplify e⁸ ÷ e²

[1]

(b)Expand and simplify (x – 3)(x + 1)

[2]

Here is a right-angled triangle.

Calculate the value of h. Give your answer correct to 3 significant figures.

h = [3]

There are 54 fish in a tank. Some of the fish are white and the rest of the fish are red.

Jeevan takes at random a fish from the tank. The probability that he takes a white fish is 49

(a)Work out the number of white fish originally in the tank.

[2]

Jeevan puts the fish he took out, back into the tank. He puts some more white fish into the tank.

Jeevan takes at random a fish from the tank. The probability that he takes a white fish is now 12

(b)Work out the number of white fish Jeevan put into the tank.

[2]

The diagram shows the front of a wooden door with a semicircular glass window.

Julie wants to apply 2 coats of wood varnish to the front of the door, shown shaded in the diagram.

250 millilitres of wood varnish covers 4 m² of the wood.

Work out how many millilitres of wood varnish Julie will need. Give your answer correct to the nearest millilitre.

_millilitres[5]

Yasmin has some identical rectangular tiles. Each tile is L cm by W cm.

Diagram NOT

Using 9 of her tiles, Yasmin makes rectangle ABCD, shown in the diagram below.

The area of ABCD is 1620 cm²

Work out the value of L and the value of W.

L = W = [5]

Alison buys 5 apples and 3 pears for a total cost of $1.96 Greg buys 3 apples and 2 pears for a total cost of $1.22

Michael buys 10 apples and 10 pears.

Work out how much Michael pays for his 10 apples and 10 pears. Show your working clearly.

$ [5]

Write 3.6 ´ 10³ as a product of powers of its prime factors. Show your working clearly.

[3]

In 2018, the population of Sydney was 5.48 million. This was 22% of the total population of Australia.

Work out the total population of Australia in 2018 Give your answer correct to 3 significant figures.

_million[3]

(i)Solve the inequalities –7  2x – 3 < 5

[3]

(ii)On the number line, represent the solution set to part (i)

[2]

A solid aluminium cylinder has radius 10 cm and height h cm.

The mass of the cylinder is 5.4 kg. The density of aluminium is 0.0027 kg/cm³

Calculate the value of h. Give your answer correct to one decimal place.

h = [5]

The table gives information about the times taken by 90 runners to complete a 10 km race.

(a)Complete the cumulative frequency table.

[1]

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

[2]

Any runner who completed the race in a time T minutes such that 42 < T  52 minutes was awarded a silver medal.

(c)Use your graph to find an estimate for the number of runners who were awarded a silver medal.

_runners[2]

The diagram shows a vertical cliff with a vertical radio mast on top of the cliff and a buoy in the sea.

The height of the cliff is 100 metres. The buoy is at the point B that is d metres from the base of the cliff. The angle of elevation from B to the top of the cliff is 20°

(a)Calculate the value of d. Give your answer correct to 3 significant figures.

d = [3]

The point A at the top of the radio mast is vertically above the top of the cliff. The angle of elevation from B to A is 25°

(b)Calculate the height of the radio mast. Give your answer correct to 3 significant figures.

_m[3]

Here is a triangle XYZ.

The length XZ and the angles YXZ and XYZ are each given correct to 2 significant figures.

Calculate the upper bound for the length YZ. Give your answer correct to one decimal place. Show your working clearly.

_cm[3]

ABCDEF and GHIJKL are regular hexagons each with centre O.

GHIJKL is an enlargement of ABCDEF, with centre O and scale factor 2 → → OA = a OB = b

(a)Write the following vectors, in terms of a and b. Simplify your answers.

→

(i)AB

[1]

→

(ii)KI

[2]

→

(iii)LD

[2]

The triangle OAB has an area of 5 cm²

(b)Calculate the area of the shaded region.

_cm²[3]

Magnus and Garry play 2 games of chess against each other.

The probability that Magnus beats Garry in any game is 29

The probability that any game between Magnus and Garry is drawn is

The result of any game is independent of the result of any other game.

(a)Complete the probability tree diagram.

Magnus loses

Magnus loses

Magnus loses[2]

For each game of chess,

the winner gets 2 points and the loser gets 0 points, when the game is drawn, each player gets 1 point.

(b)Work out the probability that, after 2 games, Magnus and Garry have the same number of points.

[3]

Magnus and Garry now play a third game of chess.

(c)Work out the probability that, after 3 games, Magnus and Garry have the same number of points.

[3]

There are 32 students in a class. In one term these 32 students each took a test in Maths (M), in English (E) and in French (F).

25 students passed the test in Maths. 20 students passed the test in English. 14 students passed the test in French. 18 students passed the tests in Maths and English. 11 students passed the tests in Maths and French. 4 students failed all three tests. x students passed all three tests.

The incomplete Venn diagram gives some more information about the results of the 32 students.

(a)Use all the given information about the results of students who passed the test in Maths to find the value of x.

x = [2]

(b)Use your value of x to complete the Venn diagram to show the number of students in each subset.

[2]

A student who passed the test in Maths is chosen at random.

(c)Find the probability that this student failed the test in French.

[1]

(a)Factorise 6y²– y – 5

[2]

(b)Make f the subject of w = 2 f8 - f

[3]

(c)Express 4x²– 8x + 7 in the form a(x + b)²+ c where a, b and c are integers.

[3]

0.4x is a recurring decimal. x is a whole number such that 1  x  9

Find, in terms of x, the recurring decimal 0.4x as a fraction. Give your fraction in its simplest form. Show clear algebraic working.

[3]

ABCED is a five-sided shape.

ABCD is a rectangle. CED is an equilateral triangle.

AB = x cm BC = y cm

The perimeter of ABCED is 100 cm. The area of ABCED is R cm²

(a)Show that R =  

[3]

(b)(i)Find the value of x for which R has its maximum value.

Give your answer in the form pq - where p and q are integers.

x = [2]

(ii)Explain why the maximum value of R is given by this value of x.

[1]

The straight line L passes through point A (– 6, 2) and point B (5, 3) The straight line M is perpendicular to L and passes through the midpoint of A and B. The line M intersects the line x = – 1 at point C.

Calculate the area of triangle ABC.

[7]