4MA1 Mathematics A: May/June 2019 Paper 1HRHigher Tier Paper 1

The diagram shows a cylinder.

The cylinder has radius 8.2 cm and height 10 cm. The cylinder is empty.

Pam pours 1.5 litres of water into the cylinder.

Work out the depth of the water in the cylinder. Give your answer correct to 1 decimal place.

cm[3]

Each interior angle of a regular polygon is 162°

Work out the number of sides the polygon has.

[3]

E = {11, 12, 13, 14, 15, 16, 17, 18, 19, 20} A = {even numbers} B = {multiples of 3}

List the members of the set

(i)A ˆ B

(ii)A ‰ B

(iii)Aމ

[3]

Solve 4x x

x = [2]

(a)Write 720 as a product of its prime factors. Show your working clearly.

[3]

(b)Find the smallest whole number that 720 can be multiplied by to give a square number.

[1]

Lorenzo increases all the prices on his restaurant menu by 8%

Before the increase, the price of a dessert was $4.25

(a)Work out the price of the dessert after the increase.

$ [3]

After the increase, the price of lasagne is $9.45

(b)Work out the price of lasagne before the increase.

$ [3]

The diagram shows isosceles triangle ABC.

AB = AC = 7.5 cm. The height of the triangle is 6 cm.

Calculate the area of the triangle.

cm²[4]

There are 10 people in a lift. These 10 people have a mean weight of 79.2 kg.

3 of these people get out of the lift. These 3 people have a mean weight of 68 kg.

Work out the mean weight of the 7 people left in the lift.

kg[3]

(a)Simplify t⁹÷ t³

[1]

(b)Simplify w⁵ q w⁷

[1]

(c)Simplify (5xy²)³

[2]

Change 22 metres per second to a speed in kilometres per hour. Show your working clearly.

km/h[3]

3 years ago, the ratio of Tom’s age to Clemmie’s age was 2 : 7 Tom is now 15 years old and Clemmie is now x years old.

Find the value of x.

x = [3]

A box, in the shape of a cuboid, is going to be put on a table.

The whole of one face of the box will be in contact with the table. The force exerted by the box on the table is always 105 newtons.

The box is 5 m by 4 m by 3 m.

The greatest pressure exerted by the box on the table is P newtons/m² The least pressure exerted by the box on the table is Q newtons/m²

Work out the value of PíQ

[3]

(a)On the Venn diagram, shade the set (A ‰ B މˆ C

[1]

(b)Use set notation to describe the shaded region in the Venn diagram below.

[1]

Each day that Barney goes to college, he either goes by bus or he walks. The probability that Barney will go to college by bus on any day is 0.3

When Barney goes to college by bus, the probability that he will be late is 0.2 When Barney walks to college, the probability that he will be late is 0.1

(a)Complete the probability tree diagram.

[2]

Barney will go to college on 200 days next year.

(b)Work out an estimate for the number of days Barney will be late for college next year.

[4]

The straight line L₁ has equation 2y = 6xí

The straight line L₂ is perpendicular to L₁ DQGSDVVHVWKURXJKWKHSRLQW í

Find an equation for L₂ Give your answer in the form ay + bx = c

[4]

A particle P is moving along a straight line. The fixed point O lies on this line.

At time t seconds, the displacement, s metres, of P from O is given by

At time t seconds, the velocity of P is v m/s.

(a)Find an expression for v in terms of t.

v = [2]

(b)Find the time at which the acceleration of the particle is 6 m/s²

_seconds[3]

The histogram shows information about the ages of all the passengers travelling on a plane. No one on the plane is older than 80 years.

24 passengers on the plane are aged between 40 years and 60 years.

(a)Work out the total number of passengers on the plane.

[3]

A passenger on the plane is picked at random.

(b)Work out an estimate for the probability that this person is older than 55 years.

[2]

(a)Expand and simplify (x + 2)(2x + 3)(x – 7) Show your working clearly.

[3]

(b)Make m the subject of p² = x + m2m − y

[3]

The 25th term of an arithmetic series is 44.5 The sum of the first 30 terms of this arithmetic series is 765

Find the 16th term of the arithmetic series. Show your working clearly.

[5]

a = 25 q 10¹⁴ⁿ where n is an integer.

Find an expression, in terms of n, for a^3/2 Give your answer in standard form.

[3]

A curve has equation y = f(x)

There is only one maximum point on the curve. The coordinates of this maximum point are (4, 3)

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

(i)y = f(x – 5)

( , )

(ii)y = 3f(x)

( , )[2]

Here is the graph of y = a sin(bx)° for 0 - x - 360

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

a =

b = [2]

Solve the simultaneous equations

y = 2x + 1

Show clear algebraic working.

[5]

B, C, D and F are points on a circle. ABC, AFD, BFE and CDE are straight lines.

Work out the size of angle x. Show your working clearly.

x = [4]

P is the point on AB such that AP : PB = 3 : 1 Q is the point on AC such that OQP is a straight line.

Use a vector method to find AQ : QC Show your working clearly.

AQ : QC = [5]

A boat sails from point X to point Y and then to point Z.

Y is on a bearing of 280° from X. Z is on a bearing of 220° from Y.

The distance from X to Y is 3.5 km. The distance from Y to Z is 6 km.

Work out the bearing of Z from X. Give your answer correct to 1 decimal place.

[5]