4MA1 Mathematics A: May/June 2022 Paper 1HRHigher Tier Paper 1

Here is a biased spinner.

When the spinner is spun once, the probabilities that it lands on red or on yellow or on green are given in the table.

(a)Work out the probability that the spinner lands on red or on yellow.

[1]

Yang is going to spin the spinner 300 times.

(b)Work out an estimate for the number of times the spinner will land on purple.

[3]

In a warehouse there are two types of shelves, type R and type S.

These two types of shelves are arranged into shelving units that form a sequence of patterns.

Here are the first three terms in the sequence.

The width of each type R shelf is 2.4 m and the width of each type S shelf is 3.5 m

(a)Work out the total width of a shelving unit that has 6 type R shelves.

m[2]

A shelving unit has n type R shelves. The total width of this shelving unit is W metres.

(b)Find an expression for W in terms of n Give your answer in its simplest form.

W = [2]

Here are five cards.

Each card has a number written on it.

The mean of the five numbers is 12

Work out the value of x

x = [3]

The language department of a college has 180 students. Each student studies exactly one of French, German, Italian or Spanish.

15 students study French. 45% of the students study German.

Express the percentage of students studying Italian or Spanish as a percentage of those studying French or German.

%[4]

(a)Expand 3c³(c + 4)

[2]

(b)(i)Factorise x²+ 8x – 9

[2]

(ii)Hence, solve x²+ 8x –9 = 0

[1]

Show that 2 23 + 3 34 = 6 512[3]

The diagram shows a solid cylinder made from iron.

The cylinder has diameter 18 cm and height 3.5 cm The mass of the cylinder is 7.04 kg

Work out the density of the iron. Give your answer in g/cm³ correct to 2 significant figures.

g/cm³[3]

Jane bought a new car for $18 000 The car depreciates in value by 15% each year.

Work out the value of the car at the end of 4 years. Give your answer correct to the nearest $

$ [3]

Solve the inequality 3 – 4x  11

[2]

Line L is drawn on the grid.

Find an equation for L Give your answer in the form y = mx + c

[3]

The diagram shows a quadrilateral ABCD

In the diagram, ABC and DAC are right‑angled triangles.

The area of quadrilateral ABCD is 31.5 cm²

Work out the length of AD

cm[6]

P = 3³ × 5² × 7 Q = 3² × 5 × 7²

(a)Write down the highest common factor (HCF) of P and Q

[1]

P = 3³ × 5² × 7 Q = 3² × 5 × 7²

(b)Work out the value of P³ × Q Give your answer in the form 3^x × 5^y × 7^z where x, y and z are positive integers.

[2]

Here is the number of runs scored by a baseball team in each of its 15 games this season.

The number of runs have been arranged in order of size.

Work out the interquartile range of the number of runs.

[2]

Solve the simultaneous equations

Show clear algebraic working.

x =

y = [4]

P, Q, R and S are points on a circle with centre O

PS is a diameter of the circle.

Angle PQR = 136°

Work out the size of angle RPS

[3]

(a)Expand and simplify (3x – 1)(x + 2)(3x + 1)

[3]

(b)Simplify fully  2x 

[3]

Here is a parallelogram PQRS, in which angle SPQ is acute.

PQ = 6.1 cm PS = 3.8 cm

The area of the parallelogram is 18 cm²

Work out the length of QS Give your answer correct to 3 significant figures.

cm[5]

The diagram shows a cube ABCDEFGH with sides of length 6 cm.

T is the midpoint of AB and V is the midpoint of CH

Work out the distance from T to V in a straight line through the cube. Give your answer in the form a cm where a is an integer.

cm[4]

The histogram gives information about the height, h cm, of each tree in part of a forest.

Frequency density

There are no trees for which h  200 and for which h > 800

The number of trees for which 300 < h  400 is 8 fewer than the number of trees for which 400 < h  500

Work out an estimate for the number of trees in this part of the forest that have a height greater than 500 cm.

[3]

The diagram shows two similar metal statues.

The volume of statue B is 20% less than the volume of statue A

The surface area of statue B is k% less than the surface area of statue A

Work out the value of k Give your answer correct to 3 significant figures.

k =

3 + 8[4]

Express₂ in the form p + q where p and q are integers. 2 − 1

Show each stage of your working clearly.

[4]

The diagram shows a sketch of part of the curve with equation y = x²– px where p is a

positive constant.

For all values of p, the curve has exactly one turning point and this turning point is a minimum shown as the point T in the sketch.

For the curve where the x coordinate of T is –3

(a)find the value of p

p = [4]

The line with equation y = k is a tangent to the curve with equation y = x²– 16x

(b)Find the value of k

k = [3]

(a)Express 2x²– 12x + 3 in the form a(x + b)²+ c where a, b and c are integers.

[3]

The curve C has equation y = 2(x + 4)²– 12(x + 4) + 3

The point M is the minimum point on C

(b)Find the coordinates of M

( , )[2]

Elliot has x counters.

Each counter has one red face and one green face.

Elliot spreads all the counters out on a table and sees that the number of counters showing a red face is 5

Elliot then picks at random one of the counters and turns the counter over. He then picks at random a second counter and turns the counter over.

The probability that there are still 5 counters showing a red face is 1932

Work out the value of x Show clear algebraic working.

x = [5]

The sum of the first 10 terms of an arithmetic series is 4 times the sum of the first 5 terms of the same series.

The 8th term of this series is 45

Find the first term of this series. Show clear algebraic working.

[5]