4MA1 Mathematics A: January 2022 Paper 2HHigher Tier Paper 2

(a)Expand and simplify ( y + 4)(2 − y)

[2]

(b)Factorise fully 15b⁵ c − 35b³ c⁹

[2]

Show that 6 34 ÷ 2 47 = 2 58[3]

Triangle ABC is similar to triangle PQR

(a)Calculate the length of BC

_cm[2]

(b)Write down an expression for y in terms of x

y = [1]

Each side of a regular octagon has a length of 18 mm, correct to the nearest 0.5 mm

(a)Write down the lower bound of the length of each side of the octagon.

_mm[1]

(b)Write down the upper bound of the length of each side of the octagon.

_mm[1]

The scale diagram shows the position on a map of a house, A

House C is on a bearing of 110° from A The distance from A to C is 700 m

(a)Mark the position of C on the diagram with a cross (×) Label your cross C[3]

(b)Write the scale of the map in the form 1 : n

1 : [1]

A bag contains only pink sweets, white sweets, green sweets and red sweets.

The table gives each of the probabilities that, when a sweet is taken at random from the bag, the sweet will be green or the sweet will be red.

number of pink sweets : number of white sweets = 2 : 1

There are 28 red sweets in the bag.

Work out the number of white sweets in the bag.

[5]

Find the lowest common multiple (LCM) of 28, 42 and 63 Show your working clearly.

[3]

The table gives information about the average house price in England in 2018 and in 2019

(a)Work out the percentage increase in the average house price from 2018 to 2019 Give your answer correct to one decimal place.

_%[2]

The average house price in 2019 was 7.7% greater than the average house price in 2017

(b)Work out the average house price in 2017 Give your answer correct to 3 significant figures.

£ [3]

The frequency table gives information about the number of points scored by a player.

The mean number of points scored is 2

Work out the value of x

x = [4]

Solve the simultaneous equations

Show clear algebraic working.

x =

y = [3]

The diagram shows a regular 10-sided polygon, ABCDEFGHIJ

Show that x = y[4]

a = 6 × 10⁴⁰

Work out the value of a³ Give your answer in standard form.

[3]

The shaded region in the diagram is bounded by three lines. The equation of one of the lines is given.

Write down three inequalities that define the shaded region.

[3]

A zip wire is shown as the dashed line AC in the diagram.

The zip wire is supported by two vertical posts AB and CD standing on horizontal ground.

The zip wire makes an angle x with the horizontal, as shown in the diagram. The design of the zip wire requires the angle x to be at least 5°

Work out the least possible height of the post AB Give your answer correct to 3 significant figures.

_m[3]

Diyar recorded the distance, in kilometres, that he cycled each day for 11 days. Here are his results.

Find the interquartile range of his results.

_km[3]

D, E, F and G are points on a circle, centre O

EOG is a diameter of the circle.

Angle EGD = 42°

Calculate the size of angle DFG Give a reason for each stage of your working.

Angle DFG = [4]

Show that 123 + 2

can be written in the form a + b where a and b are integers.[3]

Prove that when the sum of the squares of any two consecutive odd numbers is divided by 8, the remainder is always 2 Show clear algebraic working.[3]

PTQ is a diameter of a circle. RTS is a chord of the circle.

Calculate the radius of the circle.

_cm[3]

The histogram gives information about the heights, h cm, of some tomato plants.

Frequency density

There are 12 tomato plants for which 75 < h  85 One of the tomato plants is selected at random.

Find an estimate for the probability that this tomato plant has a height greater than 82.5 cm

[4]

Part of the graph of y = 2x² − 4x − 1 is shown on the grid.

(a)Use the graph to find estimates for the solutions of the equation 2x² − 4x − 1 = 0 Give your solutions correct to one decimal place.

[2]

(b)By drawing a suitable straight line on the grid, find estimates for the solutions of the equation x² − x − 1 = 0 Show your working clearly. Give your solutions correct to one decimal place.

[3]

Here is a rectangle.

Given that the area of the rectangle is less than 75 cm²

find the range of possible values of x

[5]

The diagram shows triangle PQR

Given that angle PQR is obtuse,

work out the area of triangle PQR Give your answer correct to 3 significant figures.

_cm²[6]

A particle P moves along a straight line that passes through the fixed point O

The displacement, x metres, of P from O at time t seconds, where t  0 , is given by

The direction of motion of P reverses when P is at the point A on the line.

The acceleration of P at the instant when P is at A is a m /s²

Find the value of a

a = [5]

The function g is defined as

(a)Express the inverse function g⁻¹ in the form g⁻¹ : x  …

g⁻¹ : x  [4]

(b)State the domain of g⁻¹

[1]

An arithmetic series has first term a and common difference d, where d is a prime number.

The sum of the first n terms of the series is S_n and

S_2m = 320

Find the value of d and the value of m Show clear algebraic working.

d =

m = [5]