4MA1 Mathematics A: January 2023 Paper 2HRHigher Tier Paper 2

Show that 4 23 ÷ 1 56 = 2 611[3]

(a)On the grid, rotate triangle A 180° about (1, –1) Label the new triangle B[2]

(b)On the grid, translate triangle A by the vector  −7

Label the new triangle C[1]

–8 < 2y  2

y is an integer.

(a)Find all the possible values of y

[2]

(b)Write down the inequality shown on the number line.

[1]

Using ruler and compasses only, construct the bisector of angle BAC You must show all your construction lines.

[2]

On the grid, draw the graph of 5x + 2y = 10 for values of x from –2 to 4

[3]

In a bag, there are only red counters, blue counters, green counters and yellow counters.

The total number of counters in the bag is 80

In the bag

the number of red counters is x + 7 the number of blue counters is x – 11 the number of green counters is 3x

Jude takes at random a counter from the bag.

The probability that he takes a red counter is 14

Work out the probability that Jude takes a yellow counter.

[4]

(a)Find the highest common factor (HCF) of 200 and 420

[2]

A = 2³ × 3 × 5 ×7² B = 2 × 3² × 7 C = 3 × 5² × 11

(b)Find the lowest common multiple (LCM) of A, B and C Write your answer as a product of powers of prime factors.

[2]

60 students sat a Mathematics exam.

The mean mark for the 32 students in Class A was 55 The mean mark for the 28 students in Class B was 52

Find the mean mark for all 60 students.

[3]

Teresa invests $2000 for 3 years in a savings account. She gets 4% each year compound interest.

(a)How much money will Teresa have in her savings account at the end of 3 years? Give your answer correct to the nearest dollar.

$ [3]

Sam invested $T The value of his investment decreased by 9% each year.

At the end of the first year, the value of Sam’s investment was $1365

(b)Work out the value of T

[3]

The diagram shows two solids, A and B, made from two different metals.

Solid A is in the shape of a cylinder with radius 3 cm and height 7 cm Solid A has a mass of 2000 g

Solid B has a mass of 3375 g Solid B has a volume of 450 cm³

All of the metal from Solid A and Solid B is melted down to make a uniform Solid C

Given that there is no change to mass or volume during this process

work out the density of Solid C Give your answer correct to one decimal place.

_g / cm³[3]

A

AB, BC, CD, DE and EF are five sides of a regular polygon.

RST, SCU and BCV are straight lines. RST is parallel to CD Angle RSC = 128° Angle UCV = 32°

Work out how many sides the polygon has. Show your working clearly.

[4]

(a)Simplify 2y₀

[1]

(b)Simplify fully (16a⁴ )^3/4

[2]

(c)Expand and simplify 5x(3x + 4)(2x – 1)

[3]

A rectangle has length L and width W

L is increased by 20% W is decreased by 35%

Calculate the percentage reduction in the area of the rectangle.

%[3]

A, B and C are points on a circle, centre O

B

The radius of the circle is 8.5 cm Angle ABC = 132°

Work out the perimeter of the shaded sector AOC Give your answer correct to 3 significant figures.

_cm[3]

Here are the numbers of aces that Rutger served in each of 11 tennis matches.

(a)Find the interquartile range of the numbers of aces. Show your working clearly.

[2]

Kim also plays in 11 tennis matches.

For Kim the median number of aces is 11 the interquartile range of the numbers of aces is 5

(b)State, giving a reason, whether Rutger or Kim

(i)served more aces on average,

[1]

(ii)was more consistent with the number of aces served.

[1]

Here are two vectors.

Find AC as a column vector.

[2]

The table gives information about the time taken by each student in Year 11 to complete a homework task.

(a)On the grid, draw a histogram for this information.

[3]

One of these students who took 50 minutes or less and more than 25 minutes to complete this homework task is chosen at random.

(b)Find an estimate for the probability that this student took 45 minutes or less to complete this homework task.

[2]

A statue and a model of the statue are mathematically similar.

The statue has a total surface area of 3600 cm² The model has a total surface area of 625 cm²

The volume of the model is 750 cm³

Work out the volume of the statue.

_cm³[3]

Prove algebraically that, for any three consecutive even numbers,

the sum of the squares of the smallest even number and the largest even number is 8 more than twice the square of the middle even number.[3]

A, B and C are three sets.

n(A ∩ B ∩ C ) = 5

n(A ∩ B ∩ C ′) = 2

n(A ∩ C ) = 5

n(A) = 17

n([A ∪ B ∪ C ] ) = 3

n(A′ ∩ B ∩ C ′) = 6

n(B ∩ C ) = 7

n(C ) = 14

Complete the Venn diagram to show the number of elements in each region.

C [4]

The diagram shows the graph of y = f(x)

The point P has x coordinate –2

Use the graph to find an estimate for the gradient of the curve at P

[3]

Solve the simultaneous equations

Show clear algebraic working.

[5]

AEC and BED are chords of a circle.

Angle DAE = 48°

Work out the size of angle ADE Give your answer correct to one decimal place.

[5]

The diagram shows a solid cone and a solid sphere.

The cone has base radius r, slant height l and perpendicular height h The sphere has radius r

The base radius of the cone is equal to the radius of the sphere.

Given that

show that the total surface area of the cone can be written in the form

where a is a constant to be found.[6]

ABCD is a trapezium with AB parallel to DC

A is the point with coordinates (–4, 6) B is the point with coordinates (2, 3) D is the point with coordinates (–1, 8)

The trapezium has one line of symmetry. The line of symmetry intersects CD at the point E

Work out the coordinates of the point E

( , )[6]

Write

as a single fraction in its simplest form.

[4]