4MA1 Mathematics A: January 2020 Paper 1HRHigher Tier Paper 1

Brendon, Asha and Julie share some money in the ratios 3 : 2 : 6 The total amount of money that Asha and Julie receive is $36

Work out the amount of money that Brendon receives.

$ [3]

Show that 3 15 × 2 58 = 8 25[3]

(a)Make a the subject of d = g + 2ac

[2]

(b)Factorise fully 9ef − 12f

[2]

(c)Expand and simplify (x + 2)(x − 5)

[2]

(d)Simplify fully × nn₅

[2]

B = {b, l, u, e}

G = {g, r, e, y}

W = {w, h, i, t, e}

(a)List all the members of the set

(i)B ∪ G

(ii)W ∩ G′

[2]

Serena writes down the statement B ∩ G ∩ W = ∅

(b)Is Serena’s statement correct? You must give a reason for your answer.

[1]

The diagram shows Yuen’s garden.

The garden is in the shape of a semicircle of radius 7.2 m. Yuen is going to cover his garden with grass seed.

Yuen has 12 boxes of grass seed. Each box of grass seed contains enough seed to cover 6 m² of the garden.

Has Yuen enough grass seed for his garden? Show your working clearly.[3]

Solve x² − 5x − 36 = 0 Show clear algebraic working.

[3]

In a sale, the normal price of a hat is reduced by 15% The sale price of the hat is 20.40 euros.

Work out the normal price of the hat.

_euros[3]

5 children are playing on a trampoline. The mean weight of the 5 children is 28 kg.

2 of the children get off the trampoline. The mean weight of these 2 children is 26.5 kg.

Work out the mean weight of the 3 children who remain on the trampoline.

_kg[3]

Pablo made a solid gold statue.

He melted down some gold blocks and used the gold to make the statue. Each block of gold was a cuboid, as shown below.

The mass of the statue is 5.73 kg. The density of gold is 19.32 g/cm³

Work out the least number of gold blocks Pablo melted down in order to make the statue. Show your working clearly.

[5]

The diagram shows a regular hexagon, ABCDEF, and an isosceles triangle, GHJ.

The perimeter of the hexagon is equal to the perimeter of the triangle.

Find the length of each side of the hexagon. Show clear algebraic working.

_cm[5]

The weight of a cat is 4.3 kg correct to 2 significant figures.

(a)Write down the upper bound of the weight of the cat.

_kg[1]

(b)Write down the lower bound of the weight of the cat.

_kg[1]

G = e − f

e = 17 correct to the nearest integer f = 9.4 correct to one decimal place

(c)Work out the upper bound for the value of G.

[2]

The cumulative frequency graph gives information about the waiting times, in minutes, of people with appointments at Hospital A.

(a)Use the graph to find an estimate of the median waiting time at Hospital A.

_minutes[1]

(b)Use the graph to find an estimate of the interquartile range of the waiting times at Hospital A.

_minutes[2]

At a different hospital, Hospital B, the median waiting time is 28 minutes and the interquartile range of the waiting times is 19 minutes.

(c)Compare the waiting times at Hospital A with the waiting times at Hospital B.

[2]

(a)Use algebra to show that 0.572 = 63110[2]

Given that y is a prime number,

(b)express 32 − in the form a + bc − y where a, b and c are integers.

[2]

A, B and C are points on a circle, centre O. Angle ABC = 38°

Work out the size of angle OAC. Give a reason for each stage of your working.

[4]

The diagram shows two right-angled triangles, DEF and EFG.

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

_cm[4]

Steffi is going to play one game of tennis and one game of chess.

The probability that she will win the game of tennis is 0.6 The probability that she will win both games is 0.42

Work out the probability that she will not win either game.

[4]

The function f is such that f(x) = (x − 4)² for all values of x.

(a)Find f(1)

[1]

(b)State the range of the function f.

[1]

The function g is such that g(x) = 4x + 3 x ≠ −3

(c)Work out fg(2)

[2]

The diagram shows the graph of y = f(x) for −4  x  12

The point P on the curve has x coordinate 2

(a)(i)Use the graph to find an estimate for the gradient of the curve at P.

[3]

(ii)Hence find an equation of the tangent to the curve at P. Give your answer in the form y = mx + c

[2]

The equation f(x) = k has exactly two different solutions for −4  x  12

(b)Use the graph to find the two possible values of k.

_, [2]

Frequency density

The histogram gives information about the heights of all the Year 11 students at a school.

There are 160 students in Year 11 with a height between 155 cm and 170 cm.

Work out the total number of students in Year 11 at the school.

[4]

The diagram shows a frustum of a cone and a sphere.

The frustum is made by removing a small cone from a large cone. The cones are similar.

The height of the small cone is h cm. The height of the large cone is 2h cm. The radius of the base of the large cone is r cm.

The radius of the sphere is r cm.

Given that the volume of the frustum is equal to the volume of the sphere,

find an expression for r in terms of h. Give your expression in its simplest form.

r = [5]

The diagram shows the prism ABCDEF with cross section triangle ABC.

Angle BEC = 40° and angle ACB is obtuse. AC = 6 cm and CE = 13 cm

The area of triangle ABC is 22 cm²

Calculate the length of AB. Give your answer correct to one decimal place.

_cm[6]

The graph of y = a cos(x + b)° for 0  x  360 is drawn on the grid. y 3

2

1

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

a =

b = [2]

Another curve C has equation y = f(x) The coordinates of the minimum point of C are (4, 5)

(b)Write down the coordinates of the minimum point of the curve with equation

(i)y = f(2x)

( , )

(ii)y = f(x) − 7

( , )[2]

A particle moves along a straight line. The fixed point O lies on this line. The displacement of the particle from O at time t seconds, t  0, is s metres where

At time T seconds the velocity of P is V m/s where V  −5

Find an expression for T in terms of V.

Give your expression in the form where k and m are integers to be found.

T = [6]