4MA1 Mathematics A: January 2023 Paper 1HHigher Tier Paper 1

80 students entered a dancing competition.

The table gives information about the length of time, in minutes, for which each student spent dancing.

12 < m  24 25

24 < m  36 23

36 < m  48 15

48 < m  60 6

Work out an estimate for the mean length of time the students spent dancing.

_minutes[4]

Solve 3(2 – 4x) = 5 – 8x Show clear algebraic working.

x = [3]

Use ruler and compasses only to construct the perpendicular bisector of line AB You must show all your construction lines.

[2]

The diagram shows a pentagon.

Work out the value of x

x = [3]

In a box, there are only green sweets, orange sweets and yellow sweets.

There are 280 sweets in the box so that

the number of green sweets : the number of orange sweets = 2 : 3 and the number of orange sweets : the number of yellow sweets = 1 : 5

Work out how many green sweets there are in the box.

[3]

Shane bought a car.

The amount Shane paid for the car was $32 000

Theresa also bought a car. To pay for this car, Theresa paid a deposit of $18 000 together with 14 monthly payments of $1160

Theresa paid more for her car than Shane paid for his car.

(a)Work out how much more Theresa paid as a percentage of the amount Shane paid.

%[4]

Kylie bought a van.

After 1 year, the value of the van was $39 865 During this year, the value of the van decreased by 15%

(b)Work out the value of the van when Kylie bought it.

$ [3]

Some members of a library were asked to name the type of book that they each liked to read the best.

One of the members is chosen at random. The table shows information about the probability of the type of book that this member answered.

48 members answered comedy books.

Work out how many of the members answered mystery books.

[4]

The diagram shows a triangle ABC inside a semicircle.

A, B and C are points on the semicircle.

AB is the diameter of the semicircle.

Angle ACB = 90° Angle BAC = 50° AC = 18 cm

Work out the perimeter of the semicircle. Give your answer correct to 2 significant figures.

cm[5]

(a)Write 6.25 × 10⁻⁴ as an ordinary number.

[1]

(b)Work out (2.4 × 10¹²) ÷ (9.6 × 10⁴) Give your answer in standard form.

[2]

(a)Factorise y²– 2y – 48

[2]

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

[1]

(c)Solve the inequality 7w + 6 > 12w + 14

[3]

The region R, shown shaded in the diagram, is bounded by the straight lines with equations

3y + 2x = 4

Write down the three inequalities that define R

[3]

3^1/2 × 3^2/5 = 3^m

(a)Work out the value of m

m = [1]

5^–10÷ 5^–4= 5ⁿ

(b)Work out the value of n

n = [1]

Expand and simplify 3x(2x – 5)² Show clear algebraic working.

[3]

(a)Complete the table of values for y  2 x  5  1 x 

[1]

(b)On the grid, draw the graph of y  2 x  5  1 x  for 0.5  x  5

[2]

Here are 9 cards. Each card has either a number on it or a letter on it.

Tomas is playing a game.

Tomas takes at random one of the cards and keeps it.

Tomas then takes at random another card and keeps it.

(a)Complete the probability tree diagram.

letter

letter

letter[2]

(b)Work out the probability that each of the two cards has a number on it.

[2]

(c)Work out the probability that there will be one card with a number on it and one card with a letter on it.

[3]

Here is a shape formed from two triangles ABC and CDE ACD and BCE are straight lines.

AC = 24 cm BC = 31 cm CE = 19 cm CD = 16 cm

Angle BAC = 64°

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

cm[5]

y is inversely proportional to x

y = c⁴ when x = c² where c is a positive constant.

Find a formula for y in terms of x and c Give your answer in its simplest form.

[3]

The function f is such that f(x) = kx where x ≠ 0 and k is an integer.

(a)Express the inverse function f⁻¹ in the form f⁻¹(x) = ...

f⁻¹(x) = [1]

The function g is such that g(x) = 2 – 3x⁴ where x ≠ 0

The function h is such that h(x) = 3x2 − x where x ≠ 2

(b)(i)Find g(−2)

[1]

(ii)Express the composite function hg in the form hg(x) = ... Give your answer in its simplest form.

hg(x) = [2]

The acceleration, a, of an object is given by

v = 45.23 correct to 2 decimal places u = 5.12 correct to 2 decimal places t = 8.5 correct to 2 significant figures

By considering bounds, work out the value of a to a suitable degree of accuracy. Show your working clearly and give a reason for your answer.

a = [5]

The radius of a right circular cylinder is x cm.

The height of the cylinder is  800π x − x cm.

The volume of the cylinder is V cm³

Find the maximum value of V Give your answer correct to the nearest whole number.

[5]

The diagram shows the cross section of a circular water pipe.

OABC is a sector of the circle, centre O

The shaded region in the diagram represents the water flowing in the pipe.

The water flows at 14 cm/s in the pipe.

Work out the volume of water that has flowed through the pipe in 3 minutes. Give your answer in cm³ correct to 3 significant figures.

cm³[5]

The first term of an arithmetic series is (2t + 1) where t > 0 The nth term of this arithmetic series is (14t – 5)

The common difference of the series is 3

The sum of the first n terms of the series can be written as p(qt – 1)^r where p, q and r are integers.

Find the value of p, the value of q and the value of r Show clear algebraic working.

p = q = r = [4]

ABCD is a kite.

AB = AD and CB = CD

The point B has coordinates (k, 1) where k is a negative constant. The point D has coordinates (8, 7)

The straight line L passes through the points B and D

The straight line L is parallel to the line with equation 5y – 3x = 6

Find an equation of AC Give your answer in the form px + qy = r where p, q and r are integers. Show your working clearly.

[6]

OAED is a quadrilateral.

OA = 2a OB = 2b DE = 7a + 3b

AB : BD = 1 : 2

The point C on AB is such that OCE is a straight line.

Use a vector method to find the ratio of OC : CE

[5]