4MA1 Mathematics A: January 2021 Paper 1HRHigher Tier Paper 1

Show that 3 15 × 1 56 = 5 1315[3]

Given that a < b < c

the four whole numbers a, a, b and c have

a mode of 7 a median of 8.5 a mean of 9

Work out the value of a, the value of b and the value of c.

a =

b =

c = [4]

(a)On the number line, show the inequality –2  y < 1

[2]

n is an integer.

(b)Write down all the values of n that satisfy –3.4 < n  2

[2]

A train journey from Paris to Amsterdam took 3 hours 24 minutes. The total distance the train travelled was 433.5 km.

Work out the average speed of the train. Give your answer in kilometres per hour.

km/h[3]

The diagram shows a cuboid and a cylinder.

The dimensions of the cuboid are x cm by 12 cm by 5 cm. The volume of the cuboid is 270 cm³

The radius of the cylinder is x cm. The height of the cylinder is 2x cm.

(a)Work out the volume of the cylinder. Give your answer correct to the nearest whole number.

cm³[3]

(b)Change 1 m³ to cm³

_cm³[1]

(a)Make c the subject of A = cy – 5z

[2]

(b)Write down the value of g⁰

[1]

(c)Factorise x²– 11x + 24

[2]

Kuro invests 50 000 yen for 3 years in a savings account. She gets 2.4% per year compound interest.

Work out how much money Kuro will have in her savings account at the end of the 3 years. Give your answer correct to the nearest yen.

_yen[3]

The diagram shows a regular pentagon, ABCDE, a regular hexagon, CFGHID, and a quadrilateral, EDIJ.

AEJ and HIJ are straight lines.

Work out the size of the angle marked x. Show your working clearly.

[5]

A = 2⁸ × 3⁵ × 11⁴ B = 2⁶ × 3 × 11⁸

(a)Find the highest common factor (HCF) of A and B.

[2]

(b)Find the lowest common multiple (LCM) of 2A and 3B. Give the LCM as a product of powers of its prime factors.

[2]

The diagram shows one face of a wall. This face is in the shape of a pentagon with exactly one line of symmetry.

Omondi is going to paint this face of the wall once. He has to buy all the paint that he needs to use.

The paint in each tin of paint Omondi is going to buy will cover 16 m² of the face of the wall.

Work out the least number of tins of paint Omondi will need to buy. Show your working clearly.

[5]

The manager of a call centre asked the 120 people, who rang the call centre last week, how long they each waited before their call was answered.

The table gives information about their replies.

Time waited (t minutes) Frequency

(a)Complete the cumulative frequency table.

[1]

(b)On the grid below, draw a cumulative frequency graph for your table.

[2]

(c)Use your graph to find an estimate for the median of the times waited.

_minutes[1]

(d)Using your graph, find an estimate for the percentage of the 120 people who said that they waited longer than 23 minutes before their call was answered. Show your working clearly.

%[2]

(a)Simplify (16e¹⁰ f⁶)^1/2

[2]

(b)Write 2x + 14 + x − 23 as a single fraction in its simplest form.

[3]

Given that 4^k+3= 16 × 2^k

(c)find the value of k. Show your working clearly.

k = [4]

Here are two vectors.

Find, as a column vector, AC

[2]

B, D, E and F are points on a circle.

ABC is the tangent at B to the circle. Angle ABD = 39° Angle EFD = 18°

Work out the size of angle BDE. Give reasons for your working.

[4]

(a)Use algebra to show that 4.57 = 4 1933[2]

(b)Show that 26 − 3 can be written in the form a +b

where a and b are integers. Show your working clearly.[3]

(a)Expand and simplify (x + 4)(x – 2)(x + 1)

[3]

(b)Express x²– 10x + 40 in the form (x + a)²+ b, where a and b are integers.

[2]

Solve the simultaneous equations

Show clear algebraic working.

[5]

The histogram and the table give some information about the amounts of time, in hours, that Year 11 students at Bergdesh Academy spent, in total, on their homework last week. No student in Year 11 spent longer than 9 hours on their homework.

Frequency

0 < t  2 28

2 < t  4

4 < t  5

5 < t  6

6 < t  9

Using the information in the histogram and in the table, work out an estimate for the mean amount of time the Year 11 students spent on their homework last week. Give your answer in hours correct to 3 significant figures.

_hours

t[5]

k =

t = 14 correct to 2 significant figures a = 7.8 correct to 2 significant figures h = 3.4 correct to 2 significant figures

Work out the lower bound for the value of k. Show your working clearly.

[3]

A particle P is moving along a straight line. The fixed point O lies on the line.

At time t seconds (t  0), the displacement of P from O is s metres where

Find the minimum speed of P.

_m/s[5]

The nth term of an arithmetic series is u_n where u_n > 0 for all n The sum to n terms of the series is S_n

Given that u₄= 6 and that S₁₁= (u₆)²+ 18

find the value of u₂₀

[6]

ABC is an isosceles triangle with AB = AC.

B is the point with coordinates (–1, 5) C is the point with coordinates (2, 10) M is the midpoint of BC.

Find an equation of the line through the points A and M. Give your answer in the form py + qx = r where p, q and r are integers.

[5]