4MA1 Mathematics A: Oct/Nov 2021 Paper 2HHigher Tier Paper 2

(a)Write down the value of m, given that 3⁴ ´ 3⁵ = 3^m

m = [1]

(b)Write down the value of n, given that (5³)⁷= 5ⁿ

n = [1]

(c)Find the value of p, given that  77_p  7

p = [2]

Here are two rectangles, rectangle A and rectangle B.

The area of rectangle B is twice the area of rectangle A.

Work out the value of x. Show your working clearly.

x = [4]

The table gives information about the amounts of money, in euros, that 70 of Anjali’s friends spent last Saturday.

Money spent (S euros) Frequency

0 < S  8

8 < S  16

One of Anjali’s 70 friends is going to be chosen at random.

(a)Find the probability that this friend spent more than 24 euros last Saturday.

[1]

(b)Work out an estimate for the mean amount of money spent by Anjali’s friends last Saturday. Give your answer correct to 2 decimal places.

_euros[4]

ABC and DEF are similar triangles.

(a)Work out the length of AB.

_cm[2]

Given that BC = 54 cm,

(b)work out the length of EF.

_cm[2]

The diagram shows a regular octagon ABCDHIJK and a pentagon DEFGH.

Angle GHD = angle FGH.

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

[5]

Victor buys 12 bottles of apple juice for a total cost of $21 Victor sells all 12 bottles at $2.45 each bottle.

Work out Victor’s percentage profit.

%[3]

Ali and Badia each have 25 000 dollars to invest.

4.5% compound interest per year Receive 1150 dollars interest each year

Ali invests in the Cyclone Bank for 3 years. Badia invests in the Tornado Bank for 3 years.

By the end of the 3 years, Ali will have received more interest than Badia.

How much more? Show your working clearly. Give your answer correct to the nearest dollar.

_dollars[4]

(a)Simplify (3x² y)⁰

[1]

(b)(i)Factorise x²– 5x – 36

[2]

(ii)Hence solve x²– 5x – 36 = 0

[1]

A rainwater tank contains 2.4 ´ 10⁷ raindrops. The rainwater tank also contains 1.75 ´ 10⁶ bacteria.

(a)Work out the number of bacteria per raindrop in the tank. Give your answer in standard form correct to 2 significant figures.

[3]

A drop of rainwater contains 5.01 ´ 10²¹ atoms.

In a drop of rainwater the number of atoms is 3 times the number of molecules.

(b)Work out the number of molecules in the rainwater tank. Give your answer in standard form correct to one significant figure.

_molecules[2]

ABC is an isosceles triangle with BA = BC.

N is the point on AC such that AN = 9.3 cm and BN is perpendicular to AC.

Work out the perimeter of triangle ABC. Give your answer correct to 3 significant figures.

_cm[4]

E

B, D, E and F are points on a circle, centre O. ABC is a tangent to the circle. ODC is a straight line.

BOE is a diameter of the circle.

Angle BCD = 48°

Find the size of angle DFE.

[3]

(a)Simplify (64p⁹ q¹²)^2/3

[2]

(b)Write as a single fraction 23x + 45x - 910x

Give your answer in its simplest form.

[2]

(c)Expand and simplify 4x(x – 5)(2x + 3) Show your working clearly.

[3]

The region R, shown shaded in the diagram, is bounded by three straight lines.

Write down the three inequalities that define R.

[3]

Manuel collected information about the flights that arrived late at an airport last month.

The table gives information about the number of minutes that these flights were late.

Minutes late (L minutes) Frequency

10 < L  15

15 < L  25

25 < L  40

40 < L  60

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

[3]

Manuel selected at random a flight that was late by 25 minutes or less from his results.

(b)Work out an estimate for the probability that this flight was late by 5 minutes or less.

[2]

The functions f and g are such that

(a)State the value of x that cannot be included in any domain of g

[1]

(b)Find gf(x) Simplify your answer.

gf(x) = [2]

(c)Express the inverse function g⁻¹ in the form g⁻¹(x) = ...

g⁻¹(x) = [3]

A box contains 15 counters.

There are 4 red counters, 5 green counters and the rest are yellow counters.

Niklas takes at random a counter from the box and writes down the colour of his counter. He then puts the counter back into the box.

Sasha then takes at random a counter from the box and writes down the colour of her counter.

Work out the probability that the counters taken by Niklas and Sasha both have the same colour.

[3]

Express 85 - 1 in the form a + b where a and b are integers.

Show each stage of your working clearly.

[3]

Here is a quadrilateral ABCD.

Calculate the area of quadrilateral ABCD. Give your answer correct to 3 significant figures. Show your working clearly.

_cm²[5]

The straight line L has equation x – y = 3 The curve C has equation 3x²– y²+ xy = 9

L and C intersect at the points P and Q.

Find the coordinates of the midpoint of PQ. Show clear algebraic working.

( , )[6]

Here are the first four terms of an arithmetic series.

Given that the 15th term of the series is (90 + 2k),

calculate the sum of the first 30 terms of the series.

[5]

The curve C has equation y = f(x) where f(x) = 9 – 3(x + 2)² The point A is the maximum point on C.

(a)Write down the coordinates of A.

( , )[1]

The curve C is transformed to the curve S by a translation of  

(b)Find an equation for the curve S.

[1]

The curve C is transformed to the curve T. The curve T has equation y = 3(x + 2)²– 9

(c)Describe fully the transformation that maps curve C onto curve T.

[1]

The graph of y = acos (x – b)° + c for −180  x  360 is drawn on the grid below.

(d)Find the value of a, the value of b and the value of c.

a =

b =

c = [3]

The diagram shows a sphere of diameter x cm and a pyramid ABCDE with a horizontal rectangular base BCDE.

The vertex A of the pyramid is vertically above the centre O of the base so that AB = AC = AD = AE.

BC = x cm, CD = 2x cm and AO = 5x cm.

The volume of the sphere is 288π cm³

Calculate the total surface area of the pyramid. Give your answer correct to the nearest cm²

_cm²[6]