4MA1 Mathematics A: January 2022 Paper 2HRHigher Tier Paper 2

A tin contains tea bags with a choice of four different flavours of tea. The four flavours of tea are Assam or Darjeeling or Nilgiri or Rize.

Sara takes at random a tea bag from the tin.

The table shows each of the probabilities that the flavour of the tea Sara takes is Assam or Darjeeling or Rize.

(a)Work out the probability that the flavour of the tea Sara takes is Nilgiri.

[2]

(b)Work out the probability that the flavour of the tea Sara takes is either Darjeeling or Rize.

[2]

Mary saves for a holiday each year.

In 2020 she saved a total of $720 In 2021, each month she saved $78

The total amount Mary saved in 2021 was P% more than the total she saved in 2020

(a)Work out the value of P

[4]

Roberto is going to go on holiday.

He has two coupons that will save him money on his holiday.

Coupon A Coupon B

the cost of the accommodation is $1600 the cost of the flights is $800

Roberto can only use one of the coupons. He wants to save as much money as he can.

(b)Which of the two coupons, A or B, should he use? Show your working clearly.[3]

(a)Solve 4y + 5 > 12

[2]

(b)Solve 6x  5  4x  72

Show clear algebraic working.

x = [3]

The diagram shows a regular octagon ABCDEFGH and a regular pentagon ABIJK

F E

Work out the size of the angle x

[4]

Shane invests 7200 dollars for 3 years in a savings account. He gets 2.5% per year compound interest.

How much money will Shane have in his savings account at the end of 3 years? Give your answer to the nearest dollar.

_dollars[3]

(a)Write down the value of x⁰

[1]

Given that 2^–3 × 2⁹= 2ⁿ

(b)find the value of n

n = [1]

7^206m₃ Given that  77₂₁₄  7

(c)find the value of m

m = [2]

(a)Write down an equation of the straight line with gradient –3 and which passes through the point with coordinates (0, 5)

[2]

(b)Show, by shading on the grid, the region defined by all three of the inequalities

Label the region R

[3]

A scientist is investigating the weight of 50 tigers.

Here is some information about these tigers.

Type of tiger

Mean weight of tigers (kg) 260

The mean weight of all 50 tigers is 218 kg

Work out the mean weight of the Bengal tigers.

_kg[3]

In the diagram, ABC is a right-angled triangle and DEF is a semicircular arc.

The points D and F lie on AC so that DF is the diameter of the semicircular arc DEF The radius of the semicircular arc is 3 cm.

Work out the length of AFEDC Give your answer correct to 2 significant figures.

_cm[5]

The table gives information about the population and the total amount of money, in dollars, spent on healthcare for two countries in 2016

Work out how much more was spent per person on healthcare in Luxembourg than in Austria. Give your answer correct to the nearest whole number.

_dollars[3]

(a)Complete the table of values for y = 6x

[2]

(b)On the grid, draw the graph of y = 6x for 0.5  x  6

y

[2]

The diagram shows two vertical phone masts, AB and CD, on horizontal ground.

Work out the size of the angle of depression of B from D Give your answer correct to one decimal place.

[3]

a = 8 + 4

b = 8 − 4

(a – b) (a + b) can be written in the form y 4 y

Find the value of y Show your working clearly.

y = [3]

A group of 60 students each sat an algebra test and a geometry test. Each test was marked out of 110

The cumulative frequency graph gives information about the marks gained by the 60 students in the algebra test.

(a)Use the graph to find an estimate for the median mark in the algebra test.

[1]

(b)Use the graph to find an estimate for the number of students who gained 58 marks or less in the algebra test.

[1]

(c)Use the graph to find an estimate for the interquartile range of the marks gained in the algebra test.

[2]

The interquartile range of the marks gained in the geometry test is 9

Luis says

“The students’ marks are more spread out in the algebra test than in the geometry test.”

(d)Is Luis correct? Give a reason for your answer.

[1]

To be awarded a grade A in the algebra test, a student had to gain a mark greater than 64

Two students are to be selected at random from the 60 students in the group.

(e)Use the graph to find an estimate for the probability that both of these students were awarded a grade A in the algebra test.

[3]

Make t the subject of n²  4d  tt₃

[4]

The diagram shows quadrilateral ABCD

The angle BCD is acute.

Given that the area of triangle BCD = 405 cm²

work out the size of angle ABD Give your answer correct to one decimal place.

[5]

Part of the curve with equation y = f (x) is shown on the grid.

Find an estimate for the gradient of the curve at the point where x = 2 Show your working clearly.

[3]

The line with equation 2y = x + 1 intersects the curve with equation 3y²+ 7y + 16 = x²– x at the points A and B

Find the coordinates of A and the coordinates of B Show clear algebraic working.

( , ) and ( , )[5]

ABCD is a horizontal rectangular field.

A vertical pole, AE, is placed at the corner A of the field.

Calculate the size of the angle between EC and the plane ABCD Give your answer correct to one decimal place.

[3]

y is inversely proportional to x x is directly proportional to T³

Given that y = 8 when T = 25

find the exact value of T when y = 27

T = [4]

The diagram shows a solid made from a cylinder and a hemisphere. The cylinder and the hemisphere are both made from the same metal.

The plane face of the hemisphere coincides with the upper plane face of the cylinder.

The radius of the cylinder and the radius of the hemisphere are both x cm. The height of the cylinder is 3x cm.

The total surface area of the solid is 81π cm² The mass of the solid is 840 grams.

The following table gives the density of each of four metals.

The metal used to make the solid is one of the metals in the table.

Determine the metal used to make the solid. Show your working clearly.

[6]

ABC is a triangle in which angle ABC = 90°

p and q are integers such that

the coordinates of A are ( p, 10 ) the coordinates of B are ( –1, –5 ) the coordinates of C are ( 8, q )

Given that the gradient of AC is – 67

work out the value of p and the value of q

p =

q = [5]

The functions f and g are such that

The function h is such that h(x) = fg(x)

The domain of h is {x : x  6}

Express the inverse function h^–1 in the form h^–1(x) = ...

h^–1(x) = [4]