#### Example Tasks

To test your practical skills we prepared tasks from British A-levels, which is equivalent of Polish matura exams. You can solve the tasks on your own, and check answer by clicking "show answer" button.

#### November 2021 A-Level Mathematics (7357/1)

###### Question 6a

The ninth term of an arithmetic series is 3 The sum of the first n terms of the series is Sn and S21 = 42 Find the first term and common difference of the series.

###### Question 6b

A second arithmetic series has first term 18 and common difference ¾ The sum of the first n terms of this series is Tn Find the value of n such that Tn = Sn

###### Question 8a

Given that 9 sin2α + sin2α = 8 show that 8 cot2 - 2cotα - 1 = 0

###### Question 8b

Hence, solve 9 sin2α + sin2α = 8 in the interval 0 < α < 2π Give your answers to two decimal places.

###### Question 8c

Solve 9 sin2 (2x π/4) + sin (4x π/2) = 8 in the interval 0 < x < π/2 Give your answers to one decimal place

#### November 2021 A-Level Mathematics (7357/2)

###### Question 1

Four possible sketches of y = ax2 + bx + c are shown below. Given b2 - 4ac = 0 and a, b and c are non-zero constants, which sketch is the only one that could possibly be correct?

###### Question 6

Show that the solution of the equation 5x = 3x+4 can be written as x = ln 81/ (ln 5 - ln 3) Fully justify your answer.

###### Question 8a

Kai is proving that n3 n is a multiple of 3 for all positive integer values of n. Kai begins a proof by exhaustion.

Step 1
n3 n = n(n2 - 1)

Step 2
When n = 3m, where m is a
non-negative integer
n3 n = 3m(9m2 - 1)
which is a multiple of 3

Step 3
When n = 3m + 1
n3 n = (3m + 1)((3m + 1)2 - 1)

Step 4
= (3m + 1)(9m2) = 3(3m + 1)(3m2) which is a multiple of 3

Step 5
Therefore n3 n is a multiple of 3 for all positive integer values of n

Explain the two mistakes that Kai has made after Step 3.

###### Question 8b

Correct Kai’s argument from Step 4 onwards.

#### November 2021 A-Level Mathematics (7357/3)

###### Question 2

Simplify fully
(x + 3)(6 + 2x)

(x - 3)(3 + x)
for x ≠ +/-3
Circle your answer.
-2
2
(6 - 2x)

(x - 3)
(2x - 6)

(x - 3)

###### Question 3

f (x) ¼ 3x2
Obtain lim h! 0 f(x + h) - f(x)/ h
Circle your answer.

3h2/h
x3
3(x + h)2 - 3x2

h
6x

###### Question 5a(i)

A gardener is creating flowerbeds in the shape of sectors of circles. The gardener uses an edging strip around the perimeter of each of the flowerbeds. The cost of the edging strip is £1.80 per metre and can be purchased for any length. One of the flowerbeds has a radius of 5 metres and an angle at the centre of 0.7 radians as shown in the diagram below.

Find the area of this flowerbed.

###### Question 5a(ii)

Find the cost of the edging strip required for this flowerbed.

###### Question 5b(i)

A flowerbed is to be made with an area of 20 m2
Show that the cost, £C, of the edging strip required for this flowerbed is given by
C = 18/5 (20/r+r)
where r is the radius measured in metres.

###### Question 5b(ii)

Hence, show that the minimum cost of the edging strip for this flowerbed occurs when r ≈ 4.5 Fully justify your answer.

###### Question 7 a

Solve A building has a leaking roof and, while it is raining, water drips into a 12 litre bucket. When the rain stops, the bucket is one third full. Water continues to drip into the bucket from a puddle on the roof. In the first minute after the rain stops, 30 millilitres of water drips into the bucket. In each subsequent minute, the amount of water that drips into the bucket reduces by 2%. During the nth minute after the rain stops, the volume of water that drips into the bucket is Wn millilitres.
Find W2

###### Question 7b

Explain why
Wn = A x 0.98n-1
and state the value of A.

###### Question 7c

Find the increase in the water in the bucket 15 minutes after the rain stops. Give your answer to the nearest millilitre.

###### Question 7c

Assuming it does not start to rain again, find the maximum amount of water in the bucket.

###### Question 7e

After several hours the water has stopped dripping. Give two reasons why the amount of water in the bucket is not as much as the answer found in part (d).