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TUTOR WORLD ASHFORD GCSE MATHEMATICS

Sample Test 1 (Non-Calculator)

Time: 1 hour 30 minutes

Higher Tier

Read the following carefully.

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INSTRUCTIONS

o Use black ink only.

o Calculators must not be used.

o Fill in your name at the bottom of the page before you start the test.

o Answer all questions.

o Answer the questions in the space provided.

o You must show all your working out.

INFORMATION

o The total mark for this paper is 80.

o The marks for each question are displayed in brackets.

ADVICE

o Read each question carefully before answering it.

o Check your answers at the end if you have time.

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NAME: _________________________________ MARK: _________

GRADE: _________

TUTOR WORLD ASHFORD SAMPLE TEST 1

© Tutor World Ashford, 2017

QUESTION 1

The diagram shows three shapes joined together along their edges.

Shape A is a regular hexagon, shape B is a square, and

shape C is a parallelogram.

Work out the size of the angle marked 𝑥?

𝑥

𝐴

𝐵

𝐶

DIAGRAM NOT

ACCURATELY DRAWN

Total for Question 1 is 4 marks



QUESTION 2

𝑓(𝑥) =𝑥+7

2

Find the value of 𝑎 such that 𝑓(𝑎) =3

4

QUESTION 3

Sketch the graph of 𝑦 = sin(𝑥) + 1 on the grid below, for 0𝑜 ≤ 𝑥 ≤ 450𝑜.





QUESTION 4

Find the gradient of the line which passes through the coordinates (0,1) and (√5, √5).

Give your answer as a fraction in its simplest form.

QUESTION 5

Prove that (3𝑛 + 1)2 − (𝑛 − 1)2 is always even, where 𝑛 is an integer.





QUESTION 6

The two solid cones on the right are mathematically similar.

Cone A has a radius of 4 𝑐𝑚 and a volume of 32𝜋 𝑐𝑚3.

Cone B has a height of 3 𝑐𝑚.

Find the volume of Cone B.

Give you answer in terms of 𝝅.

A

B

4𝑐𝑚

3𝑐𝑚

DIAGRAMS NOT

ACCURATELY DRAWN


https://www.google.co.uk/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwjNkvy0u8DMAhVFthQKHS6TBmIQjRwIBw&url=http://www.glogster.com/farhatchowdhury1/lateral-and-surface-area-of-cones-and-pyramids/g-6k0rk7o0vdsqm08ntr9oarp&psig=AFQjCNEHrOwCJfQX1_F_1GurLYI2CEXGqA&ust=1462452809291112

https://www.google.co.uk/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwjNkvy0u8DMAhVFthQKHS6TBmIQjRwIBw&url=http://www.glogster.com/farhatchowdhury1/lateral-and-surface-area-of-cones-and-pyramids/g-6k0rk7o0vdsqm08ntr9oarp&psig=AFQjCNEHrOwCJfQX1_F_1GurLYI2CEXGqA&ust=1462452809291112



QUESTION 7

The value (𝑉), in pounds, of a certain car is inversely proportional to one more than its age (A), in

years.

The value of the car is £5000, when it is 4 years old.

(a) Find a formula for 𝑉 in terms of 𝐴.

(b) How old is the car when its value is £2500?

[2]

[2]




QUESTION 8

Solve the equation 2𝑥2 = 5𝑥 + 3.

QUESTION 9

On Adam’s Farm, the ratio of the number of cows, pigs and sheep is 2:3:4

The ratio of the number of pigs and goats is 4:5

If there are 60 goats on the farm, how many animals are there altogether?





QUESTION 10

(a) What type of sequence is the following?

𝑏𝑥, 𝑏, 𝑏

𝑥

𝑏

𝑥2 …

(b) Find a formula for the 𝑛𝑡ℎ term of the following quadratic sequence.

−1, 5, 13, 23, …

[1]

[3]




QUESTION 11

Toby has a stamp collection.

The Venn diagram shows information about his stamp collection.

ξ = {𝑇𝑜𝑏𝑦′𝑠 𝑓𝑢𝑙𝑙 𝑐𝑜𝑙𝑙𝑒𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 720 𝑠𝑡𝑎𝑚𝑝𝑠}

𝐵 = {𝐵𝑟𝑖𝑡𝑖𝑠ℎ 𝑠𝑡𝑎𝑚𝑝𝑠}

𝐶 = {𝑠𝑡𝑎𝑚𝑝𝑠 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 20𝑡ℎ 𝑐𝑒𝑛𝑡𝑢𝑟𝑦}

(a) How many British stamps does Toby own?

Toby chooses a stamp at random from his collection.

(b) Find the probability that the stamp is in the set 𝐵′ ∩ 𝐶, simplifying your answer.

ξ

𝐵 𝐶

3𝑥 + 2 𝑥 7𝑥 − 9

11𝑥 + 1

[3]

[3]




QUESTION 12

The speed of an average snail is 1.3×10−2 m/s.

How long does it take an average snail to travel 65cm, in seconds?

QUESTION 13

𝑔(𝑥) = 2𝑥2 − 9𝑥 + 1

Express 𝑔(𝑥 − 3) in the form 𝑎𝑥2 + 𝑏𝑥 + 𝑐.





QUESTION 14

OABC is a parallelogram.

𝑀 is the midpoint of 𝑂𝐴.

𝑂𝐴⃗⃗⃗⃗ ⃗ = 𝒂

𝑂𝐶⃗⃗⃗⃗ ⃗ = 𝒄

The ratio of MN:NB is 1:2

Show that the ratio of AN:NC is also 1:2

𝑂

𝐴 𝐵

𝐶

𝑀

𝑁

Diagram NOT

accurately drawn




QUESTION 15

(a) Write 𝑦 = 𝑥2 + 8𝑥 − 15 in completed square form.

(b) Hence, state the coordinates of the turning point of the graph of 𝑦 = 𝑥2 + 8𝑥 − 15.

QUESTION 16

The line of equation 𝑦 = (𝑘 + 2)𝑥 is perpendicular to the line of equation 𝑦 = 𝑘𝑥.

Find the value of 𝑘.

[3]

[1]





QUESTION 17

Calculate, 44

5×2

1

3 giving your answer as a mixed number in its simplest form.

QUESTION 18

Work out an estimate for (11.83 × 7.01)0.5





QUESTION 19

Simplify (𝑥2+2𝑥+1

𝑦2 )−0.5

QUESTION 20

Expand and simplify, (√𝑚 + √2𝑚)(√𝑚 + 3√2𝑚).





QUESTION 21

The diagram shows 3 identical rectangles connected

together to form a new shape.

The perimeter of this new shape is 66𝑐𝑚.

Calculate its area.

DIAGRAM NOT

ACCURATELY DRAWN


10𝑐𝑚



QUESTION 22

1

𝑥=

𝑦

2+

1

𝑧

Make 𝑧 the subject of the formula above.

QUESTION 23

A straight line with the equation 𝑦 + 3𝑥 = 𝑐 passes through the points (1, −4) and (5, 𝑞).

Find the value of 𝑞.



END OF TEST

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