questions - maths english science tutors leicester large tin has a radius of 6.5 cm and a height of...

Questions Q1.

The equation

x3 – 6x = 72

has a solution between 4 and 5

Use a trial and improvement method to find this solution. Give your answer correct to one decimal place. You must show all your working.

x = . . . . . . . . . . . . . . . . . . . . . .

(Total for Question is 4 marks)

Q2.

Write down the value of (i) 7°

. (ii) 5−1

. (iii) 9½


Q3.

Rationalise the denominator

. . . . . . . . . . . . . . . . . . . . . .


Q4.

Rationalise the denominator of

Give your answer in its simplest form.

.............................................................................................................................................


Q5. Dan does an experiment to find the value of π. He measures the circumference and the

diameter of a circle.

He measures the circumference, C, as 170 mm to the nearest millimetre. He measures the diameter, d, as 54 mm to the nearest millimetre.

Dan uses π = C⁄d to find the value of π.

Calculate the upper bound and the lower bound for Dan's value of π.

.............................................................................................................................................. (Total for Question is 4 marks)

Q6. A solid sphere has

a mass of 1180 g measured to the nearest gram and a radius of 6.2 cm measured to the nearest millimetre.

Given that

find the upper bound for the density of the sphere. Give your answer to 3 significant figures.

. . . . . . . . . . . . . . . . . . . . . g/cm3


Q7. k = 3e + 5

(a) Work out the value of k when e = –2

.............................................................................................................................................(2)

(b) Solve 4y + 3 = 2y + 14

y =. . . . . . . . . . . . . . . . . . . . .

(2)

(c) Solve 3(x – 5) = 21

x =. . . . . . . . . . . . . . . . . . . . .

(2)

–3 < n < 4, n is an integer.

(d) Write down all the possible values of n.

............................................................................................................................................. (2)

Q8. (a) Expand and simplify (p + 9)(p – 4)

............................................................................................................................................. (2)

(b) Solve = 4w + 2

w = . . . . . . . . . . . . . . . . . . . . . .

(3)

(c) Factorise x2 – 49

..............................................................................................................................................

(1)

(d) Simplify (9x8y3)½

............................................................................................................................................. (2)


Q9.

Solve = x


Q10.

Solve the simultaneous equations

4x + y = 25 x − 3y = 16

x =........................................................... y =...........................................................

Q11.

Solve the simultaneous equations x2 + y2 = 9 x + y = 2

Give your answers correct to 2 decimal places.

x = . . . . . . . . . . . . . . . y = . . . . . . . . . . . . . . . or x = . . . . . . . . . . . . . . . y = . . . . . . . . . . . . . . .


Q12.

Solve 3x2 – 4x – 2 = 0

Give your solutions correct to 3 significant figures.

.............................................................................................................................................


Q13. Solve 2x2 + 5x – 3 = 0

.............................................................................................................................................


Q14.

Simplify fully

.............................................................................................................................................


Q15.

Simplify

.............................................................................................................................................


Q16. Make p the subject of the formula y = 3p2 – 4

. . . . . . . . . . . . . . . . . . . . . .


Q17. Make t the subject of the formula

.............................................................................................................................................


Q18. Make t the subject of the formula 2(d – t) = 4t + 7

t = . . . . . . . . . . . . . . . . . . . . . .


Q19. Here are the first four terms of an arithmetic sequence.

3 10 17 24

(a) Find, in terms of n, an expression for the nth term of this arithmetic sequence.

...........................................................

(2)

(b) Is 150 a term of this sequence?

You must explain how you get your answer.

.............................................................................................................................................

.............................................................................................................................................

(2) (Total for Question is 4 marks)

Q20.

The first five terms of an arithmetic sequence are

2 6 10 14 18

(a) Write down an expression, in terms of n, for the nth term of this sequence.

............................................................................................................................................. (2)

An expression for the nth term of a different sequence is 20 – 5n

(b) work out the 10th term of this sequence.

.............................................................................................................................................(2)


Q21. The diagram shows shape A. All the measurements are in centimetres.

Diagram NOT accurately drawn

(a) Find an expression in terms of x for the area, in cm2, of shape A.

You must simplify your answer.

. . . . . . . . . . . . . . . . . . . . . .

(4)

Shape B is a rectangle. Shape B has the same area as shape A. Shape B has a length of (3x + 2) centimetres.

(b) Find an expression in terms of x for the width, in centimetres, of shape B.

. . . . . . . . . . . . . . . . . . . . . .

(1)


Q22. * This shape is a solid prism. The cross section of the prism is a trapezium.

Show that the total surface area of the prism is 82x2 + 32x − 12

Q23.

The diagram shows a trapezium.

All the measurements are in centimetres.

The area of the trapezium is 351 cm2.

(a) Show that 2x2 + x − 351 = 0

(2)

(b) Work out the value of x.

...........................................................

(3)

(Total for Question is 5 marks) Q24. A water trough is in the shape of a prism.

Hamish fills the trough completely.

Water leaks from the bottom of the trough at a constant rate.

2 hours later, the level of the water has fallen by 20 cm.

Water continues to leak from the trough at the same rate.

How many more minutes will it take for the trough to empty completely?

. . . . . . . . . . . . . . . . . . . . . . minutes


Q25.

A piece of card is in the shape of a trapezium.


A hole is cut in the card. The hole is in the shape of a trapezium.

Work out the area of the shaded region.

. . . . . . . . . . . . . . . . . . . . . . cm2


Q26.

The diagram shows 3 sides of a regular polygon.


Each interior angle of the regular polygon is 140°.

Work out the number of sides of the regular polygon.

.............................................................................................................................................


Q27.

The interior angle of a regular polygon is 160°.


(i) Write down the size of an exterior angle of the polygon.

. . . . . . . . . . . . . . . . . . . . . . °

(ii) Work out the number of sides of the polygon.

..............................................................................................................................................


Q28. The diagram shows a quadrilateral ABCD.

Diagram NOT

accurately drawn

AB = 16 cm. AD = 12 cm. Angle BCD = 40°. Angle ADB = angle CBD = 90°.

Calculate the length of CD. Give your answer correct to 3 significant figures.

. . . . . . . . . . . . . . . . . . . . . . cm


Q29. XYZ is a right-angled triangle.

Calculate the length of XZ.

Give your answer correct to 3 significant figures.

..............................................................................................................................................(Total for Question is 3 marks)

Q30.


LMN is a right-angled triangle. MN = 9.6 cm. LM = 6.4 cm.

Calculate the size of the angle marked x°. Give your answer correct to 1 decimal place.

. . . . . . . . . . . . . . . . . . . . . .°


Q31.

Here are some cards. Each card has a letter on it.

Rachel takes at random two of these cards.

Work out the probability that there are different letters on the two cards.

.............................................................................................................................................


Q32. The probability that Rebecca will win any game of snooker is p.

She plays two games of snooker. (a) Complete, in terms of p, the probability tree diagram.

(2) (b) Write down an expression, in terms of p, for the probability that Rebecca will win both

games.

..............................................................................................................................................

(1) (c) Write down an expression, in terms of p, for the probability that Rebecca will win

exactly one of the games.

..............................................................................................................................................(2)


Q33. The table gives information about the heights, h metres, of trees in a wood.

Height (h metres) Frequency

0 < h ≤ 2 7

2 < h ≤ 4 14

4 < h ≤ 8 18

8 < h ≤ 16 24

16 < h ≤ 20 10

Draw a histogram to show this information.


Q34.

Bob asked each of 40 friends how many minutes they took to get to work.

The table shows some information about his results.

Time taken (m minutes) Frequency

0 < m ≤ 10 3

10 < m ≤ 20 8

20 < m ≤ 30 11

30 < m ≤ 40 9

40 < m ≤ 50 9

Work out an estimate for the mean time taken.

. . . . . . . . . . . . . . . . . . . . . . minutes


Q35.

Faisel weighed 50 pumpkins.

The grouped frequency table gives some information about the weights of the pumpkins.

Weight (w kilograms) Frequency

0 < w ≤ 4 11

4 < w ≤ 8 23

8 < w ≤ 12 14

12 < w ≤ 16 2

Work out an estimate for the mean weight.

..............................................................................................................................................


Q36. The graph of y = f(x) is shown on each of the grids.

(a) On this grid, sketch the graph of y = f(x – 3)

(2)

(b) On this grid, sketch the graph of y = 2f(x)

(2)


Q37. The diagram shows a sketch of the graph of y = cos x°

(a) Write down the coordinates of the point A.

..............................................................................................................................................

(1)

(b) On the same diagram, draw a sketch of the graph of y = 2 cos x°

(1)


Q38.


ABC is a triangle.

AB = 8.7 cm. Angle ABC = 49°. Angle ACB = 64°.

Calculate the area of triangle ABC. Give your answer correct to 3 significant figures.

. . . . . . . . . . . . . . . . . . . . cm2


Q39. ABCD is a quadrilateral.


Work out the length of DC.

Give your answer correct to 3 significant figures.

. . . . . . . . . . . . . . . . . . . . . . cm


Q40. * The diagram shows the triangle PQR.

PQ = x cm PR = 2x cm Angle QPR = 30°

The area of triangle PQR = A cm2

Show that x =


Q41. ABC is a triangle.

D is a point on AB and E is a point on AC. DE is parallel to BC.

AD = 4 cm, DB = 6 cm, DE = 5 cm, AE = 5.8 cm.

Calculate the perimeter of the trapezium DBCE.

........................................................... cm

(Total for Question is 4 marks) Q42.

A, B, C and D are points on the circumference of a circle with centre O.

Angle ABC = 116°

Find the size of the angle marked x. Give reasons for your answer.

Q43. ABC is a triangle.

BC = 12.3 cm, Angle ABC = 73°

The area of triangle ABC is 50 cm2.

Work out the length of AC. Give your answer correct to 3 significant figures.

..............................................................................................................................................(Total for Question is 6 marks)

Q44. The diagram shows a large tin of pet food in the shape of a cylinder.

The large tin has a radius of 6.5 cm and a height of 11.5 cm.

A pet food company wants to make a new size of tin.

The new tin will have a radius of 5.8 cm. It will have the same volume as the large tin.

Calculate the height of the new tin. Give your answer correct to one decimal place.

........................................................... cm


Q45.

The diagram shows a solid made from a hemisphere and a cone.


The radius of the hemisphere is 4 cm. The radius of the base of the cone is 4 cm.

Calculate the volume of the solid. Give your answer correct to 3 significant figures.

...........................................................cm3


questions - maths english science tutors leicester large tin has a radius of 6.5 cm and a height of...

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