NAME: ________________________________________ DATE: _____________________ MARKS: ____________
1
(a)
Express as a fraction in its simplest form.
10𝑦 2
7
(i)
6𝑦
35
(ii)
𝑘 2 −16
2
𝑘 −2𝑘−8
÷
…………………………………… [2]
…………………………………… [3]
(b)
Solve 3(𝑥 − 4) + 5 = 7.
𝑥 = ……………………………… [2]
(c)
Solve 3𝑡 2 + 5𝑡 − 4 = 0.
Show all your working and give your answers correct to 2 decimal places.
𝑡 = ………….. or 𝑡 =…………… [3]
1
2
The diagram shows a triangular prism.
All lengths are in centimetres.
(a)
Show that the volume, 𝑉 cm3, of the prism is given by 𝑉 = (40𝑥 − 5𝑥 2 ).
[3]
(b)
On the grid on the next page, draw the graph of 𝑉 = 40𝑥 − 5𝑥 2 for 1 ≤ 𝑥 ≤ 7.
Three of the points have been plotted for you.
2
(c)
Use your graph to find the possible values of 𝑥 for one of these prisms with a volume of
50 cm3.
𝑥 = ………….. or 𝑥 =…………… [2]
(d)
A cuboid has length 4 cm, width 3 cm and height 𝑥 cm.
By drawing a suitable line on your graph, find the value of 𝑥 when the prism and the
cuboid have the same volume.
𝑥 =………………………………… [3]
3
3
The diagram shows the positions of two ports, 𝐴 and 𝐵, and a lighthouse 𝐿.
The bearing of 𝐵 from 𝐿 is 062°.
𝐴𝐵 = 13 km, 𝐵𝐿 = 14 km and 𝐴𝐿 = 8 km.
(a)
Calculate the bearing of 𝐴 from 𝐿.
…………………………………… [4]
(b)
A boat is located at 𝐶.
𝐶 is 11 km from 𝐵 and 𝐵𝐶̂ 𝐴 = 90°.
The boat travels to port 𝐴 in a straight line.
Find the distance the boat travels.
………………………………km [2]
4
(c)
The boat then travels in a straight line from port 𝐴 to port 𝐵.
It travels at an average speed of 3.75 km/h.
Calculate the time taken for the boat to travel from port 𝐴 to port 𝐵.
Give your answer in hours and minutes.
…………. hours ………… minutes [2]
5
NAME: ________________________________________ DATE: _____________________ MARKS: ____________
1
(a)
Express as a fraction in its simplest form.
(i)
6𝑦
35
÷
10𝑦 2
7



Change ÷ to × and then turn the fraction on the right upside down
Multiply the expression
Leave answer in lowest terms
…………………………………… [2]
(ii)
𝑘 2 −16
2
𝑘 −2𝑘−8



Factorise the numerator using difference of two squares
Factorise the denominator
Simplified by cancellation
…………………………………… [3]
(b)
Solve



3(𝑥 − 4) + 5 = 7.
Expand the brackets
balance the equation
make x as the subject of the equation
𝑥 = ……………………………… [2]
6
(c)
Solve 3𝑡 2 + 5𝑡 − 4 = 0.
Show all your working and give your answers correct to 2 decimal places.
 Use quadratic formula
 Substitute the values using calculator
 Round off to 2 decimal places
𝑡 = ………….. or 𝑡 =…………… [3]
2
The diagram shows a triangular prism.
All lengths are in centimetres.
(a)
Show



that the volume, 𝑉 cm3, of the prism is given by 𝑉 = (40𝑥 − 5𝑥 2 ).
Find the area of the triangle
Use the formula for volume
Simplify expression
[3]
7
(b)
On the grid on the next page, draw the graph of 𝑉 = 40𝑥 − 5𝑥 2 for 1 ≤ 𝑥 ≤ 7.
Three of the points have been plotted for you.
 Find the rest of the points,
 Draw table of values
 Plot the points
 Label your graph
(c)
Use your graph to find the possible values of 𝑥 for one of these prisms with a volume of
50 cm3.
 Draw horizontal line V= 50 cm3
 Find the points of intersection of the line and the curve
𝑥 = ………….. or 𝑥 =…………… [2]
8
(d)
A cuboid has length 4 cm, width 3 cm and height 𝑥 cm.
By drawing a suitable line on your graph, find the value of 𝑥 when the prism and the
cuboid have the same volume.
 Find the volume of the cuboid in terms of x
 Draw a linear line for the volume of the cuboid
 read off the x coordinate of the point of intersection
𝑥 =………………………………… [3]
3
The diagram shows the positions of two ports, 𝐴 and 𝐵, and a lighthouse 𝐿.
The bearing of 𝐵 from 𝐿 is 062°.
𝐴𝐵 = 13 km, 𝐵𝐿 = 14 km and 𝐴𝐿 = 8 km.
(a)
Calculate the bearing of 𝐴 from 𝐿.
 Use the cosine rule formula to find 𝐵𝐿̂𝐴
…………………………………… [4]
(b)
A boat is located at 𝐶.
9
𝐶 is 11 km from 𝐵 and 𝐵𝐶̂ 𝐴 = 90°.
The boat travels to port 𝐴 in a straight line.
Find the distance the boat travels.
 Draw a sketch of the position of C
 use Pythagoras’ Theorem to find AC
………………………………km [2]
(c)
The boat then travels in a straight line from port 𝐴 to port 𝐵.
It travels at an average speed of 3.75 km/h.
Calculate the time taken for the boat to travel from port 𝐴 to port 𝐵.
Give your answer in hours and minutes.
 Use the speed formula to find T
 Change the answer to hours and minutes
 1 hour = 60 minutes
…………. hours ………… minutes [2]
10