Question
(a) The diagram shows a cuboid.
How many planes of symmetry does this cuboid have?
Answer/Explanation
Ans: 3
(b) Write down the order of rotational symmetry for the following diagram.
Answer/Explanation
Ans: 4
Question
The diagram shows a wooden prism of height 5cm.
The cross section of the prism is a sector of a circle with sector angle 25°.
The radius of the sector is 15cm.
Calculate the total surface area of the prism
Answer/Explanation
Ans: 281 or 280.8 to 280.9…
Question
The diagram shows a solid hemisphere.
The total surface area of this hemisphere is 243π.
The volume of the hemisphere is kπ.
Find the value of k.
[The surface area, A, of a sphere with radius r is A = \(4 \pi r^2\).]
[The volume, V, of a sphere with radius r is \(V = \frac{4}{3} \pi r^3\).]
Answer/Explanation
Ans:
486 cao
Question
The diagram shows a solid prism of length 15cm.
The cross section of the prism is the trapezium ABCD.
Angle DAB = angle CDA = 90°.
AB = 9 cm, DC = 6cm and AD = 4cm.
Calculate the total surface area of the prism.
Answer/Explanation
Ans:
420
Question
The diagram shows a triangular prism of length 12cm.
Triangle ABC is a cross section of the prism.
Angle BAC = 90°, AC = 6 cm and AB = 5cm.
Calculate the angle between the line CE and the base ABED.
Answer/Explanation
Ans:
24.8 or 24.77 to 24.78
Question
A water pipe has a circular cross section of radius 0.75cm.
Water fl ows through the pipe at a rate of 16cm/s.
Calculate the time taken for 1 litre of water to fl ow through the pipe.
Answer/Explanation
Ans:
35.4 or 35.36 to 35.37 s
Question
A sphere has a volume of 80\(cm^3\).
Calculate the radius of the sphere.
[The volume, V, of a sphere with radius r is \(V = \frac{4}{3} \pi r^3\).]
Answer/Explanation
Ans:
2.67 or 2.672 to 2.67301 cm
Question
The diagram shows a child’s toy.
The shape of the toy is a cylinder of radius 5 cm and height 8 cm on top of a hemisphere of radius 5 cm.
Calculate the volume of the toy.
[The volume, V, of a sphere with radius r is \(V=\frac{4}{3}\pi r^{3}.]\)
Answer/Explanation
Ans: 890 or 890.1 to 890.2…
Question
A solid cone has base radius 4cm and height 10cm.
A mathematically similar cone is removed from the top as shown in the diagram.
The volume of the cone that is removed is \(\frac{1}{8}\) of the volume of the original cone.
(a) Explain why the cone that is removed has radius 2cm and height 5cm.
Answer/Explanation
Ans: correct working
(b) Calculate the volume of the remaining solid.
[The volume, V, of a cone with radius r and height h is \(V = \frac{1}{3}\pi r^{2}h.]\)
Answer/Explanation
Ans: 147 or 146.5 to 146.6…
Question
The diagram shows a cuboid ABCDEFGH.
AE = 5 cm, EH = 4 cm and AG = 13 cm.
Calculate the angle between the line AG and the base EFGH of the cuboid.
Answer/Explanation
Ans:
22.6 or 22.61 to 22.62
Question
The diagram shows a toy.
The shape of the toy is a cone, with radius 4cm and height 9 cm, on top of a hemisphere with radius 4cm.
Calculate the volume of the toy.
Give your answer correct to the nearest cubic centimetre.
[The volume, V, of a cone with radius r and height h is \(V = \frac{1}{3}\pi r^{2}h.\) ]
[The volume, V, of a sphere with radius r is \(V = \frac{4}{3}\pi r^{3}.\) ]
Answer/Explanation
Ans: 285 cao cm3
Question
The diagram shows a pyramid with a square base ABCD.
All the sloping edges of the pyramid are 20cm long and AC = 17cm.
Calculate the height of the pyramid.
…………………………………….. cm
Answer/Explanation
Ans:
18.1 or 18.10….
Question
(a) 
A cylinder has height 20cm.
The area of the circular cross section is 74\(cm^2\).
Work out the volume of this cylinder.
……………………………. \(cn^3\)
(b) Cylinder A is mathematically similar to cylinder B.
The height of cylinder A is 10cm and its surface area is 440\(cm^2\).
The surface area of cylinder B is 3960 \(cm^2\).
Calculate the height of cylinder B.
………………………… cm
Answer/Explanation
Ans:
(a) 1480
(b) 30
Question
The diagram shows a square-based pyramid ABCDE.
The diagonals of the square meet at M.
E is vertically above M.
AB = BC = 12cm and EM = 9cm.
Calculate the angle between the edge EC and the base, ABCD, of the pyramid.
Answer/Explanation
Ans:
46.7 or 46.68 to 46.69
Question
A water tank in the shape of a cuboid has length 1.5 metres and width 1 metre.
The water in the tank is 60 centimetres deep.
Calculate the number of litres of water in the tank.
………………………………… litres.
Answer/Explanation
900
Question
A pipe is completely full of water.
Water flows through the pipe at a speed of 1.2m/s into a tank.
The cross-section of the pipe has an area of 6cm2.
Calculate the number of liters of water flowing into the tank in 1 hour.
liters [4]
Answer/Explanation
Ans:
22 2592
Question
The diagram shows a pyramid with a square base ABCD of side length 8cm.
The diagonals of the square, AC and BD, intersect at M.
V is vertically above M and VM = 10 cm.
Calculate the angle between VA and the base.[4]
Answer/Explanation
Ans:
21 60.5 or 60.50…
Question
The volume of a cuboid is 180 cm3.
The base is a square of side length 6cm.
Calculate the height of this cuboid.
__ cm [2]
Answer/Explanation
Ans:
5 5
Question
A cone with height 14.8cm has volume 275cm3.
Calculate the radius of the cone.
[The volume, V, of a cone with radius r and height h is V\(=\frac{1}{3}\pi r^{2}h\).]
__ cm [3]
Answer/Explanation
Ans:
12 4.21 or 4.212….
Question
Calculate the total surface area of the cuboid.
__ cm2 [3]
Answer/Explanation
Ans:
10 375
Question
Simplify.
\(\frac{x^{2}-5x}{2x^{2}-50}\)
……………………………………….
Answer/Explanation
\(\frac{x}{2(x+5)}\) or \(\frac{x}{2x+10}\) final answer
Question
The diagram shows a solid made from a cylinder and a hemisphere, both of radius 7cm.
The cylinder has length 12 cm.
Work out the total surface area of the solid.
[The surface area, A, of a sphere with radius r is \(A=4\pi r^{2}.\)]
…………………………………….\( cm^{2}\).
Answer/Explanation
990 or 989.58 to 989.73
Question
The diagram shows cuboid ABCDEFGH of length 20 cm and width 5.5cm.
The volume of the cuboid is \(495 cm^{3}.\)
Find the angle between the line AG and the base of the cuboid ABCD.
………………………………
Answer/Explanation
12.2 or 12.24…
Question
Calculate the total surface area of this cuboid.
Answer/Explanation
166
Question
The diagram shows a pyramid VABCD with a rectangular base.
V is vertically above M, the intersection of the diagonals AC and BD.
AB = 12cm, BC = 10cm and VC = 14cm.
Calculate the angle that VC makes with the base ABCD.
……………………………………………
Answer/Explanation
56.1 or 56.09….
Question
Calculate the radius of a sphere with volume 1260 \((cm)^3\)
[The volume, V, of a sphere with radius r is V =\(\frac{4}{3}\pi r^3\)]
Answer/Explanation
Volume of sphere of radius r , V=\(\frac{4}{3}\pi r^3\)
Given: Volume= 1260 \((cm)^3\)
\(\frac{4}{3}\times 3.14\times r^3=1260\)
\(3.14\times r^3=315\times 3\)
\(r^3=\frac{315\times 3}{3.14}=\frac{945}{3.14}\)
\(r^3=300.955\)
r=\(\sqrt[3]{300.955}\)
r=6.70 \(\approx 6.8\)
Question
Two similar vases have heights which are in the ratio $3: 2$.
(a) The volume of the larger vase is $1080 \mathrm{~cm}^3$. Calculate the volume of the smaller vase.
(b) The surface area of the smaller vase is $252 \mathrm{~cm}^2$. Calculate the surface area of the larger vase.
▶️Answer/Explanation
(a) 320
(b) 567
Question
The diagram shows part of a fan. $O F G$ and $O A D$ are sectors, centre $O$, with radius $18 \mathrm{~cm}$ and sector angle $40^{\circ}$. $B, C, H$ and $E$ lie on a circle, centre $O$ and radius $6 \mathrm{~cm}$. Calculate the shaded area.
▶️Answer/Explanation
314