CIE Extended Mathematics (2017-18) Mensuration

1	$( 2018/s/21/img/q6) $ Calculate the area of a circle with radius 5.1 cm. [2]
2	$( 2018/w/22/img/q21) $ The diagram shows an equilateral triangle ABC with sides of length 10cm. AMN is a sector of a circle, centre A. M is the mid-point of AC. Work out the area of the shaded region. [4]
3	$( 2017/w/23/img/q20) $ The diagram shows a shape made from a square and a semi-circle. Calculate the area of the shape. Give the units of your answer. [5]
4	$( 2018/w/41/img/q10) $ The diagram shows a circle, centre O. The straight line ABC is a tangent to the circle at B. OB = 8cm, AB = 15cm and BC = 22.4cm. AO crosses the circle at X and OC crosses the circle at Y. (a) Calculate angle XOY.[5] (b) Calculate the length of the arc XBY.[2] (c) Calculate the total area of the two shaded regions.[4]
5	$( 2017/s/42/img/q25) $ The diagram shows a hollow cone with radius 3cm and slant height 10cm. (a) (i) Calculate the curved surface area of the cone.[2] [The curved surface area, A, of a cone with radius r and slant height l is $A=\pi r l$.]  (ii) Calculate the perpendicular height of the cone.[3]  (iii) Calculate the volume of the cone.[2] [The volume, V, of a cone with radius r and height h is $V =\frac13 \pi r^2 h$ .] The cone is cut along the line OP and is opened out into a sector as shown in the diagram. Calculate the sector angle $x$.[4] The diagram shows the same sector as in part (b). Calculate the area of the shaded segment.[4]
6	$( 2017/ww/42/img/q10) $ The diagram shows a sector of a circle, a triangle and a rectangle. The sector has centre O, radius x cm and angle 270°. The rectangle has length $2x$ cm. The total area of the shape is $kx^2 cm^2$ . (a) Find the value of k. [5] (b) Find the value of x when the total area is 110 cm$^2$. [2]
7	$( 2018/s/21/img/q7) $ Calculate the length of AC.[2]
8	$( 2018/w23/img/q19) $ The diagram shows a sector of a circle with radius 6cm and sector angle 72°. The perimeter of this sector is $(p+q\pi)$ cm. Find the value of p and the value of q.[3]
9	$( 2017/s/21/img/q19) $ ABCD is a rhombus with side length 10 cm. Angle ADC = 40°. DAC is a sector of a circle with centre D. BAC is a sector of a circle with centre B. Calculate the shaded area. [4]
10	$( 2017/w/22/img/q23) $ The diagram shows a sector of a circle, centre O and radius 6 cm. The sector angle is 30°. The area of the shaded segment is $(k\pi – c)\; cm^2$, where k and c are integers. Find the value of k and the value of c.[3]
11	$( 2018/$m/42/img/q2$) $ The vertices of a square $ABCD$ lie on the circumference of a circle, radius 8 cm. (a) Calculate the area of the square.cm$^2[2]$ (b) (i) Calculate the area of the shaded segment cm$^2$ [3]       (ii) Calculate the perimeter of the shaded segment cm [4]
12	$( 2017/$w/21/img/q20$) $ $(\mathbf{a})$ A cylinder has height 20 cm. The area of the circular cross section is 74 cm$^{2}$ Work out the volume of this cylinder. $\mathrm{cm}^{3}\left[1\right]$ (b) Cylinder $A$ is mathematically similar to cylinder $B$ The height of cylinder $A$ is 10 cm and its surface area is 440 cm$^{2}$ The surface area of cylinder $B$ is 3960cm$^{2}$ Calculate the height of cylinder $B.$ cm [3]
13	$( 2018/$w/21/img/q10$) $ A water tank in the shape of a cuboid has length 1.5 metres and width l metre. The water in the tank is 60 centimetres deep. Calculate the number of litres of water in the tank. litres[3]
14	$( 2017/$m/22/img/q9$) $ The diagram shows a pyramid with a square base $ABCD.$ All the sloping edges of the pyramid are 20cm long and $AC=17$ cm. Calculate the height of the pyramid. cm [3]
15	$( 2018/$s/22/img/q14$) $ The diagram shows a solid cuboid with base area 7 cm$^{2}$ The volume of this cuboid is 21 cm$^{3}$. Work out the total surface area. cm$^2[3]$
16	$( 2018/$s/22/img/q15$) $ Find the volume of a cylinder of radius 5 cm and height 8 cm. Give the units of your answer.[3]
17	$( 2018/$w/22/img/q22$) $ The diagram shows a cuboid with dimensions 5.5 cm, 8cm and 16.2 cm. Calculate the angle between the line $AB$ and the horizontal base of the cuboid. [4]
18	$( 2017/$s$/23/$img$/$q$5) $ Calculate the volume of a hemisphere with radius 3.2 cm. [The volume, $V$, of a sphere with radius $r$ is $V=\frac{4}{3}\pi r^{3}.]$ [2]
19	$( 2017/\mathrm{s/41/img/q5a) }$ (a) The diagram shows a cylindrical container used to serve coffee in a hotel. The container has a height of 50cm and a radius of 18 ctm. (i) Calculate the volume of the cylinder and show that it rounds to 50 900 cm$^{3}$, correct to 3 significant [2] (ii) 30 litres of coffee are poured into the container. Work out the height, $h$, of the empty space in the container $h=\ldots.........$cm[3] (iii) Cups in the shape of a hemisphere are filled with coffee from the container. The radius of a cup is 3.5 cm. Work out the maximum number of these cups that can be completely filled from the 30 litres of coffee in the container. [The volume, $V$, of a sphere with radius $r$ is $V=\frac{4}{3}\pi r^{3}.]$ [4] (b) The hotel also uses glasses in the shape of a cone. The capacity of each glass is 95 cm$^{3}.$ (i) Calculate the radius, $r$ and show that it rounds to 3.3 cm, correct to l decimal place, [The volume, $V$, of a cone with radius $r$ and height $h$ is $V=\frac{1}{3}\pi r^{2}h.]$ [3] (ii) Calculate the curved surface area of the cone. [The curved surface area, $A$ of a cone with radius $r$ and slant height $l$ is $A=\pi rl$ . cm$^2\left[4\right]$
20	$( 2017/\mathrm{w/41/img/q8a) }$ The diagram shows a solid made from a hemisphere and a cone. The base diameter of the cone and the diameter of the hemisphere are each 5 mm. (a) The total surface area of the solid is $\frac {115\pi}4\mathrm{mm^{2}.}$ Show that the slant height, $l$ is 6.5 mm. t height $l$ is $A=\pi rl.]$ [The curved surface area, $A$ of a cone with radius $r$ and slant height $l$ is $A=\pi rl.]$ [The surface area, $A$, of a sphere with radius $r$ is $A=4\pi r^2.]$ [4] (b) Calculate the height, $h$, of the cone. $h=...................mm~[3]$ (c) Calculate the volume of the solid. [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 $V=\frac{4}{3}\pi r^{3}.]$ [4] (d) The solid is made from gold 1 cubic centimetre of gold has a mass of 19.3 grams. The value of 1 gram of gold is $38.62 . Calculate the value of the gold used to make the solid [3]

Answer Key 

1. 47

2. 30.2 or 30.20 to \( 30.21 \ldots \)

3. 139 or 139.2 to \( 139.3 \mathrm{~cm}^{2} \)

4. (a) 132.26 to 132.28 or 132.3 (b) 18.4 or 18.5 or 18.43 to 18.48 (c) 75.7 to 75.9

5. (a)(i) 94.2 or 94.3 or 94.24 to 94.26 (ii) 9.54 or \( 9.539 \ldots \) (iii) 89.9 or 89.90 to \( 89.92 \ldots \) (b) 108 or 107.9 to 108.1 (c) 46.6 to 46.8

6. (a) 5.68 or 5.684 to 5.685 (b) \( 4.4[0] \) or 4.398 to 4.401

7. \( \frac{4}{x^{3}} \)

8. \( [p=] 12,[q=] \frac{12}{5} \)

9. 5.53 or 5.54

10. \( k=3, c=9 \)

11. (a) 128 (b) (i) 18.3 or 18.26 to \( 18.29 \ldots \) (ii) 23.9 or 23.87 to 23.882

12. (a) 1480 (b) 30

13. 900

14. 18.1 or \( 18.10 \ldots \)

15. 62

16. 628 or 628.3 to \( 628.4 \mathrm{~cm}^{3} \)

17. 25.1 or \( 25.06 \ldots \)

18. 68.6

19. (a)(i)50890 or 50893 to 50900.4 (a)(ii)20.5 or 20.52 to 20.534 (a)(iii)334 (b)(i)3.28[6..] or 3.29 (b) (ii) 93.1 to 93.6

20. (a) 6.5 (b) 6 (c) \( 72[.0 \ldots \) or 71.99 (8)(d) 53.7 or 53.65 to 53.67

21. (a) 4.79 or 4.788 to 4.789 (b)(i) \( 8.7[0] \) or 8.702 to 8.704 (ii) 6.4

22. (a) (i) Show (a) (ii) 38.1 or 38.06 to 38.08 (b) 358 or 357.9 to 358

23. (a) (i) 1070 or 1072 . .. (ii) 2.58 or 2.580 to 2.581 (b) (i) 4.24 or 4.241 to 4.242 (ii) 64

24. (a) (i) 1930 or 1940 or 1933.4 to 1935.3 (ii) 893 or 892.8 to \( 893.0 \ldots \) (b) 2.71 or 2.709 to 2.710

25. (a)(i) -1.33 < k < 0  to 0.1 ( a) 17.5 or \( 17.46 \ldots \) (a) (ii) 140 or 139.6 to \( 139.7 \ldots \) (b) (i) 2.62 or \( 2.618 \ldots \) (b) (ii) 10.2 or \( 10.20 \ldots 10 \frac{10}{49} \)

26. (a)(i)25.5 or 25.46 (a)(ii)9.85 or 9.847 (a) (iii) 952 or 952.4 (b)(i) 22.5 or 22.49 (b) (ii) 23.46 (c) 64.9 or 64.92 to 64.94

27. (a) 204 or 203.5 to 203.6 (b) (i) Show (b) (ii)[x \( =] 5.2[0] \) or \( 5.196[\mathrm{y}=] 6 \)

28. (a) (i) 427 or 427.2 to 427.3 (a) (ii) 1010 or 1005 (a) (iii) 804 or 804.2 to 804.4 or 808 

(a) (iv) 396 or 395.6 to 395.8 or 392 (b) (i) \( \frac{1}{54} \) (b) (ii) \( 972 \pi \)