CAIE IGCSE Mathematics Extended 5 Mensuration

Covered syllabus points

E5.1 Units of measure E5.2 Area and perimeter E5.3 Circles, arcs and sectors E5.4 Surface area and volume E5.5 Compound shapes and parts of shapes

[Maximum number: 3]

The diagram shows a right-angled triangle ABC.

The area of the triangle is $60 \mathrm{~cm}^{2}$ .

Show that $2 x^{2}+11 x-105=0$ .

$\frac{(x+3)(2x+5)}2=60$ $2x^2+6x+5x+15$ $2x^2+11x-105=0$

[Maximum number: 3]

NOT TO SCALE

Points A, B, C and D lie on a circle.
ABCD is a square with area $72\,\text{cm}^2$ .

Calculate the area of the circle.
Give your answer as a multiple of $\pi$ .

Figure for Question 2 — CAIE IGCSE Mathematics Extended

$36\pi$ cao.

B2 for answer 113, or 113.0 to $113.1\ldots$ , or an answer in terms of $\pi$ which rounds to 113.
M1 for a correct first step for finding d or r, e.g. $72+72=d^2$ oe, $72=r^2+r^2$ oe, $(\frac{\sqrt{72}}{2})^2+(\frac{\sqrt{72}}{2})^2=r^2$ oe, $\sqrt{72}=\frac{r}{\sin45^\circ}$ oe, or $\frac12 r^2=\frac{72}{4}$ oe.

[Maximum number: 12]

(a)

Figure for Question (a) — CAIE IGCSE Mathematics Extended

NOT TO SCALE

The diagram shows a sector of a circle that is made into a cone by joining OA to OB.
The sector angle is x^ and the radius of the sector is 7.5 cm.
The base radius of the cone is 1.5 cm.

Calculate the value of x.

x=

[ 3 ]

72 or 72.0 cao nfww
M2 for $\frac{x}{360}\times2\pi\times7.5=2\pi\times1.5$ oe
or M1 for $\frac{x}{360}\times2\pi\times7.5$ or $2\pi\times1.5$ oe
OR
M2 for $\frac{x}{360}\times\pi\times7.5^2=\pi\times1.5\times7.5$ oe
or M1 for $\frac{x}{360}\times\pi\times7.5^2$ or $\pi\times1.5\times7.5$ oe

(b)

NOT TO SCALE

The diagram shows a cylinder with radius 8 cm inside a sphere with radius 17 cm . Both ends of the cylinder touch the curved surface of the sphere.

[ 6 ]

(i)

Show that the height of the cylinder is 30 cm .

[ 2 ]

$2\sqrt{17^2-8^2}$ or $\sqrt{34^2-16^2}$ oe
M1 for $17^2=8^2+d^2$ or $34^2=16^2+k^2$

(ii)

Calculate the volume of the cylinder as a percentage of the volume of the sphere.
[The volume, V, of a sphere with radius r is V=4/3 r^3.]

[ 4 ]

29.3 or 29.30 to 29.31
M3 for $\frac{\pi\times8^2\times30}{\frac43\pi\times17^3}\times100$ oe
OR M1 for $\pi\times8^2\times30$ oe
M1 for $\frac43\pi\times17^3$ oe

(c)

NOT TO SCALE

The diagram shows a solid sphere with radius 6 cm inside a cube with side length 20 cm.
The cube contains water to a depth of 15 cm.
The sphere is removed.
Calculate the new depth of water in the cube.
[The volume, V, of a sphere with radius r is V=4/3 r^3.]

[ 3 ]

12.7 or 12.73 to 12.74
B2 for 2.26 or 2.261 to 2.262... soi
or M2 for $\frac{20^2\times15-\frac43\pi\times6^3}{20^2}$ oe, or $15-\frac{\frac43\pi\times6^3}{20^2}$ oe
or M1 for $20^2\times15-\frac43\pi\times6^3$ oe or $20^2D=\frac43\pi\times6^3$ oe
If 0 scored, SC1 for answer 11[.0] or 10.97 to 10.98

[Maximum number: 7]

(a)

The diagram shows a shape made from a major sector AOB and triangles OBC and AOD. $O B=6 \mathrm{~cm}, B C=2 \mathrm{~cm}$ , obtuse angle $A O C=135^{\circ}$ and angle $B C O=90^{\circ}$ .

[ 7 ]

(i)

Calculate the area of the major sector AOB.

[ 3 ]

64.6 or 64.55 to 64.58
M2 for $\frac{360-135-19.5}{360}\times\pi\times6^2$ oe
or M1 for $\frac{k}{360}\times\pi\times6^2$ oe

(ii)

Calculate the total area of the shape.

[ 4 ]

94.2 or 94.3 or 94.15 to 94.27... nfww
M1 for $\frac12\times6\times\text{their }OD\times\sin135$ oe
M1 for $\frac12\times6\times2\times\sin(90-19.5)$ oe or for $\frac12\times\text{their }OC\times2$
M1dep for their (a)(ii) + their two triangle areas

All question bank results loaded

CAIE IGCSE Mathematics Extended 5 Mensuration Topic Practice

Covered syllabus points

Question 1

Question 2

Question 3

Question (a)

Question (b)

Question (i)

Question (ii)

Question (c)

Question 4

Question (a)

Question (i)

Question (ii)