EduNinja

[Maximum number: 7]

(a)

Figure for Question (a) — CAIE IGCSE Mathematics Extended

NOT TO
SCALE
ABCDEF is a regular hexagon.
D F, D A and DB are diagonals.
Complete the following statements using three different triangles.
Triangle DEF is congruent to triangle ......
Triangle ...... is congruent to triangle ......

[ 2 ]

(b)

Figure for Question (b) — CAIE IGCSE Mathematics Extended

NOT TO SCALE
P and Q are points on the circle with centre O.
TP and TQ are tangents to the circle from the point T.
Complete the following statements and reasons.
In triangles OPT and OQT
O P= ...... because each is a radius of the circle
OT is a common side

Angle O P T= angle ...... =90^ because ......
Triangles OPT and OQT are congruent using the criterion ......
This proves that the tangents TP and TQ are ......

[ 5 ]

[Maximum number: 10]

(a)

The scale drawing shows two sides, AB and BC, of a field. The scale is 5 centimetres represents 200 metres.

Scale: 5 cm to 200 m

[ 5 ]

(i)

Measure angle ABC.

Angle A B C=

[ 1 ]

(ii)

X is a point on BC.
$B X=332 \mathrm{~m}$ .
Mark the point X on the diagram.

[ 2 ]

X is 8.3 cm from B2

M1 for $(332 \div 200) \times 5$ oe

(iii)

Find the scale in the form 1: n.

1:

[ 2 ]

1:4000
M1 for $\frac{200}{5}$ or $200\times100$ , both soi

(b)

A bronze statue is 4.5 m high and has a mass of 195200 kg .

The density of bronze is $8000 \mathrm{~kg} / \mathrm{m}^{3}$ .
The volume of a mathematically similar model of the statue is $0.385 \mathrm{~m}^{3}$ .
Calculate the height of the model.
[Density = Mass $\div$ Volume ]

[ 5 ]

1.13 or 1.128 to 1.129
M4 for $4.5\sqrt[3]{\frac{0.385\times8000}{195200}}$ oe, or $\sqrt[3]{\frac{4.5^3\times0.385\times8000}{195200}}$ oe
or M3 for $\sqrt[3]{\frac{0.385}{\text{their }24.4}}$ or $\sqrt[3]{\frac{\text{their }3080}{195200}}$ , or $\frac{0.385}{\text{their }24.4}=\frac{l^3}{4.5^3}$ oe
or M2 for $\frac{\text{their }24.4}{0.385}$ or $\frac{0.385}{\text{their }24.4}$ oe, or B2 for 24.4 or 3080 seen
or M1 for $195200\div8000$ or for $0.385\times8000$

[Maximum number: 3]

A lake has an area of $3 \mathrm{~km}^{2}$ .
On a map the area of the lake is $18.75 \mathrm{~cm}^{2}$ .
Find the scale of the map in the form 1: n.

40000

B2 for 1 cm to 0.4 km or 2.5 cm to 1 km or 1600000000
or M2 for $\sqrt{\frac{3\times10^k}{18.75}}$ oe where k>5
or M1 for area conversion evidence listed in markscheme

[Maximum number: 7]

NOT TO SCALE

In the diagram, S lies on PQ and T lies on PR.
ST is parallel to QR.

(a)

Explain why triangle PQR is mathematically similar to triangle PST.

Give a geometrical reason for each statement you make.

[ 3 ]

Three correct pairs with correct reasons

$\angle PQR=\angle PST$ and corresponding angles
$\angle PRQ=\angle PTS$ and corresponding angles
$\angle QPR=\angle TPS$ and common angle
OR
Two correct pairs with correct reasons and correct similarity condition, e.g. AAA

B2 for two correct pairs with correct reasons
or B1 for one correct pair with correct reason

(b)

$S T=3 \mathrm{~cm}, Q R=9 \mathrm{~cm}$ and $P S=5 \mathrm{~cm}$ .

Work out PQ.

[ 2 ]

15

M1 for $\frac39=\frac5{PQ}$ oe or better

(c)

The area of triangle PST is $2 k \mathrm{~cm}^{2}$ .

Find, in terms of k, the area of quadrilateral QRTS. $\mathrm{cm}^{2}$

[ 2 ]

16k

M1 for $3^2$ or $(\frac13)^2$ oe

CAIE IGCSE Mathematics Extended 4 Geometry Topic Practice

Question 1

Question (a)

Question (b)

Question 2

Question (a)

Question (i)

Question (ii)

Question (iii)

Question (b)

Question 3

Question 4

Question (a)

Question (b)

Question (c)