CAIE IGCSE Mathematics 6. Topic 6 Question Bank

Practise triangle geometry using Pythagoras, trigonometric ratios, exact values, graphs and non-right-angled triangle methods.

Covered syllabus points

E6.1 Pythagoras’ theorem E6.2 Right-angled triangles E6.3 Exact trigonometric values E6.4 Trigonometric functions E6.5 Non-right-angled triangles E6.6 Pythagoras’ theorem and trigonometry

[Maximum number: 9]

The quadrilateral ACDE is formed by two right-angled triangles ABE and BCD. $A C=17 \mathrm{~cm}, A E=18 \mathrm{~cm}$ and $B D=6 \mathrm{~cm}$ .

(a)

Show that $C D=10 \mathrm{~cm}$ .

[ 5 ]

$\frac{AB}{18}=\cos60^\circ$ or $AB=18\cos60^\circ$
M1
$\cos60^\circ=\frac{1}{2}$ and AB=9
A1
Correct use of Pythagoras' theorem, i.e. $CD^2=6^2+(17-\text{their }AB)^2$ oe
M1
Correct evaluation for their AB: $CD^2=36+64$ or $CD=\sqrt{36+64}$
M1
100=10
A1

(b)

Find the perimeter of the quadrilateral ACDE.

Give your answer in the form $p+k \sqrt{q}$ .

[ 4 ]

$39+9\sqrt{3}$
B3 for $BE=9\sqrt{3}$ or for answer equivalent to $39+9\sqrt{3}$ but not in required form
OR M2 for $\frac{BE}{18}=\frac{\sqrt3}{2}$ oe or better
or M1 for $\frac{BE}{18}=\sin60^\circ$ oe or better
M1 for 18+17+10+ their BE-6 oe
M2 for $\sqrt{18^2-\text{their }9^2}$ oe
or M1 for $BE^2+(\text{their }9)^2=18^2$ oe

[Maximum number: 10]

(a)

NOT TO SCALE

The diagram shows a field, ABCD with B north of A.
BD is a path across the field.
$A B=85 \mathrm{~m}, A D=72 \mathrm{~m}, B D=129 \mathrm{~m}$ , angle $B D C=39^{\circ}$ and angle $B C D=60^{\circ}$ .

[ 10 ]

(i)

Calculate CD.

[ 3 ]

147 or 147.1...
M2 for $\frac{129\sin81^\circ}{\sin60^\circ}$ oe
or M1 for $\frac{\sin81^\circ}{CD}=\frac{\sin60^\circ}{129}$ oe

(ii)

Show that angle $A B D=31.6^{\circ}$ , correct to 1 decimal place.

[ 4 ]

$[\cos=]\frac{85^2+129^2-72^2}{2\times85\times129}$
M2
M1 for $72^2=85^2+129^2-2\times85\times129\cos ABD$ $31.58\ldots$
A2
A1 for 0.851 to 0.852 or $\frac{9341}{10965}$ or equivalent fraction

(iii)

Find the shortest distance from A to BD.

[ 3 ]

44.5 or 44.51 to 44.54
M2 for implicit correct method, e.g. $\frac{d}{85}=\sin31.6^\circ$ oe
or M1 for recognition that the line from A is perpendicular to BD

[Maximum number: 7]

(a)

On the axes, sketch the graph of $y=\sin x$ for $0^{\circ} \leqslant x \leqslant 360^{\circ}$ .

Figure for Question (a) — CAIE IGCSE Mathematics Extended

[ 2 ]

(b)

Describe fully the symmetry of the graph of $y=\sin x$ for $0^{\circ} \leqslant x \leqslant 360^{\circ}$ .

[ 2 ]

(c)

Solve $4 \sin x-1=2$ for $0^{\circ} \leqslant x \leqslant 360^{\circ}$ .
x= and x=

[ 3 ]

48.6 or 48.59 to 48.60 and 131.4 or 131.40 to 131.41
B2 for one correct value, or M1 for $\sin x=0.75$ or better.
If 0 scored, SC1 for two answers adding to 180.

[Maximum number: 16]

NOT TO SCALE

The diagram shows triangle ABC on horizontal ground.
$A C=15 \mathrm{~m}, B C=8 \mathrm{~m}$ and $A B=20 \mathrm{~m}$ .
BP and CQ are vertical poles of different heights.
$B P=3 \mathrm{~m}$ and $C Q=4 \mathrm{~m}$ .
AQ and PQ are straight wires.

(a)

Show that angle $ACB=117.5^\circ$ , correct to 1 decimal place.

[ 4 ]

$\cos ACB=\frac{15^2+8^2-20^2}{2\times15\times8}$
M2
117.54 to 117.55
A2
A1 for $-\frac{37}{80}$ , $-\frac{111}{240}$ , or -0.4625

(b)

Calculate the area of triangle ABC.

[ 2 ]

53.2 or 53.19 to 53.23
M1 for $0.5\times8\times15\times\sin117.5^\circ$ oe

(c)

Calculate the length of AQ.

[ 2 ]

15.5 or 15.52 to 15.53
M1 for $15^2+4^2$ oe

(d)

Calculate the angle of elevation of Q from P.

[ 3 ]

7.1 or 7.13 or 7.125 to 7.126
M2 for $\tan P=\frac{4-3}{8}$ oe or M1 for vertical line =4-3

(e)

Another straight wire connects A to the midpoint of PQ.

Calculate the angle between this wire and the horizontal ground.

[ 5 ]

11.5 nfww or 11.48 to 11.49...
B1 for height 3.5
M2 for $15^2+4^2-2\times15\times4\cos117.5^\circ$
M1 for $\tan=\frac{3.5}{\text{their }17.216...}$ oe

All question bank results loaded

Covered syllabus points

6. Trigonometry question 1

Question (a)

Question (b)

6. Trigonometry question 2

Question (a)

Question (i)

Question (ii)

Question (iii)

6. Trigonometry question 3

Question (a)

Question (b)

Question (c)

6. Trigonometry question 4

Question (a)

Question (b)

Question (c)

Question (d)

Question (e)