CAIE IGCSE Additional Math Geometry and Trigonometry Questions

Use this geometry-and-trigonometry unit hub to connect coordinate lines and circles with circular measure, exact trigonometric values, graphs, identities and equations.

Exam points

Solve coordinate line and circle problems using gradients, distances, ratios, equations, intersections or tangents.
Use radian measure, arc lengths, sector areas and compound circular regions in exact or numerical form.
Interpret trig values and graphs, prove identities and solve equations over stated degree or radian intervals.

Covered syllabus points

7. Straight-line graphs 8. Coordinate geometry of the circle 9. Circular measure 10. Trigonometry

[Maximum number: 6]

point A has coordinates ( -2,4 ).
The point B has coordinates (6,10).
The point C has coordinates (12,2).

(a)

Find the gradients of the lines A B, A C and B C.

[ 2 ]

Grad $A B=\frac{6}{8}$ oe

Grad $A C=-\frac{2}{14}$ oe

Grad $B C=-\frac{8}{6}$ oe B1 for 2 correct unsimplified

(b)

Hence find the equation of the circle which passes through the points A, B and C.

[ 4 ]

Centre (5,3) B1 FT on their diameter, allow for
(2,7) or (9,6)
Radius ${ }^{2}=50$ oe soi M1 for use of their centre and
coordinates of either A, B or C to
find the square of the radius or the
radius. Centre must not be A, B or
C.
or
Finding the length between any 2
points A, B or C and halving it.
$(x-5)^{2}+(y-3)^{2}=50$ oe A1 Allow $(5 \sqrt{2})^{2}$ for radius ${ }^{2}$

[Maximum number: 5]

Solutions to this question by accurate drawing will not be accepted.
A circle has centre (4,2) and meets the x-axis at (-2,0).

(a)

Find the equation of the circle.

[ 2 ]

$(x-4)^2+(y-2)^2=40$ oe, isw
or
$x^2+y^2-8x-4y-20=0$ oe, isw
B1 for $r^2=40$ or $r=\sqrt{40}$ oe, soi.

(b)

Find, in exact form, the coordinates of the points where the circle meets the y-axis.

[ 3 ]

$16+(y-2)^2=40$ oe
M1 FT their equation of the circle, providing it is the equation of a circle.
$(y-2)^2=24$ or $y^2-4y-20=0$
A1 for correct quadratic equation in any form.
$(0,2\pm\sqrt{24})$ isw or equivalent exact form
A1

[Maximum number: 7]

(a)

Figure for Question (a) — CAIE IGCSE Additional Math

The diagram shows an isosceles triangle.
Find the value of $\theta$ in radians.

[ 2 ]

$2\sin^{-1}\left(\frac{4}{5}\right)$ or $\cos^{-1}\left(\frac{64-25-25}{-2(5)(5)}\right)$ oe
1.85 or 1.854 to 1.855

(b)

Figure for Question (b) — CAIE IGCSE Additional Math

This diagram shows a shape made of two arcs.
Each arc is part of a circle with radius 5 cm.
The height of the shape is 8 cm.
Use your answer to part (a) to find

[ 5 ]

(i)

the perimeter of the shape

[ 2 ]

$5\times\text{their }1.85\times 2$
M1 FT their 1.85 providing it is less than $2\pi$ .
18.5 or 18.54 to 18.55
A1

(ii)

the area of the shape.

[ 3 ]

$\frac{1}{2}\times5^2\times\text{their }1.85$
M1 FT their 1.85 providing it is less than $2\pi$ .
Correct plan, e.g. $[2\times\frac{1}{2}\times]5^2(\text{their }1.85-\sin(\text{their }1.85))$
M1 dep on previous M1.
22.4 or 22.21 to 22.38 nfww
A1

[Maximum number: 7]

(a)

The diagram shows an equilateral triangle ABC with side a.
M is the midpoint of AC and angle $AMB=90^\circ$ .
Use the diagram to find $\sec 30^\circ$ .

[ 3 ]

$\frac{2}{\sqrt{3}}$ or $\frac{2\sqrt{3}}{3}$
B2 for $[BM=]\frac{\sqrt{3}a}{2}$ or $\sec^2 30=1+\left(\frac{1}{\sqrt{3}}\right)^2$ or $\cos^2 30=1-\left(\frac{a/2}{a}\right)^2$ oe
B1 for $[BM^2=]a^2-\frac{a^2}{4}$ soi

(b)

Show that $\frac{1}{\sec x-1}+\frac{1}{\sec x+1}$ can be written as $2\operatorname{cosec}x\cot x$ .

[ 4 ]

$\frac{2\sec x}{(\sec x-1)(\sec x+1)}$ soi
B1 for adding the two fractions.
$\frac{2\sec x}{\sec^2 x-1}$
B1 dep on previous B1 for expanding and simplifying the denominator.
$\frac{2\sec x}{\tan^2 x}$ or $\frac{2}{\tan^2 x\cos x}$
B1 dep on previous B1 for use of $\sec^2 x=1+\tan^2 x$ .
$\frac{2\sec x}{\tan x}\times\cot x$ or $\frac{2\cos x}{\sin^2 x}$ oe or $\frac{2}{\tan x\sin x}$ , leading to $2\operatorname{cosec}x\cot x$ AG
B1 dep on previous B1 with at least one more correct step to get to the given answer.
Alternative method:
$\begin{aligned} \frac{1}{\frac{1}{\cos x}-1}+\frac{1}{\frac{1}{\cos x}+1}&=\frac{\cos x}{1-\cos x}+\frac{\cos x}{1+\cos x} \end{aligned}$ $=\frac{\cos x+\cos^2 x+\cos x-\cos^2 x}{1-\cos^2 x}$
(B1) dep on previous B1 for adding the two fractions.
$=\frac{2\cos x}{\sin^2 x}$
(B1) dep on previous B1 for use of $\sin^2 x=1-\cos^2 x$ .
$=2\operatorname{cosec}x\times\frac{\cos x}{\sin x}=2\operatorname{cosec}x\cot x$ AG

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Covered syllabus points

Question 1

Question (a)

Question (b)

Question 2

Question (a)

Question (b)

Question 3

Question (a)

Question (b)

Question (i)

Question (ii)

Question 4

Question (a)

Question (b)