AQA Level 2 Further Maths · Level 2 Certificate · Chapter 6: Geometry I

Mensuration, circle theorems, geometric proof and trigonometry

About 90 minutesCalculator allowed113 marks

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Date

Do Now

The first one is what today needs. The others bring back earlier work.

A1
Needed today
Work out the exact value of $sin 4 5^{\circ} cos 4 5^{\circ} + tan^{2} 3 0^{\circ}$ .
[2 marks]
Answer

A2
From chapter 4
Solve $3 y^{2} + 5 y - 2 = 0$ .
[2 marks]
Answer
A3
From chapter 1
Write $\frac{10}{5} + 45$ in the form $a 5$ , where $a$ is an integer.
[2 marks]
Answer
A4
From chapter 5
The point $P (- 2, 6)$ lies on the circle $x^{2} + y^{2} = 40$ . Work out the gradient of the tangent to the circle at $P$ .
[2 marks]
Answer
A5
GCSE Higher
A sphere has a volume of $288 π$ cm³. Work out its radius.
(The volume of a sphere is $\frac{4}{3} π r^{3}$ .)
[2 marks]
Answer $\dots\dots$ cm

Examples, then your turn

Level 2

Your teacher works through each example with you. Then try the one beside it on your own.

Example 1

Solve $4 cos^{2} x + 4 sin x - 1 = 0$ for $0^{\circ} \leq x \leq 36 0^{\circ}$ .

Hint Replace

cos^{2} x

1 - sin^{2} x

4 - 4 sin^{2} x + 4 sin x - 1 = 0

, so

4 sin^{2} x - 4 sin x - 3 = 0

, which is

(2 sin x - 3) (2 sin x + 1) = 0

sin x = \frac{3}{2}

has no solutions.

sin x = - \frac{1}{2}

gives

x = 21 0^{\circ}

x = 33 0^{\circ}

B1
Solve $3 cos^{2} x - sin x - 1 = 0$ for $0^{\circ} \leq x \leq 36 0^{\circ}$ . Give your answers to 1 decimal place where they are not exact.
[4 marks]
Answer $x = \dots\dots$

Example 2

$P A$ and $P B$ are tangents to the circle, centre $O$ , at $A$ and $B$ .

Prove that $O P$ bisects angle $A P B$ .

Hint

O A = O B

, as they are radii. Angle

O A P

= angle

O B P = 9 0^{\circ}

, as a tangent is perpendicular to the radius.

O P

is a side of both triangles. So triangles

O A P

and

O B P

are congruent (RHS), angle

A P O

= angle

B P O

, and

O P

bisects angle

A P B

B2
$T A$ and $T B$ are tangents to the circle, centre $O$ , at $A$ and $B$ . The line $O T$ meets the chord $A B$ at $M$ .
Prove that $O T$ is perpendicular to $A B$ .
[4 marks]

Example 3

Prove that $\frac{sin x}{1 - cos x} - \frac{1}{tan x} \equiv \frac{1}{sin x}$ .

Hint Write

\frac{1}{tan x}

\frac{cos x}{sin x}

and use the common denominator

sin x (1 - cos x)

. The top is

sin^{2} x - cos x + cos^{2} x = 1 - cos x

, which cancels with the

1 - cos x

underneath.

B3
Prove that $\frac{cos x}{1 - sin x} - tan x \equiv \frac{1}{cos x}$ .
[4 marks]

Practice

Level 2

Each question asks for something different. Show your working. Give answers to 3 significant figures, and angles to 1 decimal place, unless the question says otherwise.

C1
A solid cylinder has radius $r$ cm and height $(r + 4)$ cm. Its total surface area is $140 π$ cm².
Work out the value of $r$ .
[3 marks]
Not drawn accurately
Answer $r = \dots\dots$

C2
Each interior angle of a regular polygon is $7$ times its exterior angle. Work out the number of sides of the polygon.
[3 marks]
Answer
C3
$T A$ is a tangent to the circle at $A$ . $A B C D$ is a cyclic quadrilateral. Angle $T A D = (x + 18)^{\circ}$ , angle $A B D = (2 x - 7)^{\circ}$ and angle $D B C = (x + 4)^{\circ}$ .
Work out the size of angle $A D C$ . Give reasons.
[4 marks]
Not drawn accurately
Answerangle $A D C = \dots \dots^{\circ}$
C4
In triangle $A B C$ , $A B = A C$ . $D$ is the point on $A C$ such that $B D = B C$ . Angle $B A C = x$ .
Prove that angle $A B D = 9 0^{\circ} - \frac{3 x}{2}$ .
[3 marks]
Not drawn accurately
C5
$A B C D$ is an isosceles trapezium. $A B$ is parallel to $D C$ . $A B = 18$ cm, $D C = 10$ cm and $A D = B C = 7$ cm.
Not drawn accurately
(a)
Work out the size of angle $D A B$ .
[2 marks]
Answer $\dots \dots^{\circ}$
(b)
Work out the area of the trapezium.
[2 marks]
Answer $\dots\dots$ cm²
Total for C5: 4 marks
C6
Here is the graph of $y = tan x$ for $0^{\circ} \leq x \leq 36 0^{\circ}$ .
You are given that $tan 6 5^{\circ} = t$ . Work out the two values of $x$ , for $0^{\circ} \leq x \leq 36 0^{\circ}$ , for which $tan x = - t$ .
[2 marks]
Answer $x = \dots\dots$
C7
Solve $(3 tan x - 1) (5 sin x + 4) = 0$ for $0^{\circ} \leq x \leq 36 0^{\circ}$ .
[4 marks]
Answer $x = \dots\dots$
C8
Show that $\frac{sin ^{2} x - 1}{sin x cos x} \times tan x$ simplifies to an integer.
[3 marks]
Answer $\dots\dots$
C9
A sector of a circle, centre $O$ , has an angle of $15 0^{\circ}$ . Its area is $15 π$ cm².
Work out the perimeter of the sector. Give your answer in the form $a + bπ$ .
[4 marks]
Not drawn accurately
Answer $\dots\dots$ cm

Exam-style questions

AQA exam style

Answer every question in the space given. Show your working: most of the marks are for method.

D1
$A$ , $D$ and $C$ lie on a straight line. $B D$ is perpendicular to $A C$ . $A B = 43$ cm, angle $B A D = 6 0^{\circ}$ and angle $B C D = 3 0^{\circ}$ .
Not drawn accurately
(a)
Show that $B C = 12$ cm. You must show your working.
[3 marks]
(b)
Work out the exact length of $A C$ . Give your answer in the form $a 3$ .
[3 marks]
Answer $A C = \dots\dots$ cm
Total for D1: 6 marks

D2
Ben solves $tan x = - 1.2$ for $0^{\circ} \leq x \leq 36 0^{\circ}$ . He writes, " $x = - 50. 2^{\circ}$ and $x = 129. 8^{\circ}$ ."
Explain what Ben has done wrong, and give the correct solutions.
[3 marks]
Answer $x = \dots\dots$
D3
$A$ , $B$ and $C$ are points on a circle, centre $O$ . $A B$ is a diameter of the circle. $A B = 29$ cm and $A C = 20$ cm. The tangent to the circle at $B$ meets $A C$ extended at $T$ .
Not drawn accurately
(a)
Write down the size of angle $A C B$ . Give a reason.
[1 mark]
Answer $\dots \dots^{\circ}$
(b)
Work out the length of $B C$ .
[2 marks]
Answer $B C = \dots\dots$ cm
(c)
Work out the length of $B T$ .
[3 marks]
Answer $B T = \dots\dots$ cm
Total for D3: 6 marks
D4
Answer all three parts.
(a)
Show that the equation $2 sin^{2} x - cos x - 1 = 0$ can be written as $2 cos^{2} x + cos x - 1 = 0$ .
[2 marks]
(b)
Hence solve $2 sin^{2} x - cos x - 1 = 0$ for $0^{\circ} \leq x \leq 36 0^{\circ}$ .
[3 marks]
Answer $x = \dots\dots$
(c)
Hence write down the solutions of $2 sin^{2} x - cos x - 1 = 0$ for $36 0^{\circ} \leq x \leq 72 0^{\circ}$ .
[1 mark]
Answer $x = \dots\dots$
Total for D4: 6 marks
D5
A solid cone has base radius $9$ cm and vertical height $12$ cm. The top of the cone is cut off by a cut parallel to the base, $4$ cm below the vertex. The part left is a frustum.
(Curved surface area of a cone $= π r l$ . Volume of a cone $= \frac{1}{3} π r^{2} h$ .)
(a)
Show that the slant height of the whole cone is $15$ cm.
[1 mark]
(b)
Work out the curved surface area of the frustum. Give your answer as a multiple of $π$ .
[3 marks]
Answer $\dots\dots$ cm²
(c)
Work out the volume of the frustum. Give your answer as a multiple of $π$ .
[2 marks]
Answer $\dots\dots$ cm³
Total for D5: 6 marks
D6
A boat sails $12$ km from $P$ to $Q$ on a bearing of $02 5^{\circ}$ . It then sails $9$ km from $Q$ to $R$ on a bearing of $11 5^{\circ}$ .
Not drawn accurately
(a)
Show that angle $P QR = 9 0^{\circ}$ .
[2 marks]
(b)
Work out the distance $P R$ .
[2 marks]
Answer $P R = \dots\dots$ km
(c)
Work out the bearing of $R$ from $P$ . Give your answer to the nearest degree.
[2 marks]
Answer $\dots \dots^{\circ}$
Total for D6: 6 marks
D7
$P A$ is a tangent to the circle at $A$ . $B$ and $C$ are points on the circle, and $P B C$ is a straight line.
(a)
Prove that triangles $P A B$ and $P C A$ are similar.
[3 marks]
(b)
$P B = 4$ cm and $B C = 5$ cm. Hence work out the length of $P A$ .
[2 marks]
Answer $P A = \dots\dots$ cm
Total for D7: 5 marks
D8
Here is the graph of $y = 3 + 2 cos x$ for $0^{\circ} \leq x \leq 36 0^{\circ}$ .
(a)
Write down the greatest value of $y$ and the coordinates of the minimum point.
[2 marks]
Answer $\dots\dots$ ; $(\dots\dots, \dots\dots)$
(b)
Solve $3 + 2 cos x = 2.4$ for $0^{\circ} \leq x \leq 36 0^{\circ}$ .
[3 marks]
Answer $x = \dots\dots$
Total for D8: 5 marks
D9
Which of these is equal to $tan 33 5^{\circ}$ ? Circle your answer.
$tan 2 5^{\circ} - tan 2 5^{\circ} tan 6 5^{\circ} - tan 15 5^{\circ}$
[1 mark]
Answer
D10
A regular hexagon is drawn inside a circle of radius $6$ cm, centre $O$ , with its six vertices on the circle.
Show that the area of the region inside the circle but outside the hexagon is $(36 π - 543)$ cm². You must show your working.
[4 marks]

Extension

Stretch

No route is given. Spend five minutes on a problem before you look at a hint. The hints are on the last page of the worksheet, and you can use one at a time.

E1
Solve $2 sin x tan x = 3$ for $0^{\circ} \leq x \leq 36 0^{\circ}$ .
[4 marks]
Hint 1 · What to try
Write $tan x$ as $\frac{sin x}{cos x}$ and multiply both sides by $cos x$ .
Hint 2 · The first line
$2 sin^{2} x = 3 cos x$ , so $2 - 2 cos^{2} x = 3 cos x$ , which is $2 cos^{2} x + 3 cos x - 2 = 0$ .
Hint 3 · The full method
$(2 cos x - 1) (cos x + 2) = 0$ . $cos x = - 2$ is impossible, so $cos x = \frac{1}{2}$ : $x = 6 0^{\circ}$ or $30 0^{\circ}$ .
Answer $x = \dots\dots$

E2
$x$ is an acute angle and $sin x + cos x = \frac{7}{5}$ .
Work out the two possible values of $tan x$ .
[5 marks]
Hint 1 · What to try
Square both sides and use $sin^{2} x + cos^{2} x = 1$ .
Hint 2 · The first line
$1 + 2 sin x cos x = \frac{49}{25}$ , so $sin x cos x = \frac{12}{25}$ . Now you know the sum and the product of $sin x$ and $cos x$ .
Hint 3 · The full method
They are the roots of $t^{2} - \frac{7}{5} t + \frac{12}{25} = 0$ , that is $(5 t - 3) (5 t - 4) = 0$ . Either $sin x = \frac{3}{5}$ and $cos x = \frac{4}{5}$ , or the other way round.
Answer $tan x = \dots\dots$ or $\dots\dots$
E3
$A B C$ is a right-angled triangle with $A C = 12$ cm, $B C = 35$ cm and $A B = 37$ cm. A circle is drawn inside the triangle so that it touches all three sides.
Work out the radius of the circle.
[4 marks]
Not drawn accurately
Hint 1 · What to try
The radii to the two sides at the right angle make a square of side $r$ with the corner $C$ .
Hint 2 · The first line
The two tangents from a point to a circle are equal. From $A$ both are $12 - r$ ; from $B$ both are $35 - r$ .
Hint 3 · The full method
$A B$ is made of one tangent from $A$ and one from $B$ : $(12 - r) + (35 - r) = 37$ , so $r = 5$ .
Answer $\dots\dots$ cm

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can find lengths, areas and volumes with algebra, including sectors, cones, frustums and polygons.
I can use the circle theorems and give the reason for each step.
I can write a geometric proof using angle facts, circle theorems and similar triangles.
I can use trigonometry in right-angled triangles, with exact values for 30, 45 and 60 degrees.
I can use the graphs and symmetry of sine, cosine and tangent for any angle.
I can solve trigonometric equations, including quadratics in sine or cosine.
I can prove identities and solve equations using the two trigonometric identities.

Answers and mark scheme

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