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AHL 3.7—Radians - IB Questionbank | RevisionDojo

Practice AHL 3.7—Radians with authentic IB Mathematics Applications & Interpretation (AI) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like core principles, advanced applications, and practical problem-solving. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

SL & HLPaper 1

Convert $\frac{11\pi}{6}$ radians to degrees.

[2]

Determine the value of $\cos\left(\frac{11\pi}{6}\right)$ .

[3]

Explain why angles measured in radians are dimensionless quantities.

[3]

Question 2

HLPaper 1

A circle has a radius of 5 cm.

Express an angle of $240^\circ$ in radians.

[2]

Find the area of the sector subtended by the angle ${240^\circ}$ .

[4]

Determine the length of the arc subtended by the angle ${240^\circ}$ .

[3]

Question 3

HLPaper 1

The lengths of two of the sides in a triangle are 4 cm and 5 cm. Let θ be the angle betweenthe two given sides. The triangle has an area of $\frac{{5\sqrt {15} }}{2}$ cm².

Show that ${\text{sin}}\,\theta = \frac{{\sqrt {15} }}{4}$ .

[1]

Find the two possible values for the length of the third side.

[6]

Question 4

HLPaper 2

The figure shows sector OAB of a circle with centre O and radius $r$ cm. The angle subtended at the centre, $\theta$ , lies between $0 < \theta < \frac{\pi}{2}$ . A perpendicular line is drawn from the point $B$ to meet the radius $OA$ at the point $C$ , forming the right triangle $\triangle OBC$ .

Derive an expression for the area of triangle ( $\triangle OBC$ ) in terms of $r$ and $\theta$ .

[2]

Find an expression for the length of the segment $\overline{OC}$ .

[3]

Calculate the length of the arc (AB) in terms of $r$ and $\theta$ .

[2]

Question 5

HLPaper 2

A flying drone is programmed to complete a series of movements in a horizontal plane relative to an origin O and a set of x-y-axes. In each case, the drone moves to a new position represented by the following transformations:

a rotation anticlockwise of π/6 radians about O
a reflection in the line y = x/√3
a rotation clockwise of π/3 radians about O. All the movements are performed in the listed order.

Write down the matrix for each of the three transformations:

(i) rotation anticlockwise of π/6 radians about O

(ii) reflection in the line y = x/√3

(iii) rotation clockwise of π/3 radians about O

[3]

Find a single matrix P that defines a transformation that represents the overall change in position by multiplying the three matrices in the correct order.

[2]

Find P².

[1]

Hence state what the value of P² indicates for the possible movement of the drone.

[1]

Three drones are initially positioned at the points A, B and C. After performing the movements listed above, the drones are positioned at points A', B' and C' respectively.

Show that the area of triangle ABC is equal to the area of triangle A'B'C'.

[3]

Find a single transformation that is equivalent to the three transformations represented by matrix P.

[1]

Question 6

HLPaper 1

Solve the equation ${\sec ^2}x + 2\tan x = 0,{\text{ }}0 \leqslant x \leqslant 2\pi$ .

Question 7

HLPaper 1

A sector of a circle has a radius of 10 cm and subtends an angle of $\frac{\pi}{3}$ radians at the centre of the circle.

Calculate the arc length of the sector.

[2]

Find the area of the sector.

[3]

Draw a diagram of the circle, shading the given sector that subtends an angle of $\frac{\pi}{3}$ .

[2]

Question 8

HLPaper 2

Consider the function $f(x) = 2{\sin ^2}x + 7\sin 2x + \tan x - 9,{\text{ }}0 \leqslant x < \frac{\pi }{2}$ .

Let $u = \tan x$ .

Express $\sin 2x$ in terms of $u$ .

[3]

Hence show that $f(x) = 0$ can be expressed as ${u^3} - 7{u^2} + 15u - 9 = 0$ .

[2]

Determine an expression for $f’(x)$ in terms of $x$ .

[2]

Express $\sin x$ in terms of $\mu$ .

[2]

Solve the equation $f(x) = 0$ , giving your answers in the form $\arctan k$ where $k \in \mathbb{Z}$ .

[3]

Question 9

HLPaper 2

A ball is attached to the end of a string and spun horizontally. Its position relative to a givenpoint, $O$ , at time $t$ seconds, $t \geq 0$ , is given by the equation

$r = (\begin{matrix} 1.5 \cos (0.1 t^{2}) \\ 1.5 \sin (0.1 t^{2}) \end{matrix})$ where all displacements are in metres.

The string breaks when the magnitude of the ball’s acceleration exceeds $20 {ms}^{- 2}$ .

Show that the ball is moving in a circle with its centre at $O$ and state the radius ofthe circle.

[4]

Find an expression for the velocity of the ball at time $t$ .

[2]

Hence show that the velocity of the ball is always perpendicular to theposition vector of the ball.

[2]

Find an expression for the acceleration of the ball at time $t$ .

[3]

Find the value of $t$ at the instant the string breaks.

[3]

How many complete revolutions has the ball completed from $t = 0$ to theinstant at which the string breaks?

[3]

Question 10

HLPaper 2

Consider an equilateral triangle $\triangle ABC$ where each side has a length of 1 metre. The point $M$ is the midpoint of the segment $CP$ , and $P$ is the midpoint of the segment $AB$ . A circular arc, centered at $M$ , passes through the points $A$ and $B$ , forming a shaded region as shown in the diagram.

Calculate the distance from point $A(2,5)$ to the midpoint $M(4,1)$ .

[3]

Find the area of the shaded region enclosed by the circular arc and the line segment (AB).

[3]

Find the length of the circular arc (AB) formed by the arc passing through points $A$ and $B$ .

[3]

Calculate the area of the equilateral triangle ( $\triangle ABC$ ).

[2]

Find the fraction of the total area of the triangle ( $\triangle ABC$ ) covered by the shaded region.

[4]

AHL 3.7—Radians Questionbank