AHL 3.7—Radians Questionbank

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 $\theta$ be the angle between the two given sides. The triangle has an area of $\frac{5\sqrt{15}}{2}$ cm $^2$ .

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

[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 $OBC$ .

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

[2]

Find an expression for the length of the segment $OC$ .

[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 $\frac{\pi}{6}$ radians about $O$
a reflection in the line $y = \frac{x}{\sqrt{3}}$
a rotation clockwise of $\frac{\pi}{3}$ 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 $\frac{\pi}{6}$ radians about $O$

(ii) reflection in the line $y = \frac{x}{\sqrt{3}}$

(iii) rotation clockwise of $\frac{\pi}{3}$ radians about $O$

[3]

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

[2]

Find $\boldsymbol{P}^2$ .

[1]

Hence state what the value of $\boldsymbol{P}^2$ 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 $\boldsymbol{P}$ .

[1]

Question 6

HLPaper 1

Solve the equation $\sec^2 x + 2\tan x = 0$ for $0 \leqslant x \leqslant 2\pi$ .

[6]

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

Let $u = \tan x$.

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

[2]

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

[3]

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

[2]

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

[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 given point, $\text{O}$, at time $t$ seconds, $t \geq 0$, is given by the equation

$\mathbf{r} = \begin{pmatrix} 1.5 \cos(0.1t^2) \\ 1.5 \sin(0.1t^2) \end{pmatrix}$

where all displacements are in metres.

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

Show that the ball is moving in a circle with its centre at $\text{O}$ and state the radius of the 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 the position 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 the instant at which the string breaks?

[4]

Question 10

HLPaper 2

Consider an equilateral triangle $\triangle ABC$ where each side has a length of $1$ metre. Let $P$ be the midpoint of the segment $AB$ , and let $C$ be the third vertex such that $CP$ is the altitude of the triangle. The point $M$ is the midpoint of the segment $CP$ . A circular arc, centered at $M$ , passes through the points $A$ and $B$ , forming a shaded region (a circular segment) bounded by the arc and the chord $AB$ .

Calculate the radius, $r$ , of the circular arc.

[3]

Find the area of the shaded region.

[3]

Find the length of the circular arc $AB$ .

[3]

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

[2]

Find the area of the shaded region as a fraction of the area of triangle $\triangle ABC$ .

[4]