3.2 Geometry and trigonometry - AHL content Topic Practice

IB Maths AA syllabusMaths AA SL/HLFirst assessment 2025

Practise HL geometry and trigonometry through inverse functions, identities, trig equations, 3D vectors, lines, planes and products.

Exam points

sketch inverse or reciprocal trig graphs with clear domains, endpoints, and intercepts
use compound-angle identities to prove exact values or transform trig expressions before solving
work with vector equations, dot and cross products to test angles, planes, intersections and skew lines

[Maximum number: 6]

This question asks you to investigate lines normal to curves of the form $y=\frac{k^{2}}{x}$ .
The curve H has equation $y=\frac{1}{x}$ where $x \in \mathbb{R}, x \neq 0$ .

Prove that $B \hat{C} A$ is a right angle.

[Maximum number: 4]

Show that $\sin 3 \theta \equiv 3 \sin \theta-4 \sin ^{3} \theta$ .

[Maximum number: 19]

The point P(-1,1,-13) lies on the line $L_{1}$ . The line $L_{1}$ has a direction vector $\left(\begin{array}{l}7 \\ 1 \\ 2\end{array}\right)$ .

(a)

Write down a vector equation for $L_{1}$ in the form $\boldsymbol{r}=\boldsymbol{a}+\lambda \boldsymbol{b}$ .

[ 1 ]

(b)

Find a vector equation for line $L_{2}$ in the form $\boldsymbol{s}=\boldsymbol{c}+\mu \boldsymbol{d}$ , given that $L_{2}$ passes through the points A(2,-4,2) and B(7,-6,1).

[ 2 ]

(c)

Show that $L_{1}$ and $L_{2}$ are skew.

The point N lies on $L_{2}$ .

[ 5 ]

(d)

Find $\overrightarrow{\mathrm{PN}} \cdot \overrightarrow{\mathrm{AB}}$ in terms of $\mu$ .

[ 4 ]

(e)

Given that N is the point on $L_{2}$ that lies closest to P , find the coordinates of N .

[ 3 ]

(f)

Point O denotes the origin (0,0,0).

Find the equation of the plane containing points O, P and N , giving your answer in the form $\alpha x+\beta y+\gamma z=\delta$ , where $\alpha, \beta, \gamma, \delta \in \mathbb{Z}$ .

[ 4 ]

[Maximum number: 22]

The points $\mathrm{A}(1,-4,0), \mathrm{B}(-3,-6,2), \mathrm{C}(-1,-2,4)$ and D form a parallelogram, ABCD , where D is diagonally opposite B.

(a)

Find the coordinates of D.

The diagonals of the parallelogram, $[\mathrm{AC}]$ and $[\mathrm{BD}]$ , intersect at point E .

[ 2 ]

(b)

Find the coordinates of E.

[ 2 ]

(c)

Given that $\overrightarrow{\mathrm{AB}} \times \overrightarrow{\mathrm{AD}}=m\left(\begin{array}{c}-1 \\ 1 \\ -1\end{array}\right)$ , where $m \in \mathbb{Z}^{+}$ , find the value of m.

[ 4 ]

(d)

Hence, find the area of parallelogram ABCD .

The plane, $\Pi_{1}$ , contains the parallelogram ABCD.

[ 4 ]

(e)

Find the Cartesian equation of $\Pi_{1}$ .

A second plane, $\Pi_{2}$ , has Cartesian equation 5 x+y-7 z=1.
The acute angle between $\Pi_{1}$ and $\Pi_{2}$ is $\theta$ .

[ 2 ]

(f)

Show that $\cos \theta=\frac{1}{5}$ .

The line L passes through E and is perpendicular to $\Pi_{1}$ .
The line L intersects the plane $\Pi_{2}$ at point F .

[ 3 ]

(g)

Find the coordinates of F.

[ 5 ]

[Maximum number: 3]

Show that $(1+i \tan \theta)^{4}+(1-i \tan \theta)^{4} \equiv \frac{2 \cos 4 \theta}{\cos ^{4} \theta}$ .

[Maximum number: 5]

Consider the complex number $w=\frac{4 \mathrm{i}}{1+\mathrm{i} \sqrt{3}}$ .

Hence, express $\tan \frac{\pi}{12}$ in the form $p+q \sqrt{3}$ , where $p, q \in \mathbb{Z}$ .

[Maximum number: 21]

The equations of two lines, $L_{1}$ and $L_{2}$ , are given by:

\begin{aligned} & L_{1}: \boldsymbol{r}_{1}=\left(\begin{array}{l} 5 \\ 4 \\ 2 \end{array}\right)+s\left(\begin{array}{c} 1 \\ -1 \\ 1 \end{array}\right) \text { where } s \in \mathbb{R} \\ & L_{2}: \frac{x+1}{2}=y-7=\frac{z+5}{3} \end{aligned}

(a)

Show that the position vector of the point of intersection, $L_{1}$ and $L_{2}$ is $\left(\begin{array}{c}1 \\ 8 \\ -2\end{array}\right)$ . The plane $\Pi$ contains the lines $L_{1}$ and $L_{2}$ .

[ 3 ]

(b)

Write down the equation of $\Pi$ , giving your answer in the form $\boldsymbol{r}=\boldsymbol{a}+\lambda \boldsymbol{b}+\mu \boldsymbol{c}$ where $\lambda, \mu \in \mathbb{R}$ .

(c)

Given that $\boldsymbol{b} \times \boldsymbol{c}=\left(\begin{array}{c}-4 \\ -1 \\ 3\end{array}\right)$ , show that the Cartesian equation of $\Pi$ is 4 x+y-3 z=18.

The plane intersects the coordinate axes at P(4.5,0,0), Q(0, q, 0), and R(0,0, r).

[ 3 ]

(d)

Write down the value of

[ 2 ]

(i)

(ii)

[ 2 ]

(e)

Use a vector method to find the area of the triangle PQR .

Another line, $L_{3}$ , is normal to $\Pi$ and passes through the point of intersection of $L_{1}$ and $L_{2}$ .

[ 5 ]

(f)

Write down an equation for $L_{3}$ in the form $\boldsymbol{r}_{3}=\boldsymbol{m}+\gamma \boldsymbol{n}$ .

[ 2 ]

(g)

Given that the point S(-11,5,7) lies on the line $L_{3}$ , find $\gamma$ .

[ 2 ]

(h)

Hence, find the volume of pyramid PQRS .

[ 4 ]

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Exam points

Question 1(f)

Question 8(a)

Question 10

Question 10(a)

Question 10(b)

Question 10(c)

Question 10(d)

Question 10(e)

Question 10(f)

Question 11

Question 11(a)

Question 11(b)

Question 11(c)(i)

Question 11(c)(ii)

Question 11(d)

Question 11(e)

Question 11(f)

Question 11(b)

Question 11(e)

Question 12

Question 12(a)

Question 12(b)(i)

Question 12(b)(ii)

Question 12(c)

Question 12(c)(i)

Question 12(c)(ii)

Question 12(d)

Question 12(e)

Question 12(f)

Question 12(g)