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Geometry & Trigonometry - IB Questionbank | RevisionDojo

Practice Geometry & Trigonometry with authentic IB Mathematics Analysis and Approaches (AA) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like functions and equations, calculus, complex numbers, sequences and series, and probability and statistics. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

HLPaper 1

Consider two lines in three-dimensional space, given by their vector equations $L_1:\mathbf{r}_1 = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} + t \begin{pmatrix} 4 \\ 5 \\ 6 \end{pmatrix}$ and $L_2:\mathbf{r}_2 = \begin{pmatrix} 7 \\ 8 \\ 9 \end{pmatrix} + s \begin{pmatrix} a \\ 1 \\ 2 \end{pmatrix}$ .

Find the value of $a$ for which the lines are skew.

[7]

Question 2

SLPaper 1

Consider the function $f(x) = 3 \sin(kx + \frac{\pi}{4}) - 1$

For the function $f(x)$ find the amplitude of the function.

[1]

Determine the value of $k$ such that the $f(x)$ has a period of $12\pi$

[3]

Find the $x$ -coordinates of the minima and maxima of the function $f(x)$ within a period of $12\pi$ and $k>0$ under the domain $x\in[0,2\pi]$

[6]

Question 3

HLPaper 1

The motion of a particle is given by the equation $\vec{r}=\binom{3}{5}+t\binom{4}{3}$ , where $t$ is in seconds.

Find initial position of the particle and its position after 2 seconds.

[6]

Find the velocity and the speed of the particle.

[3]

How far from the origin is the particle after 2 seconds.

[4]

How far from its initial position is the particle after 2 seconds.

[4]

Question 4

HLPaper 1

Given vectors $\mathbf{u} = \begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix} 3 \\ -1 \\ 1 \end{pmatrix}$ , and a vector $\mathbf{w}$ that is perpendicular to both $\mathbf{u}$ and $\mathbf{v}$ .

Find a vector $\mathbf{w}$ that is perpendicular to both $\mathbf{u}$ and $\mathbf{v}$ .

[5]

Question 5

HLPaper 1

The position vectors of points $P$ and $Q$ are $p=3 i+2 j+k$ and $q=i+3 j-2 k$

Find the vector product $p \times q$

[4]

Using your answer to part 1., or otherwise, find the area of the parallelogram with two sides $\overrightarrow{O P}$ and $\overrightarrow{O Q}$

[5]

Question 6

HLPaper 1

Prove that: $\tan(\arcsin(x)) = \frac{x}{\sqrt{1 - x^2}}$ for $|x| < 1$

[4]

Question 7

HLPaper 1

Consider a triangle $XYZ$ such that $X$ has coordinates $(0, 0, 0), Y$ has coordinates $(0, 1, 2)$ and $Z$ has coordinates $(2b, 0, b - 1)$ where $b < 0$

Let $N$ be the midpoint of the line segment $XZ$ .

Find, in terms of $b$ , a Cartesian equation of the plane $\Pi$ containing this triangle.

[5]

Find, in terms of $b$ , the equation of the line $L$ which passes through N and is perpendicular to the plane $П$ .

[2]

Question 8

HLPaper 1

Consider the plane $\Pi_1$ given by the equation $x + 2y - z = 4$ and the plane $\Pi_2$ given by the equation $3x - y + 4z = 10$ .

Find the line of intersection of the planes $\Pi_1$ and $\Pi_2$ .

[4]

Verify if the line of intersection found is perpendicular to the normal of either plane.

[3]

Question 9

HLPaper 1

Consider a water tank in the shape of an inverted cone with a height of 10 meters and a base radius of 5 meters. Water is being pumped into the tank at a rate of 3 cubic meters per minute.

Find the rate at which the water level is rising when the water is 4 meters deep.

[5]

Question 10

SLPaper 1

Let $g(x) = \sin(x) + \cos(x)$ .

Find the derivative of the function $g(x)$ .

[1]

Find the equation of the tangent line to the graph of $g(x)$ at $x = \frac{\pi}{4}$ .

[3]

Hence, or otherwise, show that at $x=\frac{\pi}{4}$ the function attains its maximum.

[4]

Geometry & Trigonometry Questionbank