IB Math AI HL Paper 1 Exam Style Practice Geometry

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Categories IB Math AI HL, Math Applications, Math problems

IB HL Mathematics – Paper 1 Sample Questions

Unit 3: Geometry • IB Paper 1 (No Calculator)

⭐ About These Questions

The following questions are written in the style of IB Mathematics HL Paper 1.
Paper 1 questions require clear algebraic working, geometric reasoning, and exact answers where possible.
No calculator is assumed.

Question 1 — Coordinate Geometry

Points $A(2,3)$ and $B(8,7)$ lie in the coordinate plane.

(a) Find the gradient of line $AB$ .

(b) Find the equation of the perpendicular bisector of $AB$ .

(c) Determine the coordinates of the midpoint of $AB$ .

Solution

(a) Gradient

$m=\frac{7-3}{8-2}=\frac{4}{6}=\frac{2}{3}$

Answer:

$\boxed{\frac{2}{3}}$

(b) Midpoint

$M=\left(\frac{2+8}{2},\frac{3+7}{2}\right)=(5,5)$

Gradient of perpendicular line:

$-\frac{3}{2}$

Equation using point–slope form:

$y-5=-\frac{3}{2}(x-5)$

(c) Midpoint:

$\boxed{(5,5)}$

📊 Marks: M1 A1 A1

Question 2 — Vectors

Let

$\vec{a}= \begin{pmatrix} 3\\ 1 \end{pmatrix}, \qquad \vec{b}= \begin{pmatrix} 2\\ -4 \end{pmatrix}$

(a) Find $\vec{a}+\vec{b}$ .

(b) Find $2\vec{a}-\vec{b}$ .

(c) Find the magnitude of $\vec{a}$ .

Solution

(a)

$\vec{a}+\vec{b}= \begin{pmatrix} 3+2\\ 1-4 \end{pmatrix} = \begin{pmatrix} 5\\ -3 \end{pmatrix}$

(b)

$2\vec{a}= \begin{pmatrix} 6\\ 2 \end{pmatrix}$

$2\vec{a}-\vec{b}= \begin{pmatrix} 6-2\\ 2-(-4) \end{pmatrix} = \begin{pmatrix} 4\\ 6 \end{pmatrix}$

(c)

$|\vec{a}|=\sqrt{3^2+1^2}$

$=\sqrt{10}$

Answer:

$\boxed{\sqrt{10}}$

📊 Marks: M1 A1 A1

Question 3 — Vector Equation of a Line

A line passes through point

$A(1,2,3)$

and has direction vector

$\begin{pmatrix} 2\\ -1\\ 4 \end{pmatrix}$

(a) Write the vector equation of the line.

(b) Find the coordinates of the point when $t=3$ .

(c) Determine whether the point $(7,-1,15)$ lies on the line.

Solution

(a)

Vector equation:

$\vec{r}= \begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix} +t \begin{pmatrix} 2\\ -1\\ 4 \end{pmatrix}$

(b) Substitute $t=3$

$x=1+6=7$

$y=2-3=-1$

$z=3+12=15$

Point:

$(7,-1,15)$

(c)

Since the coordinates match the equation for $t=3$ :

$\boxed{\text{The point lies on the line}}$

📊 Marks: M1 A1 A1

Question 4 — Transformations

The graph of $f(x)=x^2$ is transformed to

$g(x)=2(x-3)^2+1$

(a) Describe the transformations from $f(x)$ to $g(x)$ .

(b) Find the coordinates of the vertex of $g(x)$ .

(c) Determine the axis of symmetry.

Solution

(a)

Transformations:

Horizontal translation right by 3
Vertical stretch factor 2
Vertical translation up by 1

(b)

Vertex form:

$g(x)=2(x-3)^2+1$

Vertex:

$(3,1)$

(c)

Axis of symmetry:

$x=3$

📊 Marks: M1 A1 A1

📚 Continue Learning

These questions are only a small sample of the full IB HL Mathematics Geometry unit.

For the complete lessons, worked examples, and additional IB-style practice questions, visit the full course page.

👉 Continue to the IB Math AI HL Paper 1 Exam Style Practice Course Page

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