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Geometry and Trigonometry - IB Questionbank | RevisionDojo

Practice Geometry and Trigonometry 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

HLPaper 2

A graph has the degree sequence:

\{5, 5, 4, 3, 3, 2, 2, 2\}

Is this a valid simple graph? Justify using the Havel-Hakimi algorithm.

[4]

How many edges does the graph contain?

[1]

Question 2

SLPaper 1

A builder is assessing a triangular support structure with side lengths $a=5$ cm, $b=12$ cm, and $c=13$ cm to ensure it meets right-angle specifications.

Prove that △ABC is a right-angled triangle, verifying the relationship between the sides.

[3]

Calculate the area of △ABC to determine the space it occupies.

[2]

Question 3

HLPaper 1

Prove the identity:

\sin^2 \theta - \cos^2 \theta = -\cos 2\theta

\sin^2 \theta - \cos^2 \theta = -\cos 2\theta

[4]

Question 4

HLPaper 1

A line $L$ has the vector equation

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

Find the coordinates of the point on line $L$ that is closest to the point $A(3, -2, 1)$ .

[6]

Question 5

SLPaper 2

A circular garden has a radius of 7 m. A path runs along a quarter of the perimeter.

Find the length of the path.

[3]

Question 6

SLPaper 1

A builder is using a right-angled triangular support structure with sides $a=5$ , $b=12$ , and $c=13$ , where $\theta$ is the angle opposite side $a$ .

Find the values of $\sin\theta$ , $\cos\theta$ , and $\tan\theta$ to understand the structural angles.

[3]

Use the Pythagorean theorem to verify the relationship between the sides, confirming the triangle’s right-angle properties.

[2]

Question 7

SLPaper 1

The line joining the points $A(2,-4)$ and $B(6,8)$ meets the $x$ - and $y$ -axes at the points $P$ and $Q$ , respectively.

Find the equation of line $AB$ and calculate the length of the segment $PQ$ .

[5]

Question 8

SLPaper 1

The line through the point $A(2,-3)$ is perpendicular to the line $2x + y = 5$ . This new line intersects the $x$ -axis at point $P$ and the $y$ -axis at point $Q$ .

Calculate the ratio $PA : AQ$ .

[7]

Question 9

HLPaper 1

Given vectors $\vec{a}=\begin{pmatrix}4 \\ -2\end{pmatrix}$ and $\vec{b}=\begin{pmatrix}1 \\ 3\end{pmatrix}$ , find the vector $\vec{r}$ such that $\vec{r}$ is perpendicular to both $\vec{a}$ and $\vec{b}-\vec{a}$ .

[5]

Question 10

HLPaper 1

A line $L$ is defined by the vector equation

\vec{r}=\left[\begin{array}{c} 4 \\ -1 \\ 2 \end{array}\right]+\lambda\left[\begin{array}{c} 1 \\ 3 \\ -2 \end{array}\right]

Find the value of $\lambda$ for which the point on the line lies on the plane $x + y + z = 5$ , and hence find the coordinates of this point.

[4]

Paper

Level

Question 1

HLPaper 2

A graph has the degree sequence:

\{5, 5, 4, 3, 3, 2, 2, 2\}

Is this a valid simple graph? Justify using the Havel-Hakimi algorithm.

[4]

How many edges does the graph contain?

[1]

Question 2

SLPaper 1

A builder is assessing a triangular support structure with side lengths $a=5$ cm, $b=12$ cm, and $c=13$ cm to ensure it meets right-angle specifications.

Prove that △ABC is a right-angled triangle, verifying the relationship between the sides.

[3]

Calculate the area of △ABC to determine the space it occupies.

[2]

Question 3

HLPaper 1

Prove the identity:

\sin^2 \theta - \cos^2 \theta = -\cos 2\theta

\sin^2 \theta - \cos^2 \theta = -\cos 2\theta

[4]

Question 4

HLPaper 1

A line $L$ has the vector equation

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

Find the coordinates of the point on line $L$ that is closest to the point $A(3, -2, 1)$ .

[6]

Question 5

SLPaper 2

A circular garden has a radius of 7 m. A path runs along a quarter of the perimeter.

Find the length of the path.

[3]

Question 6

SLPaper 1

A builder is using a right-angled triangular support structure with sides $a=5$ , $b=12$ , and $c=13$ , where $\theta$ is the angle opposite side $a$ .

Find the values of $\sin\theta$ , $\cos\theta$ , and $\tan\theta$ to understand the structural angles.

[3]

Use the Pythagorean theorem to verify the relationship between the sides, confirming the triangle’s right-angle properties.

[2]

Question 7

SLPaper 1

The line joining the points $A(2,-4)$ and $B(6,8)$ meets the $x$ - and $y$ -axes at the points $P$ and $Q$ , respectively.

Find the equation of line $AB$ and calculate the length of the segment $PQ$ .

[5]

Question 8

SLPaper 1

The line through the point $A(2,-3)$ is perpendicular to the line $2x + y = 5$ . This new line intersects the $x$ -axis at point $P$ and the $y$ -axis at point $Q$ .

Calculate the ratio $PA : AQ$ .

[7]

Question 9

HLPaper 1

[5]

Question 10

HLPaper 1

A line $L$ is defined by the vector equation

\vec{r}=\left[\begin{array}{c} 4 \\ -1 \\ 2 \end{array}\right]+\lambda\left[\begin{array}{c} 1 \\ 3 \\ -2 \end{array}\right]

Find the value of $\lambda$ for which the point on the line lies on the plane $x + y + z = 5$ , and hence find the coordinates of this point.

[4]

Paper

Level

Question 1

HLPaper 2

A graph has the degree sequence:

\{5, 5, 4, 3, 3, 2, 2, 2\}

Is this a valid simple graph? Justify using the Havel-Hakimi algorithm.

[4]

How many edges does the graph contain?

[1]

Question 2

SLPaper 1

A builder is assessing a triangular support structure with side lengths $a=5$ cm, $b=12$ cm, and $c=13$ cm to ensure it meets right-angle specifications.

Prove that △ABC is a right-angled triangle, verifying the relationship between the sides.

[3]

Calculate the area of △ABC to determine the space it occupies.

[2]

Question 3

HLPaper 1

Prove the identity:

\sin^2 \theta - \cos^2 \theta = -\cos 2\theta

\sin^2 \theta - \cos^2 \theta = -\cos 2\theta

[4]

Question 4

HLPaper 1

A line $L$ has the vector equation

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

Find the coordinates of the point on line $L$ that is closest to the point $A(3, -2, 1)$ .

[6]

Question 5

SLPaper 2

A circular garden has a radius of 7 m. A path runs along a quarter of the perimeter.

Find the length of the path.

[3]

Question 6

SLPaper 1

A builder is using a right-angled triangular support structure with sides $a=5$ , $b=12$ , and $c=13$ , where $\theta$ is the angle opposite side $a$ .

Find the values of $\sin\theta$ , $\cos\theta$ , and $\tan\theta$ to understand the structural angles.

[3]

Use the Pythagorean theorem to verify the relationship between the sides, confirming the triangle’s right-angle properties.

[2]

Question 7

SLPaper 1

The line joining the points $A(2,-4)$ and $B(6,8)$ meets the $x$ - and $y$ -axes at the points $P$ and $Q$ , respectively.

Find the equation of line $AB$ and calculate the length of the segment $PQ$ .

[5]

Question 8

SLPaper 1

The line through the point $A(2,-3)$ is perpendicular to the line $2x + y = 5$ . This new line intersects the $x$ -axis at point $P$ and the $y$ -axis at point $Q$ .

Calculate the ratio $PA : AQ$ .

[7]

Question 9

HLPaper 1

[5]

Question 10

HLPaper 1

A line $L$ is defined by the vector equation

\vec{r}=\left[\begin{array}{c} 4 \\ -1 \\ 2 \end{array}\right]+\lambda\left[\begin{array}{c} 1 \\ 3 \\ -2 \end{array}\right]

Find the value of $\lambda$ for which the point on the line lies on the plane $x + y + z = 5$ , and hence find the coordinates of this point.

[4]

Geometry and Trigonometry Questionbank