IB Mathematics Applications & Interpretation Topic AHL 3.9—matrix Transformations Questionbank (HL)

IB Mathematics Applications & Interpretation Topic AHL 3.9—matrix Transformations Questionbank (HL) | RevisionDojo

Practice IB Mathematics Applications & Interpretation (AI) Topic AHL 3.9—matrix Transformations with authentic exam-style questions for both SL and HL students. This question bank focuses on the exact syllabus content for AHL 3.9—matrix Transformations and mirrors Paper 1, 2, 3 style where relevant.

Get instant solutions, detailed explanations, and build confidence with questions aligned to IB examiner expectations.

Paper

Level

Question 1

HLPaper 1

Consider three points $P (1, 2, - 1)$ , $Q (4, 3, 2)$ , and $R (0, - 1, 5)$ .

Find the vector $P Q$ .

[1]

Calculate the magnitude of $P Q$ .

[2]

Determine the dot product of vectors $P Q$ and $P R$ .

[3]

Find the cosine of the angle between $P Q$ and $P R$ .

[3]

Verify if the vectors are perpendicular.

[1]

Question 2

HLPaper 2

A computer-controlled laser engraver is calibrated to perform a sequence of coordinate transformations on a design positioned in the $x$ - $y$ plane relative to the origin $O$ . The following three operations are applied to every point on the design:

an anticlockwise rotation about $O$ through $\frac{2 π}{3}$ radians
a mirror reflection across the line $y = - x$
a clockwise rotation about $O$ through $\frac{π}{6}$ radians. The computer executes the transformations in the order they are listed.

Determine the matrix for an anticlockwise rotation about $O$ through $\frac{2 π}{3}$ radians.

[1]

Determine the matrix for a mirror reflection across the line $y = - x$ .

[1]

Determine the matrix for a clockwise rotation about $O$ through $\frac{π}{6}$ radians.

[1]

Determine the composite matrix $T$ that represents the resulting mapping by finding the product of these three matrices in the appropriate order.

[2]

Calculate $T^{2}$ .

[2]

Hence state what the value of $T^{2}$ indicates about the final position of the design if the complete sequence of operations is performed twice.

[1]

Consider three points $U, V$ , and $W$ on the original design. After the laser engraver applies the transformations, these points move to $U^{'}, V^{'}$ , and $W^{'}$ respectively.

Show that the area of triangle $U V W$ is equal to the area of triangle $U^{'} V^{'} W^{'}$ .

[3]

Describe the specific geometric transformation that corresponds to matrix $T$ .

[1]

Question 3

HLPaper 1

In a game animation, each vertex with position vector $r$ is updated by $T : r \mapsto r^{'}$ , where $(x^{'} y^{'}) = (0 - 1 11) (x y) + (51)$ .

Calculate the image of the point $(- 4, 3)$ under $T$ .

[2]

Determine the coordinates $(u, v)$ of the point whose image is twice its position vector.

[3]

Let $R$ be a region in the plane. Justify why the area of $T (R)$ is equal to the area of $R$ .

[2]

Question 4

HLPaper 2

A pharmacologist is modelling the concentration of a medication in a patient's bloodstream following an intramuscular injection. The variable $x$ represents the concentration of the medication in milligrams per litre ( $mg L^{- 1}$ ) at time $t$ hours after the injection.

The initial concentration in the bloodstream is $0 mg L^{- 1}$ and the rate of increase of the concentration is initially $5 mg L^{- 1} hr^{- 1}$ .

Assume the constants in the differential equations have the appropriate units (in $hr^{- 1}$ and $hr^{- 2}$ ) so that expressions such as $e^{λ t}$ are dimensionless.

Show that the system of coupled first order differential equations: $\frac{d x}{d t} = y, \frac{d y}{d t} = - 6 x - 5 y$ can be written as the second order differential equation: $\frac{d ^{2} x}{d t ^{2}} + 5 \frac{d x}{d t} + 6 x = 0$

[2]

Find the eigenvalues of the system of coupled first order equations given in the previous part.

[3]

Hence find the exact solution for the concentration $x$ at time $t$ .

[3]

Sketch the graph of $x$ against $t$ , labelling the coordinates of the point where the concentration is at its maximum.

[2]

The medication is considered to be therapeutically effective only when the concentration in the bloodstream is at least $0.5 mg L^{- 1}$ .

Calculate the total duration for which the medication is effective.

[3]

Suggest one limitation of using this model to predict the drug concentration over a long period of time.

[1]

Question 5

HLPaper 1

In a robotics simulation, each waypoint with position vector $r$ is updated by $S : r \mapsto r^{'}$ , where $(x^{'} y^{'}) = (1 - 1 2 - 1) (x y) + (2 - 3)$ .

Calculate the image of the point $(3, - 2)$ under $S$ .

[2]

Determine the coordinates $(u, v)$ of the point whose image is twice its position vector.

[3]

Let $R$ be a region in the plane. Justify why the area of $S (R)$ is equal to the area of $R$ .

[2]

IB Mathematics Applications & Interpretation AHL 3.9—matrix Transformations Questionbank

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IB Mathematics Applications & Interpretation AHL 3.9—matrix Transformations Questionbank

IB Mathematics Applications & Interpretation (AI) Topic AHL 3.9—matrix Transformations Questionbank

Unlock all 13 questions with a free account