IB Mathematics Analysis and Approaches Topic AHL 3.15—classification of Lines Questionbank (HL)

IB Mathematics Analysis and Approaches Topic AHL 3.15—classification of Lines Questionbank (HL) | RevisionDojo

Practice IB Mathematics Analysis and Approaches (AA) Topic AHL 3.15—classification of Lines with authentic exam-style questions for both SL and HL students. This question bank focuses on the exact syllabus content for AHL 3.15—classification of Lines and mirrors Paper 1, 2, 3 style where relevant.

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Paper

Level

Question 1

HLPaper 1

Two lines in three-dimensional space are given by $L_1:\ \mathbf r = \begin{pmatrix} 0\\ 1\\ 2 \end{pmatrix} +\lambda \begin{pmatrix} 1\\ -1\\ 2 \end{pmatrix}, \qquad L_2:\ \mathbf r = \begin{pmatrix} 2\\ 3\\ 0 \end{pmatrix} +\mu \begin{pmatrix} 2\\ -2\\ 4 \end{pmatrix}.$

Write both lines in parametric and Cartesian form.

[3]

Determine whether the lines are skew, parallel, coincident, or intersecting. If they intersect, find the point of intersection.

[4]

Find the acute angle between the lines.

[2]

Let $S = (1, 0, 0)$ . Find the shortest distance from $S$ to line $L_{1}$ , and the coordinates of the foot $F$ of the perpendicular.

[5]

Let $P = (2, 3, 0)$ , a point on $L_{2}$ . Find all points $Q$ on $L_{2}$ such that $P Q$ is perpendicular to the direction of $L_{1}$ .

[4]

Question 2

HLPaper 2

Two lines are given by $L_1:\ \mathbf r = \begin{pmatrix} 1\\ 2\\ -1 \end{pmatrix} +\lambda \begin{pmatrix} 2\\ -1\\ 2 \end{pmatrix}, \qquad L_2:\ \mathbf r = \begin{pmatrix} 5\\ -1\\ 3 \end{pmatrix} +\mu \begin{pmatrix} -2\\ 1\\ 1 \end{pmatrix}.$

Write both lines in parametric and Cartesian form.

[3]

Determine the relationship between the lines and, if they intersect, find the point of intersection.

[4]

Find the acute angle between $L_{1}$ and $L_{2}$ .

[4]

Let $S = (3, 1, 0)$ . Find the shortest distance from $S$ to $L_{1}$ , and the coordinates of the foot $F$ .

[5]

Question 3

HLPaper 1

Two lines in three-dimensional space are given by $L_1: \mathbf{r}=\begin{pmatrix}1\\-2\\3\end{pmatrix}+\lambda\begin{pmatrix}2\\1\\-1\end{pmatrix},\qquad L_2: \mathbf{r}=\begin{pmatrix}5\\0\\1\end{pmatrix}+\mu\begin{pmatrix}1\\-2\\2\end{pmatrix},\qquad \lambda,\mu\in\mathbb{R}.$

Write both lines in parametric and Cartesian form.

[3]

Determine whether the lines are skew, parallel, coincident, or intersecting. If they intersect, find the point of intersection.

[3]

Find the acute angle $θ$ between $L_{1}$ and $L_{2}$ , giving an exact value for $cos θ$ and $θ$ (in degrees).

[3]

Let $S = (2, 0, 5)$ . Find the shortest distance from $S$ to the line $L_{1}$ , and the coordinates of the foot $F$ of the perpendicular from $S$ to $L_{1}$ .

[4]

Find all points $Q$ on $L_{2}$ such that the vector $P Q$ makes an angle of $4 5^{\circ}$ with the direction of $L_{1}$ , where $P$ is the intersection point from Part 2. Give exact parameter values $μ$ and coordinates of $Q$ if they exist.

[5]

Question 4

HLPaper 1

Line $L_{1}$ : $r = 123 + λ 4 - 1 2$ and plane $π$ : $2 x - y + z = 5$ .

Find the point of intersection of the line $L_{1}$ and the plane $π$ .

[4]

Determine if the line $L_{1}$ is parallel, perpendicular, or neither to the normal of the plane $π$ .

[4]

Question 5

HLPaper 1

The line $L_{1}$ has equation $r = - i + 4 j + 2 k + s (3 i - 2 j + k)$ , $s \in R$ , and $L_{2}$ has equation $r = 11 i - 3 j + 13 k + t (- 2 i + j - 3 k)$ , $t \in R$ . Show that the lines $L_{1}$ and $L_{2}$ intersect and find the coordinates of their point of intersection.

[8]

IB Mathematics Analysis and Approaches AHL 3.15—classification of Lines Questionbank

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IB Mathematics Analysis and Approaches AHL 3.15—classification of Lines Questionbank

IB Mathematics Analysis and Approaches (AA) Topic AHL 3.15—classification of Lines Questionbank

Unlock all 16 questions with a free account