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Exam-Style Questions.Problems adapted from questions set for previous Mathematics exams. |
1. | IB Standard |
Consider two perpendicular vectors \(p\) and \(q\).
(a) Let \(r=p-q\). Draw a diagram to show what this relationship might look like.
(b) If \(p=\begin{pmatrix} 4 \\ 1 \\ -3 \\ \end{pmatrix}\) and \(q=\begin{pmatrix} 3 \\ n \\ -5 \\ \end{pmatrix}\), where \(n\in \mathbb Z\), find \(n\).
2. | IB Standard |
The line \(L_1\) passes through the points A(3, 5, 1) and B(3, 6, 0).
(a) Show that \(AB=\begin{pmatrix} 0 \\ 1 \\ -1 \\ \end{pmatrix}\)
(b) Find a direction vector for \(L_1\)
(c) a vector equation for \(L_1\)
Another line \(L_2\) has equation \(\begin{pmatrix} 8 \\ 3 \\ -2 \\ \end{pmatrix} + t \begin{pmatrix} -1 \\ 1 \\ 0 \\ \end{pmatrix}\). The lines \(L_1\) and \(L_2\) intersect at the point P.
(d) Find the coordinates of P.
(e) Find a direction vector for \(L_2\).
(f) Hence, find the angle between \(L_1\) and \(L_2\).
3. | IB Standard |
Two points \(A\) and \(B\) have coordinates (1 , 3 , 6) and (8 , 7 , 10) respectively.
(a) Find \( \overrightarrow{AB} \) in terms of the unit vectors \(i, j\) and \(k\).
(b) Find \(\mid\overrightarrow{AB} \mid\)
Let \( \overrightarrow{AC} = 5i + 2j - k\)
(c) Find the angle between \(AB\) and \(AC\).
(d) Find the area of triangle \(ABC\).
(e) Hence or otherwise find the shortest distance from \(C\) to the line through \(A\) and \(B\).
4. | IB Analysis and Approaches |
Consider the vectors \(\mathbf{a}\) and \(\mathbf{b}\) such that \(\mathbf{a} = \begin{pmatrix} 16 \\ -12 \end{pmatrix} \) and \( |\mathbf{b}| = 11\).
(a) Find the possible range of values for \(|\mathbf{a+b}|\).
Consider the vector \(\mathbf{p}\) such that \(\mathbf{p=a+b}\).
(b) Given that \(|\mathbf{a+b}|\) is a minimum, find \(\mathbf{p}\).
Consider the vector q such that \(\mathbf{q} = \begin{pmatrix}x \\ y \end{pmatrix} \) , where \(x,y \in \mathbb{R} \).
(c) Find \(\mathbf{q}\) such that \(\mathbf{|q| = |b|}\) and \(\mathbf{q}\) is perpendicular to \(\mathbf{a}\).
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The exam-style questions appearing on this site are based on those set in previous examinations (or sample assessment papers for future examinations) by the major examination boards. The wording, diagrams and figures used in these questions have been changed from the originals so that students can have fresh, relevant problem solving practice even if they have previously worked through the related exam paper.
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