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Exam-Style Questions.Problems adapted from questions set for previous Mathematics exams. |
1. | IB Standard |
George and Hugo like to fly model airplanes. On one day George's plane takes off from level ground and shortly after that Hugo's plane takes off.
The position of George’s plane \(s\) seconds after it takes off is given by \(\begin{pmatrix} 1 \\ 2 \\ 0 \\ \end{pmatrix} + s\begin{pmatrix} 5 \\ -2 \\ 6 \\ \end{pmatrix} \) where the distances are in metres.
(a) Find the speed of George’s plane to the nearest integer.
(b) Find the height of George’s plane after four seconds.
The position of Hugo’s airplane \(t\) seconds after it takes off is given by \(\begin{pmatrix} 4 \\ -4 \\ 0 \\ \end{pmatrix}+t\begin{pmatrix} 7 \\ -2 \\ 9 \\ \end{pmatrix} \) where the distances are in metres.
(c) Show that the paths of the planes are not perpendicular.
The two airplanes collide at the point \((46, -16, 54)\).
(d) How long after George’s plane takes off does Hugo’s plane take off ?
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 Analysis and Approaches |
Two lines, \(L_1\) and \(L_2\), intersect at point \(P\).
Point \(A(3t, 6, 5)\), where \(t \neq 0\), lies on \(L_2\).
The acute angle between the two lines is \(\frac{\pi}{4}\).
The direction vector of \(L_1\) is \(\begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}\), and \(\overrightarrow{PA} = \begin{pmatrix} 3t \\ 0 \\ 5 + t \end{pmatrix}\).
(a) Find the value of \(t\).
(b) Hence or otherwise, find the shortest distance from \(A\) to \(L_1\).
A plane, \(\Pi\), contains \(L_1\) and \(L_2\).
(c) Find a normal vector to \(\Pi\).
The base of a right cone lies in \(\Pi\), centred at \(A\) such that \(L_1\) is a tangent to its base. The volume of the cone is \(300 \pi\) cubic units.
(e) Find the two possible positions of the vertex of the cone.
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