Past Exam Questions

The lines \(l_{1}\) and \(l_{2}\) have vector equations
\(\mathbf{r}=4 \mathbf{i}-2 \mathbf{j}+\lambda(2 \mathbf{i}+\mathbf{j}-4 \mathbf{k}) \quad \text { and } \quad \mathbf{r}=4 \mathbf{i}-5 \mathbf{j}+2 \mathbf{k}+\mu(\mathbf{i}-\mathbf{j}-\mathbf{k})\)
respectively.
(i) Show that \(l_{1}\) and \(l_{2}\) intersect.
(ii) Find the perpendicular distance from the point \(P\) whose position vector is \(3 \mathbf{i}-5 \mathbf{j}+6 \mathbf{k}\) to the plane containing \(l_{1}\) and \(l_{2}\).
(iii) Find the perpendicular distance from \(P\) to \(l_{1}\).

The lines \(l_{1}\) and \(l_{2}\) have equations
\(l_{1}: \mathbf{r}=6 \mathbf{i}+5 \mathbf{j}+4 \mathbf{k}+\lambda(\mathbf{i}+\mathbf{j}+\mathbf{k}) \quad \text { and } \quad l_{2}: \mathbf{r}=6 \mathbf{i}+5 \mathbf{j}+4 \mathbf{k}+\mu(4 \mathbf{i}+6 \mathbf{j}+\mathbf{k}) .\)

Find a cartesian equation of the plane \(\Pi\) containing \(l_{1}\) and \(l_{2}\).

Find the position vector of the foot of the perpendicular from the point with position vector \(\mathbf{i}+10 \mathbf{j}+3 \mathbf{k}\) to \(\Pi\).

The line \(l_{3}\) has equation \(\mathbf{r}=\mathbf{i}+10 \mathbf{j}+3 \mathbf{k}+v(2 \mathbf{i}-3 \mathbf{j}+\mathbf{k})\). Find the shortest distance between \(l_{1}\) and \(l_{3}\).

[Question 11 is printed on the next page.]

Find a cartesian equation of the plane \(\Pi\) containing the lines
\(\mathbf{r}=3 \mathbf{i}+\mathbf{k}+s(2 \mathbf{i}+\mathbf{j}-\mathbf{k}) \quad \text { and } \quad \mathbf{r}=3 \mathbf{i}-7 \mathbf{j}+10 \mathbf{k}+t(\mathbf{i}-3 \mathbf{j}+4 \mathbf{k})\)

The line \(l\) passes through the point \(P\) with position vector \(6 \mathbf{i}-2 \mathbf{j}+\mathbf{k}\) and is parallel to the vector \(2 \mathbf{i}+\mathbf{j}-4 \mathbf{k}\). Find
(i) the position vector of the point where \(l\) meets \(\Pi\),

(ii) the perpendicular distance from \(P\) to \(\Pi\),

(iii) the acute angle between \(l\) and \(\Pi\).

The line \(l_{1}\) is parallel to the vector \(4 \mathbf{j}-\mathbf{k}\) and passes through the point \(A\) whose posin. \(2 \mathbf{i}+\mathbf{j}+4 \mathbf{k}\). The variable line \(l_{2}\) is parallel to the vector \(\mathbf{i}-(2 \sin t) \mathbf{j}\), where \(0 \leqslant t\lt 2 \pi\), through the point \(B\) whose position vector is \(\mathbf{i}+2 \mathbf{j}+4 \mathbf{k}\). The points \(P\) and \(Q\) are on \(l_{1}\) respectively, and \(P Q\) is perpendicular to both \(l_{1}\) and \(l_{2}\).
(i) Find the length of \(P Q\) in terms of \(t\).

(ii) Hence find the values of \(t\) for which \(l_{1}\) and \(l_{2}\) intersect.

(iii) For the case \(t=\frac{1}{4} \pi\), find the perpendicular distance from \(A\) to the plane \(B P Q\), giving your answer correct to 3 decimal places.

The vectors \(\mathbf{a}\), \(\mathbf{b}\), \(\mathbf{c}\) and \(\mathbf{d}\) in \(\mathbb{R}^{3}\) are given by

\(\mathbf{a}=\begin{pmatrix}2\\ -1\\ 1\end{pmatrix},\quad \mathbf{b}=\begin{pmatrix}1\\ 1\\ 1\end{pmatrix},\quad \mathbf{c}=\begin{pmatrix}0\\ 1\\ -1\end{pmatrix},\quad \mathbf{d}=\begin{pmatrix}3\\ -2\\ 0\end{pmatrix}.\)

Show that \(\\{\mathbf{a},\mathbf{b},\mathbf{c}\\}\) is a basis for \(\mathbb{R}^{3}\).

Express \(\mathbf{d}\) in terms of \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\).

The points \(A, B\) and \(C\) have position vectors \(\mathbf{i}+2 \mathbf{j}+2 \mathbf{k}, 2 \mathbf{i}+4 \mathbf{j}+5 \mathbf{k}\) and \(2 \mathbf{i}+3 \mathbf{j}+4 \mathbf{k}\) respectively. Find \(\overrightarrow{A B} \times \overrightarrow{A C}\).

Deduce, in either order, the exact value of
(i) the area of the triangle \(A B C\),
(ii) the perpendicular distance from \(C\) to \(A B\).

The position vectors of points \(A, B, C\), relative to the origin \(O\), are \(\mathbf{a}, \mathbf{b}, \mathbf{c}\), where
\(\mathbf{a}=3 \mathbf{i}+2 \mathbf{j}-\mathbf{k}, \quad \mathbf{b}=4 \mathbf{i}-3 \mathbf{j}+2 \mathbf{k}, \quad \mathbf{c}=3 \mathbf{i}-\mathbf{j}-\mathbf{k} .\)

Find \(\mathbf{a} \times \mathbf{b}\) and deduce the area of the triangle \(O A B\).

Hence find the volume of the tetrahedron \(O A B C\), given that the volume of a tetrahedron is \(\frac{1}{3} \times\) area of base × perpendicular height.

DO NOT USE A CALCULATOR IN THIS QUESTION. In this question all lengths are in centimetres.

The diagram shows two similar triangles. The height of the smaller triangle is \(1+7 \sqrt{5}\) and the height of the larger triangle is \(a+b \sqrt{5}\), where \(a\) and \(b\) are integers.

Find the values of \(a\) and \(b\).

Solutions to this question by accurate drawing will not be accepted.

The points \(A\) and \(B\) are \((-8,8)\) and \((4,0)\) respectively.

(i) Find the equation of the line \(AB\).

(ii) Calculate the length of \(AB\).

The point \(C\) is \((0,7)\) and \(D\) is the mid-point of \(AB\).

(iii) Show that angle \(ADC\) is a right angle.

The point \(E\) is such that \(\overrightarrow{AE}=\begin{pmatrix}4\\-7\end{pmatrix}\).

(iv) Write down the position vector of the point \(E\).

(v) Show that \(ACBE\) is a parallelogram.

The coordinates of points \(A, B, C\) and \(D\) are as follows. \(A(-4,3) \quad B(6,-9) \quad C(15,10) \quad D(14,-1)\)

The line \(L\) has equation \(y=11 x-75\). The perpendicular bisector of the line \(A B\) meets \(L\) at the point \(E\). Find the area of triangle \(C D E\).

The point \(A\) has coordinates \((1,4)\) and the point \(B\) has coordinates \((5,6)\). The perpendicular bisector of \(A B\) intersects the \(x\)-axis at the point \(C\) and the \(y\)-axis at the point \(D\). Given that \(O\) is the origin, find the area of triangle \(O C D\).

The points \(A\) and \(B\) have coordinates \((2,5)\) and \((10,-15)\) respectively. The point \(P\) lies on the perpendicular bisector of \(AB\). The \(y\)-coordinate of \(P\) is \(-9\).

(a) Find the coordinates of \(P\).

(b) The point \(R\) is the reflection of \(P\) in the line \(AB\). Find the coordinates of \(R\).

The straight line \(y=3x-11\) and the curve \(xy=4-3x-2x^2\) intersect at the points \(A\) and \(B\). The point \(C\), with coordinates \((a,-8)\), where \(a\) is a constant, lies on the perpendicular bisector of the line \(AB\). Find the value of \(a\).

The perpendicular bisector of the line joining the points \((-3,\frac23)\) and \((6,-\frac73)\) passes through the point \((2,k)\). Find the value of \(k\).

A line, \(L\), has equation \(4x+5y=9\). Points \(A\) and \(B\) have coordinates \((-6,7)\) and \((1,9)\), respectively. Find the equation of the line parallel to \(L\) which passes through the midpoint of \(AB\).

The points \(A\), \(B\) and \(C\) have coordinates \((2,6)\), \((6,1)\) and \((p,q)\), respectively. The point \(B\) is the midpoint of \(AC\). Find the equation of the line through \(C\) that is perpendicular to \(AB\), giving your answer in the form \(ax+by=c\), where \(a\), \(b\) and \(c\) are integers.

The line \(AB\) is such that the points \(A\) and \(B\) have coordinates \((-4,6)\) and \((2,14)\) respectively.

(a) The point \(C\), with coordinates \((7,a)\), lies on the perpendicular bisector of \(AB\). Find the value of \(a\).

(b) Given that the point \(D\) also lies on the perpendicular bisector of \(AB\), find the coordinates of \(D\) such that the line \(AB\) bisects the line \(CD\).

The line

\(x+2y=10\)

intersects the two lines satisfying the equation

\(|x+y|=2\)

at the points \(A\) and \(B\).

(a) Show that the point \(C(-5,20)\) lies on the perpendicular bisector of the line \(AB\).

(b) The point \(D\) also lies on this perpendicular bisector. \(M\) is the mid-point of \(AB\). The distance \(CD\) is three times the distance \(CM\). Find the possible coordinates of \(D\).

The points \(A(5,-4)\) and \(C(11,6)\) are such that \(AC\) is the diagonal of a square, \(ABCD\).

(a) Find the length of the line \(AC\).

(b)

(i) The coordinates of the centre, \(E\), of the square are \((8,y)\). Find the value of \(y\).

(ii) Find the equation of the diagonal \(BD\).

(iii) Given that the \(x\)-coordinate of \(B\) is less than the \(x\)-coordinate of \(D\), write \(\overrightarrow{EB}\) and \(\overrightarrow{ED}\) as column vectors.

The curves \(y=x^2+x-1\) and \(2y=x^2+6x-2\) intersect at the points \(A\) and \(B\).

(a) Show that the mid-point of the line \(AB\) is \((2,9)\).

The line \(l\) is the perpendicular bisector of \(AB\).

(b) Show that the point \(C(12,7)\) lies on the line \(l\).

(c) The point \(D\) also lies on \(l\), such that the distance of \(D\) from \(AB\) is two times the distance of \(C\) from \(AB\). Find the coordinates of the two possible positions of \(D\).