Past Exam Questions

Solutions by accurate drawing will not be accepted.

The points \(A\) and \(B\) have coordinates \((-2,4)\) and \((6,10)\) respectively.

(a) Find the equation of the perpendicular bisector of the line \(AB\), giving your answer in the form \(ax+by+c=0\), where \(a\), \(b\) and \(c\) are integers.

The point \(C\) has coordinates \((5,p)\) and lies on the perpendicular bisector of \(AB\).

(b) Find the value of \(p\).

It is given that the line \(AB\) bisects the line \(CD\).

(c) Find the coordinates of \(D\).

The line \(y=5x+6\) meets the curve \(xy=8\) at the points \(A\) and \(B\).

(a) Find the coordinates of \(A\) and of \(B\).

(b) Find the coordinates of the point where the perpendicular bisector of the line \(AB\) meets the line \(y=x\).

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

The points \(A\) and \(B\) are \((4,3)\) and \((12,-7)\) respectively.

(a) Find the equation of the line \(L\), the perpendicular bisector of the line \(AB\).

(b) The line parallel to \(AB\) which passes through the point \((5,12)\) intersects \(L\) at the point \(C\). Find the coordinates of \(C\).

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

Find the equation of the perpendicular bisector of the line joining the points \((4,-7)\) and \((-8,9)\).

(a) Find the equation of the perpendicular bisector of the line joining the points \((12,1)\) and \((4,3)\), giving your answer in the form \(y=mx+c\).

(b) The perpendicular bisector cuts the axes at points \(A\) and \(B\). Find the length of \(AB\).

The points \(A\) and \(B\) have coordinates \((p,3)\) and \((1,4)\) respectively, and the line \(L\) has equation \(3x+y=2\).

(i) Given that the gradient of \(AB\) is \(\frac13\), find \(p\).

(ii) Show that \(L\) is the perpendicular bisector of \(AB\).

(iii) Given that \(C(q,-10)\) lies on \(L\), find \(q\).

(iv) Find the area of triangle \(ABC\).

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

\(P\) is the point \((8,2)\) and \(Q\) is the point \((11,6)\).

(i) Find the equation of the line \(L\), which passes through \(P\) and is perpendicular to the line \(PQ\).

The point \(R\) lies on \(L\) such that the area of triangle \(PQR\) is \(12.5\) units\(^2\).

(ii) Showing all your working, find the coordinates of each of the two possible positions of point \(R\).

Find the equation of the perpendicular bisector of the line joining the points \((1,3)\) and \((4,-5)\). Give your answer in the form \(ax+by+c=0\), where \(a\), \(b\) and \(c\) are integers.

Find the equation of the perpendicular bisector of the line joining the points \((1,3)\) and \((4,-5)\). Give your answer in the form \(ax+by+c=0\), where \(a\), \(b\) and \(c\) are integers.

(a) A straight line passes through the points \((4,23)\) and \((-8,29)\). Find the point of intersection, \(P\), of this line with the line \(y=2x+5\).

(b) Find the distance of \(P\) from the origin.

The points \(P\) and \(Q\) have coordinates \((5,-12)\) and \((15,-6)\) respectively. The point \(R\) lies on the line \(l\), the perpendicular bisector of the line \(PQ\). The \(x\)-coordinate of \(R\) is \(7\).

(a) Find the \(y\)-coordinate of \(R\).

(b) The point \(S\) lies on \(l\) such that its distance from \(PQ\) is \(3\) times the distance of \(R\) from \(PQ\). Find the coordinates of the two possible positions of \(S\).

The coordinates of points \(A\) and \(B\) are \((-5,6)\) and \((4,-6)\) respectively. The point \(C\) lies on the line \(AB\), between \(A\) and \(B\), such that

\(\frac{AC}{CB}=\frac12.\)

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

(b) The line \(CD\) is perpendicular to \(AB\). Find the equation of \(CD\) in the form \(y=mx+c\).

(c) The length of \(BD\) is \(\sqrt{125}\). Find the coordinates of the two possible positions of point \(D\).

Points \(A\) and \(C\) have coordinates \((-4,6)\) and \((2,18)\) respectively. The point \(B\) lies on the line \(AC\) such that \(\overrightarrow{AB}=\frac23\overrightarrow{AC}\).

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

(b) Find the equation of the line \(l\), which is perpendicular to \(AC\) and passes through \(B\).

(c) Find the area enclosed by the line \(l\) and the coordinate axes.

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

The points \(A(3,2)\), \(B(7,-4)\), \(C(2,-3)\) and \(D(k,3)\) are such that \(CD\) is perpendicular to \(AB\). Find the equation of the perpendicular bisector of \(CD\).

The points \(A\), \(B\) and \(C\) have coordinates \((4,7)\), \((-3,9)\) and \((6,4)\) respectively.

(i) Find the equation of the line \(L\), that is parallel to the line \(AB\) and passes through \(C\). Give your answer in the form \(ax+by=c\), where \(a\), \(b\) and \(c\) are integers.

(ii) The line \(L\) meets the \(x\)-axis at the point \(D\) and the \(y\)-axis at the point \(E\). Find the length of \(DE\).

The diagram shows the points \(A(-3,5)\) and \(B(5,-1)\). The midpoint of \(AB\) is \(M\), and the line \(PM\) is perpendicular to \(AB\). The point \(P\) has coordinates \((r,s)\).

(i) Find the equation of the line \(PM\) in the form \(y=mx+c\), where \(m\) and \(c\) are exact constants.

(ii) Hence find an expression for \(s\) in terms of \(r\).

(iii) Given that the length of \(PM\) is \(10\) units, find the value of \(r\) and of \(s\).

Do not use a calculator in this question.

The diagram shows the trapezium \(ABCD\) in which angle \(ADC\) is \(90^\circ\) and \(AB\) is parallel to \(DC\). It is given that

\(AB=4+3\sqrt5,\qquad DC=11+2\sqrt5,\qquad AD=7+\sqrt5.\)

(i) Find the perimeter of the trapezium, giving your answer in simplest surd form.

(ii) Find the area of the trapezium, giving your answer in simplest surd form.

The line \(y=2x+1\) intersects the curve \(xy=14-2y\) at the points \(P\) and \(Q\). The midpoint of \(PQ\) is \(M\).

(i) Show that the point \(\left(-10,\dfrac{23}{8}\right)\) lies on the perpendicular bisector of \(PQ\).

(ii) The line \(PQ\) intersects the \(y\)-axis at \(R\). The perpendicular bisector of \(PQ\) intersects the \(y\)-axis at \(S\). Find the area of triangle \(RSM\).

Variables \(x\) and \(y\) are such that, when \(\lg(2y+1)\) is plotted against \(x^2\), a straight line graph passing through the points \((1,1)\) and \((2,5)\) is obtained.

(a) Find \(y\) in terms of \(x\).

(b) Find the value of \(y\) when \(x=\frac{\sqrt3}{2}\).

(c) Find the value of \(x\) when \(y=2\).

Variables \(x\) and \(y\) are such that when \(\mathrm e^{4y}\) is plotted against \(x\), a straight line of gradient \(\frac25\), passing through \((10,2)\), is obtained.

(a) Find \(y\) in terms of \(x\).

(b) Find the value of \(y\) when \(x=45\), giving your answer in the form \(\ln p\).

(c) Find the values of \(x\) for which \(y\) can be defined.