1. $(2011 /$ june $/$ paper01/q8)
The points $A$ and $B$ have coordinates $(1,5)$ and $(9,7)$ respectively.
(a) Find an equation of $A B$, giving your answer in the form $y=a x+b$, where $a$ and $b$ are rational numbers.
The line $l$ is the perpendicular bisector of $A B$.
(b) Find an equation of $l .$ (4)
The point $C$ has coordinates $(3, q)$. Given that $C$ lies on $l$
(c) find the value of $q$.
The line $l$ meets the $x$-axis at the point $D$.
(d) Find the exact area of the kite $A C B D$.
2. $(2012 / \mathrm{jan} /$ paper02 $/ \mathrm{q} 3)$
Find the coordinates of the points of intersection of the curve with equation $y=3+6 x-x^{2}$ and the line with equation $y-x=7$
3. $(2012 / \mathrm{jan} / \mathrm{paper} 02 / \mathrm{q} 7)$
The points $A, B$ and $C$ have coordinates $(3,5),(7,8)$ and $(6,1)$ respectively.
(a) Show, by calculation, that $A B$ is perpendicular to $A C$.
(b) Find an equation for $A C$ in the form $a x+b y+c=0$, where $a, b$ and $c$ are integers whose values must be stated.
The point $D$ is on $A C$ produced and $A C: C D=1: 2$
(c) Find the coordinates of $D$ (2)
(d) Calculate the area of triangle $A B D$. (4)
4. (2012/june/paper01/q10)
The point $A$ has coordinates $(-3,4)$ and the point $C$ has coordinates $(5,2) .$ The mid-point of $A C$ is $M$. The line $l$ is the perpendicular bisector of $A C$.
(a) Find an equation of $l$.
(b) Find the exact length of $A C$.
The point $B$ lies on the line $l .$ The area of triangle $A B C$ is $17 \sqrt{2}$
(c) Find the exact length of $B M$.
(d) Find the exact length of $A B$.
(e) Find the coordinates of each of the two possible positions of $B$. (6)
5. (2013/jan/Paper01/q7)
The point $C$ with coordinates $(2,1)$ is the centre of a circle which passes through the point $A$ with coordinates $(3,3)$
(a) Find the radius of the circle.
The line $A B$ is a diameter of the circle.
(b) Find the coordinates of $B$. (2)
The points $D$ with coordinates $(0,2)$ and $E$ with coordinates $(4,0)$ lie on the circle.
(c) Show that $D E$ is a diameter of the circle.
The point $P$ has coordinates $(x, y)$.
(d) Find an expression, in terms of $x$ and $y$, for the length of $C P$. $(2)$
Given that the point $P$ lies on the circle,
(e) show that $x^{2}+y^{2}-4 x-2 y=0$ (2)
6. $(2013 / \mathrm{jan} / \mathrm{paper} 02 / \mathrm{q} 7)$
The line $/$ passes through the points with coordinates $(1,6)$ and $(3,2)$.
(a) Show that an equation of $l$ is $y+2 x=8$
The curve $C$ has equation $x y=8$
(b) Show that $l$ is a tangent to $C$.
Given that $l$ is the tangent to $C$ at the point $A$,
(c) find the coordinates of $A$. (2)
(d) Find an equation, with integer coefficients, of the normal to $C$ at $A$.
7. (2013/june/paper02/q8)
The equation of line $l_{1}$ is $2 x+3 y+6=0$
(a) Find the gradient of $l_{1}$
The line $l_{2}$ is perpendicular to $l_{1}$ and passes through the point $P$ with coordinates $(7,2)$.
(b) Find an equation for $l_{2}$ (3)
The lines $l_{1}$ and $l_{2}$ intersect at the point $Q$.
(c) Find the coordinates of $Q$. (3)
The line $l_{3}$ is parallel to $l_{1}$ and passes through the point $P$.
(d) Find an equation for $l_{3}$
The line $l_{1}$ crosses the $x$-axis at the point $R$.
(e) Show that $P Q=Q R$. (3)
The point $S$ lies on $l_{3}$
The line $P R$ is perpendicular to $Q S$.
(f) Find the exact area of the quadrilateral $P Q R S$. (3)
8. (2014/jan/paper02/q1)
The points $A$ and $B$ have coordinates $(5,9)$ and $(9,3)$ respectively. The line $l$ is the perpendicular bisector of $A B$.
Find an equation for $l$ in the form $a x+b y+c=0$, where $a, b$ and $c$ are integers.
9. (2014/june/paper01/q9)
The points $A$ and $B$ have coordinates $(2,5)$ and $(16,12)$ respectively. The point $D$ with coordinates $(8,8)$ lies on $A B$.
(a) Find, in the form $p: q$, the ratio in which $D$ divides $A B$ internally.
The line $l$ passes through $D$ and is perpendicular to $A B$.
(b) Find an equation of $l$. (4)
The point $E$ with coordinates ( $e, 6)$ lies on $l$.
(c) Find the value of $e$.
The line $E D$ is produced to $F$ so that $E D=D F$.
(d) Find the coordinates of $F$.
(e) Find the area of the kite $A E B F$. (3)
10. (2014/june/paper02/q4)
(a) Find the coordinates of the points where the line with equation $y=4 x-4$ meets the curve with equation $y=x^{2}-3 x+6$
(b) Hence, or otherwise, find the set of values of $x$ for which $x^{2}-3 x+6 \geqslant 4 x-4$
11. (2015/jan/paper01/q10)
The points $A, B$ and $C$ have coordinates $(-2,3),(2,5)$ and $(4,1)$ respectively.
(a) Show, by calculation, that $A B$ is perpendicular to $B C$.
(b) Show that the length of $A B=$ the length of $B C$.
The midpoint of $A C$ is $M$.
(c) Find the coordinates of $M$.
(d) Find the exact length of the radius of the circle which passes through the points $A, B$ and $C$.
The point $P$ lies on $B M$ such that $B P: P M=2: 1$
(e) Find the coordinates of $P$.
The point $Q$ lies on $A P$ produced such that $A P: P Q=2: 1$
(f) Find the coordinates of $Q$.
(g) Show that $Q$ lies on $B C$.
12. (2015/june/paper02/q9)
The points $A$ and $B$ have coordinates $(2,9)$ and $(10,3)$ respectively.
The point $M$ is the midpoint of $A B$.
(a) Find the coordinates of $M$. (2)
(b) Find the length of $A B$
The line $l$ is the perpendicular bisector of $A B$.
(c) Find an equation for $l$ giving your answer in the form $a y=b x+c$, where $a, b$ and $c$ are integers.
The point $D$ lies on $l$ and has coordinates $(d, 2)$.
(d) Find the value of $d$.
The point $E$ lies on $l$ and is such that $D M: M E=1: 2$
(e) Find the coordinates of $E$,
(f) Find the area of the kite $A E B D$.
13. (2016/jan/paper01/q11)
$$\mathrm{f}(x)=4+3 x-x^{2}$$
(a) Write $\mathrm{f}(x)$ in the form $P-Q(x+R)^{2}$, where $P, Q$ and $R$ are rational numbers.
The curve $C$ has equation $y=4+3 x-x^{2}$
(b) Find the coordinates of the maximum point of $C$.(1)
The line $l_{1}$ is a tangent to $C$ at the point where $x=1$
(c) Find an equation for $l_{1}$
Another line $l_{2}$ is perpendicular to $l_{1}$ and is also a tangent to $C$.
The lines $l_{1}$ and $l_{2}$ intersect at the point $A$.
(d) Find the coordinates of $A$.$(5)$
The point $B$ with coordinates $(-3,2)$ lies on $l_{1}$
(e) Find the exact length of $A B$.$(2)$
The point $D$ with coordinates $(8,0)$ lies on $l_{2}$
(f) Find the exact area of triangle $A B D$.(3)
14. (2016/june/paper01/q10)
The points $A$ and $B$ have coordinates $(2,4)$ and $(5,-2)$ respectively.
The point $C$ divides $A B$ in the ratio $1: 2$
(a) Find the coordinates of $C$. (2)
The point $D$ has coordinates $(1,1)$
(b) Show that $D C$ is perpendicular to $A B$. (3)
(c) Find the equation of $D C$ in the form $p y=x+q$
The point $E$ is such that $D C E$ is a straight line and $D C=C E$.
(d) Find the coordinates of $E$. (2)
(e) Calculate the area of quadrilateral $A D B E$.
15. (2017/jan/paper01/q11)
The curve $C$ has equation $y=p x+q x^{2}$ where $p$ and $q$ are integers.
The curve $C$ has a stationary point at $(3,9)$.
(a) (i) Show that $p=6$ and find the value of $q$.
(ii) Determine the nature of the stationary point at $(3,9)$. (7)
The straight line $l$ with equation $y+x-10=0$ intersects $C$ at two points.
(b) Determine the $x$ coordinate of each of these two points of intersection.
The finite region bounded by the curve $C$ and the straight line $l$ is rotated through $360^{\circ}$ about the $x$-axis.
(c) Use algebraic integration to find the volume of the solid formed. Give your answer in terms of $\pi$.
16. (2017/jan/paper02/q9)
The points $P$ and $Q$ have coordinates $(-2,5)$ and $(2,-3)$ respectively.
(a) Find an equation for the line $P Q$. $(2)$
The point $N$ is such that $P N Q$ is a straight line and $P N: N Q=3: 1$
The straight line $l$ passes through $N$ and is perpendicular to $P Q$.
(b) Find
(i) the coordinates of $N$,
(ii) an equation for $l$. (5)
The points $S$ and $T$ lie on $l$ and have coordinates $(3, s)$ and $(t,-2)$ respectively.
(c) Find
(i) the value of $s$,
(ii) the value of $t$. (2)
(d) Find the area of the quadrilateral $P S Q T$.
17. (2017/june/paper01/q8)
The points $A$ and $B$ have coordinates $(1,7)$ and $(13,1)$ respectively.
(a) Find the exact length of $A B$, (2)
The point $C$ divides $A B$ in the ratio $1: 2$
(b) Find the coordinates of $C$. (2)
The line $l$ passes through $C$ and is perpendicular to $A B$.
(c) Find an equation of $l$, giving your answer in the form $y=a x+b$ where $a$ and $b$ are integers.
The point $D$ with coordinates $(9, d)$ lies on $l$.
(d) Find the value of $d$.
The point $E$ is the midpoint of $C D$.
(e) Find the exact value of the area of the quadrilateral $A D B E$. (5)
18. (2018/jan/paper02/q10)
The point $A$ has coordinates $(-6,-4)$ and the point $B$ has coordinates $(4,1)$ The line $l$ passes through the point $A$ and the point $B$.
(a) Find an equation of $l$. (2)
The point $P$ lies on $l$ such that $A P: P B=3: 2$
(b) Find the coordinates of $P$ (2)
The point $Q$ with coordinates $(m, n)$ lies on the line through $P$ that is perpendicular to $l$. Given that $m<0$ and that the length of $P Q$ is $3 \sqrt{5}$
(c) find the coordinates of $Q$. (5)
The point $R$ has coordinates $(-13,0)$
(d) Show that
(i) $A B$ and $R Q$ are equal in length,
(ii) $A B$ and $R Q$ are parallel.
(e) Find the area of the quadrilateral $A B Q R$.
19. (2018/june/paper02/q9)
The points $A, B$ and $C$ have coordinates $(-4,4),(1,6)$ and $(-2,-1)$ respectively.
(a) Show, by calculation, that $A B$ is perpendicular to $A C$.
(b) Find an equation for $B C$ in the form $p x+q y+r=0$, where $p, q$ and $r$ are integers.
The line $l$ is the perpendicular bisector of $A B$.
(c) Find an equation for $l$.
The line $l$ and the line $B C$ intersect at the point $E$.
(d) Find the coordinates of $E$. (2)
(e) Calculate the area of triangle $A E C$. (4)
20. (2019/june/paper02/q8)
The point $A$ has coordinates $(2,6)$, the point $B$ has coordinates $(6,8)$ and the point $C$ has coordinates $(4,2)$
(a) Find the exact length of
(i) $A B$
(ii) $B C$
(iii) $A C$
(b) Find the size of each angle of triangle $A B C$ in degrees.
The points $A, B$ and $C$ lie on a circle with centre $P$.
(c) Find the coordinates of $P$.
(d) Find the exact length of the radius of the circle in the form $\sqrt{a}$, where $a$ is an integer. (2)
21. (2019/juneR/paper01/q11)
The points $A$ and $B$ have coordinates $(-1,3)$ and $(5,6)$ respectively.
(a) Find an equation for the line $A B$. (2)
The point $P$ divides $A B$ in the ratio $2: 1$
(b) Show that the coordinates of $P$ are $(3,5)$
The point $C$ with coordinates $(m, n)$, where $m>0$, is such that $C P$ is perpendicular to the line $A B$.
Given that the radius of the circle which passes through $A, P$ and $C$ is 5
(c) find the value of $m$ and the value of $n$
The point $D$ with coordinates ( $p, q)$ is such that the line $A D$ is perpendicular to the line $A B$ and the line $D C$ is parallel to the line $A B$.
(d) Find the value of $p$ and the value of $q$. (3)
(e) Find the area of trapezium $A B C D$.
Answer
1.(a) $y=\frac{1}{4} x+\frac{19}{4}$ (b) $y=-4 x+26$ (c) $q=14$ (d) $59 \frac{1}{2}$
2.) $(1,8),(4,11)$
3.(a) Show (b) $3 y+4 x-27=0$ (c) $(12,-7)$ (d) $37 \displaystyle\frac{1}{2}$
4.(a) $y=4 x-1$ (b) $A C=\sqrt{68}$ (c) $B M=\sqrt{34}$ (d) $A B=\sqrt{51}$ (e) $(1+\sqrt{2}, 3+4 \sqrt{2}),(1-\sqrt{2}, 3-4 \sqrt{2})$
5.(a) $r=\sqrt{5}$ (b) $B(1,-1)$ (c) $D E=2 \sqrt{5}$ (d) $C P=\sqrt{(x-2)^{2}+(y-1)}$
6(a) show (b) show (c) $(2,4)$ (d) $2 y=x+6$
7.(a) $-\displaystyle\frac{2}{3}$ (b) $2 y=3 x-17$ (c) $Q(3,-4)$ (d) $3 y+2 x=20$ (e) Show (f) 52
8. $2 x-3 y+4=0$
9.(a) $\displaystyle\frac{3}{4}$ (b) $y+2 x=24$ (c) $e=9$ (d) $F(7,10)$ (e) 35
10.(a) $(2,4),(5,16)$ (b) $x \leqslant 2$ or $x \geqslant 5$
11.(a) Show (b) Show (c) $M(1,2)$ (d) $\sqrt{10}$ (e) $\left(\displaystyle\frac{4}{3},3\right)$ (f) (3,3) (g) Show
12.(a)(6,6) (b) 10 (c) $3 y=4 x-6$ (d) $d=3$ (e) (12,14) (f) 75
13.(a) $ \displaystyle\frac{25}{4}-\left(x-\displaystyle\frac{3}{2}\right)^{2}$ (b) $\left(\displaystyle\frac{3}{2}, \displaystyle\frac{25}{4}\right)$ (c) $y=x+5$ (d) $\left(\displaystyle\frac{3}{2}, \displaystyle\frac{13}{2}\right)$ (e) $A B=\displaystyle\frac{9 \sqrt{2}}{2}$ (f) $A=\displaystyle\frac{117}{4}$
14.(a) $C=(3,2)$ (b) Show (c) $2 y=x+1$ (d) $E=(5,3)$ (e) 15
15.(a) (i) Show (ii) $\max$(b) $x=2,5$(c) $V=\displaystyle\frac{333 \pi}{5}$
16. (a) $y=-2 x+1(b)(i)(1,-1)$ (ii) $2 y=x-3$ (c) (i) $s=0$ (ii) $t=-1$ (d) 20
17. $(a) A B=6 \sqrt{5}$ (b) $C=(5,5)$ (c) $y=2 x-5$ (d) $d=13$ (e) 30
18.(a) $y=\displaystyle\frac{1}{2}x-1$ (b) $(0,-1)$ (c) $(-3,5)$ (d) Show
19.(a) Show (b) $7x-3y+11=0$ (c) $4y=-10x+5$ (d) $\left(-\displaystyle\frac{1}{2},\displaystyle\frac{5}{2}\right)$ (e) 7.25
20. $(a)\left(\begin{array}{lll}\text { i }) A B=\sqrt{20} & \text { (ii) } B C=\sqrt{40} & \text { (iii) } A C=\sqrt{20}\end{array}\right.$ (b) $\angle A=90^{\circ}, \angle B=\angle C=45^{\circ}$ (c) $P=(5,5)$ (d) radius $=\sqrt{10}$
21.(a) $2y=x+7$ (b) Show (c) $m=7,n=-3$ (d) $p=3,q=-5$ (e) $50$
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