1. (2011/june/paper01/q1)
Solve the equations
$$
\begin{aligned}
&y=x^{2}-3 x+2 \\
&y-x=7
\end{aligned}
$$ (5 marks)
2. (2012/jan/paper01/q1)
Show that the two lines with equations
$$
\begin{aligned}
&6 x+4 y=-15 \\
&10 x-15 y=9
\end{aligned}
$$
are perpendicular.
(4 marks)
3. (2012/jan/paper01/q2)
Solve the equation
$$
\frac{x}{x+1}-\frac{1}{x+2}=2
$$
Give your answers correct to 3 significant figures. (4 marks)
4. (2012/june/paper02/q3)
Solve the equations
$$
\begin{gathered}
2 x^{2}+x y-y^{2}=36 \\
x+2 y=1
\end{gathered}
$$ (6 marks)
5. (2014/jan/paper02/q3)
Solve the equations
$$
\begin{aligned}
&x^{2}+x y-3 x=2 \\
&5 y+6 x=22
\end{aligned}
$$ (6 marks)
6. (2016/jan/paper02/q3)
Solve the equations
$$
\begin{aligned}
3 y &=12-4 x \\
(x+1)^{2}+(y-2)^{2} &=4
\end{aligned}
$$ (7 marks)
7. (2017/june/paper02/q2)
Solve the equations
$$
\begin{aligned}
y &=x^{2}-6 x+5 \\
y+x &=11
\end{aligned}
$$ (5 marks)
8. (2019/june/paper02/q5)
Use algebra to solve the equations
$$
\begin{array}{r}
x y=36 \\
x y+x+2 y=53
\end{array}
$$ (6 marks)
9. (2013/jan/Paper01/q1)
(a) On the axes below sketch the lines with equations
(i) $y=8$
(ii) $y+x=6$
(iii) $y=3 x-4$
Show the coordinates of the points where each line crosses the coordinate axes.
(3 marks)
(b) Show, by shading, the region $R$ which satisfies $y \geqslant 3 x-4, y+x \geqslant 6, x \geqslant 0$ and $y \leqslant 8$ (1 mark)
10. (2014/june/paper01/q1)
(a) On the axes below, sketch the lines with equations $y=x+3$ and $y+2 x=7$ On your sketch mark the coordinates of the points where the lines cross the $y$-axis. (2 marks)
(b) Show, by shading on your sketch, the region $R$ defined by the inequalities $$ y \leqslant x+3, y+2 x \leqslant 7, x \geqslant 0 \text { and } y \geqslant 0 $$ (1 mark)
(c) Determine, by calculation, whether or not the point with coordinates $(2,2)$ lies in $R$. (2 marks)
11. (2015/jan/paper01/q5)
(a) On the axes opposite, draw the lines with equations
(i) $y=-x-1$
(ii) $y=3 x-9$
(iii) $2 y=x+7$ (4 marks)
(b) Show, by shading, the region $R$ defined by the inequalities $$ y \geqslant-x-1, \quad y \geqslant 3 x-9 \quad \text { and } \quad 2 y \leqslant x+7 $$ (1 mark) For all points in $R$, with coordinates $(x, y)$, $$ P=y-2 x $$
(c) Find
(i) the greatest value of $P$,
(ii) the least value of $P$. (4 marks)
12. (2017/jan/paper02/q1)
(a) On the axes below, sketch the lines with equations $x=3, y=x+1$ and $2 y+x=5$ On your sketch, mark the coordinates of any points where the lines cross the axes. (3 marks)
(b) Show, by shading on your sketch, the region $R$ defined by the inequalities $$ x \leqslant 3, y \leqslant x+1 \text { and } 2 y+x \geqslant 5 $$ (1 mark)
13. (2017/june/paper02/q1)
(a) On the grid opposite, draw the graphs of the lines with equations
(i) $y=2 x$
(ii) $y=6-x$
(iii) $2 y=x-2$
(3 marks)
(b) Show, by shading on the grid, the region $R$ defined by the inequalities
$$
y \leqslant 2 x, \quad y \leqslant 6-x, \quad 2 y \geqslant x-2, \quad y \geqslant 0
$$
For all points in $R$, with coordinates $(x, y)$,
$$
P=y+2 x
$$ (1 mark)
(c) Find the greatest value of $P$. (1 mark)
14. (2018/jan/paper01/q2)
(a) On the grid opposite, draw
(i) the line with equation $y=3 x-3$
(ii) the line with equation $3 x+2 y=12$
(2 marks)
(b) Show, by shading, the region $R$ defined by the inequalities
$$
y \leqslant 3 x-3 \quad 3 x+2 y \leqslant 12 \quad y \geqslant-1
$$
For all points in $R$ with coordinates $(x, y)$
$$
P=4 x-y
$$ (2 marks)
(c) Find the greatest value of $P$. (4 marks)
15. (2019/juneR/paper02/q5)
(a) On the grid opposite, draw the graphs of the lines with equations
$$
2 x+3 y=24 \quad y=2 x \quad 3 y=2 x-12
$$ (3 marks)
(b) Show, by shading on the grid, the region $R$ defined by the inequalities
$$
2 x+3 y \leqslant 24 \quad y \leqslant 2 x \quad 3 y \geqslant 2 x-12 \quad y \geqslant 0
$$ (1 mark)
For all points in $R$, with coordinates $(x, y)$
$$
F=2 x+5 y
$$
(c) Find the greatest value of $F$. (3 marks)
Answers
1. $\quad x=5, y=12$ or $x=-1, y=6$
2. Show
3. $x=-3.62,-1.38$
4. $x=5, y=-2 ; \quad x=-\frac{24}{5}, y=\frac{17}{5}$
5. $x=2, y=2$ or $x=5, y=-\frac{8}{5}$
6. $\quad x=\frac{3}{5}, \quad y=\frac{16}{5}$
7. $x=6, y=5$ or $x=-1, y=12$
8. $x=9, y=4$ or $x=8, y=4 \frac{1}{2}$
9. 16. Fig
10. (a) Graph (b) Graph (c) lies in $R$.
11. (a) Graph (b) Graph (c)(i) 8 (ii) $-7$
12. Graph
13. (a) Figure (b) Figure (c) $P_{\max }=10 \frac{2}{3}$
14. (a) Graph (b) Graph (c) $\frac{59}{3}$
15. (a) Graph (b) Graph (c) 36
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