# FPM Algebra and Inequality (Chapter 3)

$\def\frac{\dfrac}$

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)

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