FPM (Alg, Int, Linear Programming)

$\def\frac{\dfrac}$

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/june/paper02/q3)

(a) (i) Find $\displaystyle\int\left(1+3 x-\frac{2}{x^{2}}\right) \mathrm{d} x$

(ii) Hence show that $\displaystyle\int_{1}^{2}\left(1+3 x-\frac{2}{x^{2}}\right) \mathrm{d} x=4 \frac{1}{2}$ (4 marks)

(b) (i) Find $\displaystyle\int 3 \sin 2 x d x$

(ii) Hence show that $\displaystyle\int_{0}^{\frac{\pi}{6}} 3 \sin 2 x \mathrm{~d} x=\frac{3}{4}$ (4 marks)

10. (2015/june/paper01/q1)
The region enclosed by the curve with equation $y=4 x^{2}-9$, the positive $x$-axis and the negative $y$-axis is rotated through $360^{\circ}$ about the $x$-axis. Use algebraic integration to find, to 3 significant figures, the volume of the solid generated. (5 marks)

11. (2018/june/paper01/q5)

(a) (i) Find $\displaystyle\int\left(3-x+\frac{1}{x^{3}}\right) \mathrm{d} x$

(ii) Hence evaluate $\displaystyle\int_{1}^{2}\left(3-x+\frac{1}{x^{3}}\right) \mathrm{d} x$ (4 marks)

(b) (i) Find $\displaystyle\int 6 \sin 3 x \mathrm{~d} x$

(ii) Hence evaluate $\displaystyle\int_{\frac{\pi}{9}}^{\frac{\pi}{6}} 6 \sin 3 x \mathrm{~d} x$

(4 marks)

12. (2018/jan/paper02/q3)

The region enclosed by the circle with equation $x^{2}+y^{2}=5$ and the straight line with equation $x=1$, shown shaded in Figure 2, is rotated through $360^{\circ}$ about the $y$-axis. Use algebraic integration to find the exact volume of the solid generated. (5 marks)

13. (2018/june/paper02/q10)
The curve $C$ has equation $y^{2}=16 x$ where $y \geqslant 0$ Given that the point $A$ with coordinates $(a, 2 a)$ where $a \neq 0$ lies on $C$,

(a) find the value of $a$. (2 marks)
The line $l$ passes through $A$ and has gradient $-2$ Given that $l$ crosses the $x$-axis at the point $B$,

(2 marks) (b) find the $x$ coordinate of $B$.
The finite region enclosed by $C, l$ and the $x$-axis is rotated through $360^{\circ}$ about the $x$-axis.

(c) Using algebraic integration, find, to 3 significant figures, the volume of the solid generated. (5 marks)

14. (2019/juneR/paper02/q9)
The finite region $R$ enclosed by the $y$-axis, the straight line with equation $y+2 x=13$ and the curve with equation $y=x^{2}-2$, is defined for points with coordinates $(x, y)$ with $x \geqslant 0$ The region $R$ is rotated through $360^{\circ}$ about the $y$-axis. Use algebraic integration to find the volume of the solid generated. Give your answer in terms of $\pi$. (9 marks)

15. (2013/jan/paper02/q5)
$\cos (A+B)=\cos A \cos B-\sin A \sin B$

(a) Use the above identity to show that $2 \sin ^{2} A=1-\cos 2 A$ (3 marks)

(b) Hence find the value of $k$ such that $\sin ^{2} 2 A=k(1-\cos 4 A)$ (1 mark)

Figure 2 shows part of the curve with equation $y=3 \sin 2 x$. The region $R$, bounded by the curve, the positive $x$-axis and the line $x=\frac{\pi}{6}$, is rotated through $360^{\circ}$ about the $x$-axis.

(c) Use calculus to find, to 3 significant figures, the volume of the solid generated. (6 marks)

16. (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)

17. (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)

18. (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)

19. (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 $$ (1 mark) x \leqslant 3, y \leqslant x+1 \text { and } 2 y+x \geqslant 5 $$

20. (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)

21. (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)

22. (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)

23. (2019/june/paper01/q2)
Given that $\frac{4+2 \sqrt{3}}{5-2 \sqrt{3}}$ can be written in the form $\frac{a+b \sqrt{3}}{c}$ where $a$ and $b$ are integers and $c$ is prime, find the value of $a$, the value of $b$ and the value of $c$. Show your working clearly. (3 marks)

24. (2019/june/paper01/q6)
Figure 2 shows a lawn $A B C D E F$, where $A B D E$ is a rectangle of length $y$ metres and width $2 x$ metres. Each end of the lawn is a semicircle of radius $x$ metres. The lawn has perimeter $90 \mathrm{~m}$ and area $S \mathrm{~m}^{2}$

(a) Show that $S$ can be written in the form $$S=k x-\pi x^{2}$$ where $k$ is a constant. State the value of $k$. (4 marks)

(b) Use calculus to find, to 4 significant figures, the value of $x$ for which $S$ is a maximum, justifying that this value of $x$ gives a maximum value of $S$. (5 marks)

(c) Find, to the nearest whole number, the maximum value of $S$. (2 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. (a)(i) $x+\frac{3 x^{2}}{2}+\frac{2}{x}+c$ (ii) show $(b)$ (i) $\frac{-3 \cos 2 x}{2}+c$

10. $V=204$

11. (a)(i) $3x-\frac{1}{2}x^2-\frac{1}{2x^2}+C$ (ii) $1\frac{7}{8}$ (b)(i) $-2\cos3x +C$ (ii) 1

12. $\frac{32\pi}{3}$

13. (a) $a=4$ (b) $x=8$ (c) 670

14. $\frac{117}{2}\pi$

15. (a) show (b) $k=\frac{1}{2}$ (c) $4.34$

16. Fig

17. (a) Graph (b) Graph (c) lies in $R$. 

18. (a) Graph (b) Graph (c)(i) 8 (ii) $-7$

19. Graph

20. (a) Figure (b) Figure (c) $P_{\max }=10 \frac{2}{3}$

21. (a) Graph (b) Graph (c) $\frac{59}{3}$

22. (a) Graph (b) Graph (c) 36

23. $\frac{32+18 \sqrt{3}}{13}$

24. (a) Show (b) $x=14.3$ (c) $S_{\max }=645$