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)
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)
(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) $S_{\max }=645$
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