(a) On the grid opposite, draw the line with equation

(i) $3x - 4y = 12$

(ii) $y + 6 + 3x = 0$

(iii) $3y = 18 - x$

(b) Show, by shading on the grid, the region $R$ defined by the inequalities

For all points in $R$, with coordinates $(x, y)$

Using values from your graph,

(c) find the least value of $P$ and the greatest value of $P$

Only use this grid if you need to redraw your graph.

(a) On the grid opposite, draw the graph of the line with equation

(b) Show, by shading on the grid, the region $R$ defined by the inequalities

For all points in $R$, with coordinates $(x, y)$

(c) find the least value of $P$

Only use this grid if you need to redraw your graph.

Figure 1 shows a rectangle with width $x$ cm and length $(2x + 3)$ cm.

The perimeter of the rectangle is $P$ cm and the area of the rectangle is $A$$\mathrm{cm}^2$

Find the set of possible values for $x$

Give your answer in the form $a < x < b$ where $a$ and $b$ are rational numbers.

Show clear algebraic working.

(a) Use algebra to find the $x$ coordinates of the points where the curve with equation $y = 3x^2 + 9x - 17$ intersects the line with equation $y = 3 - 2x$

(b) Hence, or otherwise, find the set of values of $x$ for which $3 - 2x \leq 3x^2 + 9x - 17$

Find the set of values for $x$ for which

(a) $9 - 3x > 11x + 2$

(b) $10x^2 + 7x < 12$

(c) both $9 - 3x > 11x + 2$ and $10x^2 + 7x < 12$

The remainder when $f(x)$ is divided by $(x - 2)$ is 4 times the remainder when

$f(x)$ is divided by $(x + 1)$

(a) Show that $c = 6$

Given that $(x + 2)$ is a factor of $f(x)$

(b) show that the equation $f(x) = 0$ has only one real root.

(a) On the grid below, draw the line with equation

(i) $3x + 4y = 24$      (ii) $2x - 5y + 10 = 0$

(b) Show, by shading on the grid, the region $R$ defined by the inequalities

Label the region $R$

Given that $(3x - 2)$ is a factor of $f(x)$

(a) show that $a = 21$

(b) Hence show algebraically that the curve $y = f(x)$ has only one intersection with the $x$-axis.

The length of rectangle $R$ is 2 cm greater than its width.

The area of $R$ is greater than $8\mathrm{cm}^2$ and the perimeter of $R$ is less than $30\mathrm{cm}$.

Given that the width of $R$ is $w$ cm,

find the set of possible values of $w$

Give your answer in the form $a < w < b$ where $a$ and $b$ are rational numbers.

$f(x) = px^3 + qx^2 - 37x - 12q$ where $p$ and $q$ are constants.

When $f'(x)$ is divided by $(x + 2)$ the remainder is $-33$

Given that $(x + 5)$ is a factor of $f(x)$

(a) (i) show that $p = 2$

(ii) find the value of $q$

(b) Hence, use algebra to factorise $f(x)$ completely.

(c) Hence solve the equation $f(x) = 0$

The curve $y = g(x)$ passes through the point with coordinates $(1, 20)$

Given that $(x-5)$ is a factor of $g(x)$

(a) show that $g(x) = 2x^3 - 5x^2 - 37x + 60$

(b) Hence, or otherwise, use algebra to solve the equation $g(x) = 0$

One solution to the following simultaneous equations

is $\left(-\frac{1}{2}, q\right)$, where $p$ is an integer and $q$ is a prime number.

(a) Find the value of $p$ and the value of $q$

(b) Hence find the other solution to the equations.

(a) On the axes opposite, draw the line with equation

(b) Show, by shading on your graph, the region $R$ defined by the inequalities

For all points in $R$, with coordinates $(x, y)$

(c) Find

(i) the greatest value of $P$

(ii) the least value of $P$

Only use this grid if you need to redraw your graph.

(a) On the grid opposite draw the line with equation

(b) Show, by shading, the region $R$ defined by the inequalities

For all points in $R$ with coordinates $(x, y)$

(c) Using your graph, find the least value of $P$

Only use this grid if you need to redraw your graph.

Find the set of values of $x$ for which

Figure 2 shows a rectangle with width $x$ metres and length $(x+4)$ metres. Its perimeter is $P$ metres and its area is $A\,\mathrm{m}^2$.

(a) Find, in terms of $x$, an expression for

(i) $P$,

(ii) $A$.

The perimeter has to be less than $30$ metres. The area has to be greater than $12\,\mathrm{m}^2$.

(b) Find the set of possible values for $x$. Give your answer in the form

where $p$ and $q$ are integers. $(x+1)$ is a factor of $f(x)$. The remainder when $f(x)$ is divided by $(x+2)$ is $-5$.

(a) Find the value of $p$ and the value of $q$.

(b) Hence show that $f(x)=0$ has only one real root.

Find the set of values of $x$ for which

(a) $$8x-7<5x+5,$$

(b) $$2x^2-5x-3>0,$$

(c) both $8x-7<5x+5$ and $2x^2-5x-3>0$.

where $p$ and $q$ are integers.

Given that $(x-3)$ is a factor of $f(x)$,

(a) show that

Given that $(x+p)$, where $p>0$, is also a factor of $f(x)$,

(b) show that

(c) Hence find the value of $p$ and the value of $q$.

(d) Using your values of $p$ and $q$, factorise $f(x)$ completely.

(a) Using the axes below, sketch the lines with equations

(i) $y=6$,

(ii) $y+x=10$,

(iii) $y=2x-5$.

Show the coordinates of any point where each line crosses the coordinate axes.

(b) Show, by shading on your sketch, the region $R$ defined by the inequalities

where $p$ and $q$ are constants. The remainder when $f(x)$ is divided by $(x-1)$ is $-12$. The remainder when $f(x)$ is divided by $(x-4)$ is $30$.

(a) Find the value of $p$ and the value of $q$.

Using your values of $p$ and $q$,

(b) show that $f(3)=0$,

(c) express $f(x)$ as a product of linear factors,

(d) hence solve $f(x)=0$.

(a) Factorise $x^{2}-x-2$.

(b) Hence, or otherwise, show that $(x^{2}-x-2)$ is a factor of $f(x)$.

Use algebra to solve the equations

(a) Using the axes below, sketch the line with equation

(i) $y+2x=-5$

(ii) $y=x+4$

Show the coordinates of the points where each line crosses the coordinate axes.

(b) Show, by shading, the region $R$ defined by the inequalities

(a) On the grid opposite, draw the graphs of the lines with equations

(b) Show, by shading on the grid, the region $R$ defined by the inequalities

For all points in $R$, with coordinates $(x,y)$

(c) Find the greatest value of $F$.

(a) On the grid opposite, draw

(i) the line with equation $y=3x-3$

(ii) the line with equation $3x+2y=12$ (2)

(b) Show, by shading, the region $R$ defined by the inequalities

For all points in $R$ with coordinates $(x,y)$

(c) Find the greatest value of $P$. (4)

$f(x)=2x^3-3px^2+x+4p$ where $p$ is an integer.

Given that $(x-4)$ is a factor of $f(x)$

(a) show that the value of $p$ is 3 (2)

Using this value of $p$,

(b) find the remainder when $f(x)$ is divided by $(x+2)$ (2)

(c) factorise $f(x)$ completely (3)

(d) solve the equation $2x^3-3px^2+x+4p=0$ (2)

Use algebra to find the set of values of $x$ for which $(3x-1)(x-1)<2(3x-1)$. (5)

(a) On the axes below, sketch the lines with equations $x=3$, $y=x+1$ and $2y+x=5$. On your sketch, mark the coordinates of any points where the lines cross the axes. (3)

(b) Show, by shading on your sketch, the region $R$ defined by the inequalities

(a) On the grid opposite, draw the graphs of the lines with equations

(i) $y=2x\qquad$ (ii) $y=6-x\qquad$ (iii) $2y=x-2$ (3)

(b) Show, by shading on the grid, the region $R$ defined by the inequalities

For all points in $R$, with coordinates $(x,y)$,

(c) Find the greatest value of $P$. (1)

(a) Show that $(x-2)$ is a factor of $f(x)$. (2)

(b) Hence, or otherwise, factorise $f(x)$ completely. (3)

Find the set of values of $x$ for which

When $f(x)$ is divided by $(x-2)$ the remainder is $-20$. Given that $(x-3)$ is a factor of $f(x)$

(a) find the value of $p$ and the value of $q$. (7)

(b) Hence use algebra to solve the equation $f(x)=0$. (5)

Solve the equations

where $a$ and $b$ are constants.

The remainder when $f(x)$ is divided by $x-1$ is $-12$. The remainder when $f(x)$ is divided by $x+1$ is 48.

(a) Find the value of $a$ and the value of $b$. (6)

(b) Show that $f\!\left(\dfrac12\right)=0$. (1)

(c) Express $f(x)$ as a product of linear factors. (4)

(d) Hence solve $f(x)=0$. (1)

(a) On the axes opposite, draw the lines with equations

(i) $y=-x-1$

(ii) $y=3x-9$

(iii) $2y=x+7$ (4)

(b) Show, by shading, the region $R$ defined by the inequalities

For all points in $R$ with coordinates $(x,y)$, $P=y-2x$.

(c) Find

(i) the greatest value of $P$,

(ii) the least value of $P$. (4)