| Exam Board | AQA |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2012 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Inequalities |
| Type | Combined linear and quadratic inequalities |
| Difficulty | Moderate -0.8 This is a straightforward C1 question testing basic inequality manipulation and quadratic solving. Parts (a) and (b) are simple algebraic rearrangements shown step-by-step, part (c) is routine factorization/quadratic formula, and part (d) requires combining results with the constraint x>0. All techniques are standard with no novel insight required, making this easier than average. |
| Spec | 1.02c Simultaneous equations: two variables by elimination and substitution1.02d Quadratic functions: graphs and discriminant conditions1.02g Inequalities: linear and quadratic in single variable1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03e Complete the square: find centre and radius of circle |
| Answer | Marks | Guidance |
|---|---|---|
| \(\Rightarrow 2x > 11\) | B1 (1 mark) | AG (be convinced) condone \(11 < 2x\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\Rightarrow x^2 + 4x - 96 < 0\) | B1 (1 mark) | AG must see this line OE |
| Answer | Marks | Guidance |
|---|---|---|
| \((x+12)(x-8)\) | M1 | correct factors or correct quadratic equation formula |
| Critical values \(8, -12\) | A1 | |
| sketch or sign diagram | M1 | |
| \(\Rightarrow -12 < x < 8\) | A1cso (4 marks) | accept \(x < 8\) AND \(x > -12\); but not \(x < 8\) OR \(x > -12\); nor \(x < 8\), \(x > -12\) |
| Answer | Marks |
|---|---|
| \(5\frac{1}{2} < x < 8\) | B1 (1 mark) |
# Question 6:
## Part 6(a):
Sides are $x$ and $x + 4$
$\Rightarrow x + x + x + 4 + x + 4 > 30$
or $2x + 2x + 8 > 30$
or $2(2x+4) > 30$
or $4x + 8 > 30$
$(\Rightarrow 4x > 22)$
$\Rightarrow 2x > 11$ | B1 (1 mark) | AG (be convinced) condone $11 < 2x$
## Part 6(b):
$x(x+4) < 96$
$\Rightarrow x^2 + 4x - 96 < 0$ | B1 (1 mark) | AG must see this line OE
## Part 6(c):
$(x+12)(x-8)$ | M1 | correct factors or correct quadratic equation formula
Critical values $8, -12$ | A1 |
sketch or sign diagram | M1 |
$\Rightarrow -12 < x < 8$ | A1cso (4 marks) | accept $x < 8$ AND $x > -12$; but **not** $x < 8$ OR $x > -12$; **nor** $x < 8$, $x > -12$
## Part 6(d):
$5\frac{1}{2} < x < 8$ | B1 (1 mark) |
---6 A rectangular garden is to have width $x$ metres and length $( x + 4 )$ metres.
\begin{enumerate}[label=(\alph*)]
\item The perimeter of the garden needs to be greater than 30 metres. Show that $2 x > 11$.
\item The area of the garden needs to be less than 96 square metres. Show that $x ^ { 2 } + 4 x - 96 < 0$.
\item Solve the inequality $x ^ { 2 } + 4 x - 96 < 0$.
\item Hence determine the possible values of the width of the garden.\\
$7 \quad$ A circle with centre $C$ has equation $x ^ { 2 } + y ^ { 2 } + 14 x - 10 y + 49 = 0$.\\
(a) Express this equation in the form
$$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }$$
(b) Write down:
\begin{enumerate}[label=(\roman*)]
\item the coordinates of $C$;
\item the radius of the circle.\\
(c) Sketch the circle.\\
(d) A line has equation $y = k x + 6$, where $k$ is a constant.
\begin{enumerate}[label=(\roman*)]
\item Show that the $x$-coordinates of any points of intersection of the line and the circle satisfy the equation $\left( k ^ { 2 } + 1 \right) x ^ { 2 } + 2 ( k + 7 ) x + 25 = 0$.
\item The equation $\left( k ^ { 2 } + 1 \right) x ^ { 2 } + 2 ( k + 7 ) x + 25 = 0$ has equal roots. Show that
$$12 k ^ { 2 } - 7 k - 12 = 0$$
\item Hence find the values of $k$ for which the line is a tangent to the circle.
\end{enumerate}
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C1 2012 Q6 [7]}}