Question 7 - A-Level Maths

AQA C1 2009 June — Question 7 9 marks

Exam Board	AQA
Module	C1 (Core Mathematics 1)
Year	2009
Session	June
Marks	9
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Inequalities
Type	Line-curve intersection conditions
Difficulty	Standard +0.3 This is a standard C1 question on line-curve intersection using the discriminant. Part (a) is routine algebraic manipulation (equating expressions). Part (b) requires knowing that two distinct intersections means b²-4ac > 0, then solving a quadratic inequality—all standard techniques with no novel insight required. Slightly above average difficulty due to the inequality solving in (b)(ii), but still a textbook exercise.
Spec	1.02d Quadratic functions: graphs and discriminant conditions 1.02g Inequalities: linear and quadratic in single variable 1.02q Use intersection points: of graphs to solve equations

7 The curve $C$ has equation $y = k \left( x ^ { 2 } + 3 \right)$, where $k$ is a constant.
The line $L$ has equation $y = 2 x + 2$.

Show that the $x$-coordinates of any points of intersection of the curve $C$ with the line $L$ satisfy the equation $$k x ^ { 2 } - 2 x + 3 k - 2 = 0$$
The curve $C$ and the line $L$ intersect in two distinct points.
1. Show that $$3 k ^ { 2 } - 2 k - 1 < 0$$
2. Hence find the possible values of $k$.

Show mark scheme Show mark scheme source

Question 7:

Part (a)

Answer	Marks	Guidance
$k(x^2 + 3) = 2x + 2 \Rightarrow kx^2 - 2x + 3k - 2 = 0$	B1 [1]	AG OE all terms on one side and $= 0$

Part (b)(i)

Answer	Marks	Guidance
Discriminant $= (-2)^2 - 4k(3k-2)$	M1	Condone one slip (including $x$ as one slip); condone $2^2$ or $4$ as first term
$= 4 - 12k^2 + 8k$	A1	Condone recovery from missing brackets
Two distinct real roots $\Rightarrow b^2 - 4ac > 0$: $4 - 12k^2 + 8k > 0$	B1$\checkmark$	"their discriminant in terms of $k$" $> 0$; not simply the statement $b^2 - 4ac > 0$
$\Rightarrow 12k^2 - 8k - 4 < 0$; $\Rightarrow 3k^2 - 2k - 1 < 0$	A1 [4]	Change from $> 0$ to $< 0$ and divide by 4; AG CSO

Part (b)(ii)

Answer	Marks	Guidance
$(3k+1)(k-1)$	M1	Correct factors or correct use of formula
Critical values $1$ and $-\dfrac{1}{3}$	A1	May score M1, A1 for correct critical values seen as part of incorrect final answer
Use of sign diagram or sketch	M1	If previous A1 earned, sign diagram or sketch must be correct for M1; otherwise M1 may be earned for attempt using their critical values
$\Rightarrow -\dfrac{1}{3} < k < 1$ or $1 > k > -\dfrac{1}{3}$	A1 [4]	Full marks for correct final answer with or without working; $\leqslant$ loses final A mark; condone $-\dfrac{1}{3} < k$ AND $k < 1$ for full marks but not OR or "," instead of AND

## Question 7:

**Part (a)**
$k(x^2 + 3) = 2x + 2 \Rightarrow kx^2 - 2x + 3k - 2 = 0$ | B1 [1] | AG OE all terms on one side and $= 0$

**Part (b)(i)**
Discriminant $= (-2)^2 - 4k(3k-2)$ | M1 | Condone one slip (including $x$ as one slip); condone $2^2$ or $4$ as first term

$= 4 - 12k^2 + 8k$ | A1 | Condone recovery from missing brackets

Two distinct real roots $\Rightarrow b^2 - 4ac > 0$: $4 - 12k^2 + 8k > 0$ | B1$\checkmark$ | "their discriminant in terms of $k$" $> 0$; not simply the statement $b^2 - 4ac > 0$

$\Rightarrow 12k^2 - 8k - 4 < 0$; $\Rightarrow 3k^2 - 2k - 1 < 0$ | A1 [4] | Change from $> 0$ to $< 0$ and divide by 4; AG CSO

**Part (b)(ii)**
$(3k+1)(k-1)$ | M1 | **Correct** factors or **correct** use of formula

Critical values $1$ and $-\dfrac{1}{3}$ | A1 | May score M1, A1 for correct critical values seen as part of incorrect final answer

Use of sign diagram or sketch | M1 | If previous A1 earned, sign diagram or sketch must be correct for M1; otherwise M1 may be earned for attempt using their critical values

$\Rightarrow -\dfrac{1}{3} < k < 1$ or $1 > k > -\dfrac{1}{3}$ | A1 [4] | Full marks for correct final answer with or without working; $\leqslant$ loses final A mark; condone $-\dfrac{1}{3} < k$ AND $k < 1$ for full marks but not OR or "," instead of AND

Show LaTeX source

7 The curve $C$ has equation $y = k \left( x ^ { 2 } + 3 \right)$, where $k$ is a constant.\\
The line $L$ has equation $y = 2 x + 2$.
\begin{enumerate}[label=(\alph*)]
\item Show that the $x$-coordinates of any points of intersection of the curve $C$ with the line $L$ satisfy the equation

$$k x ^ { 2 } - 2 x + 3 k - 2 = 0$$
\item The curve $C$ and the line $L$ intersect in two distinct points.
\begin{enumerate}[label=(\roman*)]
\item Show that

$$3 k ^ { 2 } - 2 k - 1 < 0$$
\item Hence find the possible values of $k$.
\end{enumerate}\end{enumerate}

\hfill \mbox{\textit{AQA C1 2009 Q7 [9]}}

This paper (7 questions)

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Q1 8 Q2 7 Q3 13 Q4 17 Q5 11 Q6 10 Q7 9

Answer	Marks	Guidance
Discriminant \(= (-2)^2 - 4k(3k-2)\)	M1	Condone one slip (including \(x\) as one slip); condone \(2^2\) or \(4\) as first term
\(= 4 - 12k^2 + 8k\)	A1	Condone recovery from missing brackets
Two distinct real roots \(\Rightarrow b^2 - 4ac > 0\): \(4 - 12k^2 + 8k > 0\)	B1\(\checkmark\)	"their discriminant in terms of \(k\)" \(> 0\); not simply the statement \(b^2 - 4ac > 0\)
\(\Rightarrow 12k^2 - 8k - 4 < 0\); \(\Rightarrow 3k^2 - 2k - 1 < 0\)	A1 [4]	Change from \(> 0\) to \(< 0\) and divide by 4; AG CSO

Answer	Marks	Guidance
\((3k+1)(k-1)\)	M1	Correct factors or correct use of formula
Critical values \(1\) and \(-\dfrac{1}{3}\)	A1	May score M1, A1 for correct critical values seen as part of incorrect final answer
Use of sign diagram or sketch	M1	If previous A1 earned, sign diagram or sketch must be correct for M1; otherwise M1 may be earned for attempt using their critical values
\(\Rightarrow -\dfrac{1}{3} < k < 1\) or \(1 > k > -\dfrac{1}{3}\)	A1 [4]	Full marks for correct final answer with or without working; \(\leqslant\) loses final A mark; condone \(-\dfrac{1}{3} < k\) AND \(k < 1\) for full marks but not OR or "," instead of AND