CAIE P1 2009 June — Question 2 4 marks

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2009
SessionJune
Marks4
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicInequalities
TypeLine-curve intersection conditions
DifficultyStandard +0.3 This is a standard discriminant problem requiring students to set up a quadratic equation from the intersection condition, then apply b²-4ac > 0 for two distinct roots. It's slightly easier than average as it's a routine technique with straightforward algebra and no geometric insight required.
Spec1.02d Quadratic functions: graphs and discriminant conditions1.02q Use intersection points: of graphs to solve equations
2 Find the set of values of \(k\) for which the line \(y = k x - 4\) intersects the curve \(y = x ^ { 2 } - 2 x\) at two distinct points.
AnswerMarks Guidance
\(kx - 4 = x^2 - 2x \rightarrow x^2 - (2+k)x + 4 = 0\)M1 Complete elimination of \(y\) (or \(x\))
Use of \(b^2 - 4ac\)M1 Any use (\(=0\), \(<0\), \(>0\))
\((2+k)^2 = 16\) \(\rightarrow\) \(k = 2\) or \(-6\)A1 For the values of \(k\), however used
\(k > 2\) or \(k < -6\)A1 Correct only
[4] 
$kx - 4 = x^2 - 2x \rightarrow x^2 - (2+k)x + 4 = 0$ | M1 | Complete elimination of $y$ (or $x$)
Use of $b^2 - 4ac$ | M1 | Any use ($=0$, $<0$, $>0$)
$(2+k)^2 = 16$ $\rightarrow$ $k = 2$ or $-6$ | A1 | For the values of $k$, however used
$k > 2$ or $k < -6$ | A1 | Correct only
| [4] |
2 Find the set of values of $k$ for which the line $y = k x - 4$ intersects the curve $y = x ^ { 2 } - 2 x$ at two distinct points.

\hfill \mbox{\textit{CAIE P1 2009 Q2 [4]}}
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