CAIE P1 2014 November — Question 5 5 marks

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2014
SessionNovember
Marks5
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 above average difficulty due to the parameter k appearing in both equations, requiring careful algebraic manipulation, but follows a well-practiced technique taught in P1.
Spec1.02d Quadratic functions: graphs and discriminant conditions1.02q Use intersection points: of graphs to solve equations
5 Find the set of values of \(k\) for which the line \(y = 2 x - k\) meets the curve \(y = x ^ { 2 } + k x - 2\) at two distinct points.
Question 5:
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(x^2 + x(k-2) + (k-2)(=0)\)M1 Equate and move terms to one side of equation
\((k-2)^2 - 4(k-2)(> 0)\)M1 Apply \(b^2 - 4ac\ (>0)\). Allow \(\geq\) at this stage
\((k-2)(k-6)(> 0)\)DM1
\(k < 2\) or \(k > 6\) (condone \(\leq,\ \geq\))A2 Attempt to factorise or solve or find 2 solutions. SCA1 for \(2, 6\) seen with wrong inequalities
Allow \(\{-\infty, 2\} \cup \{6, \infty\}\) etc.[5] 
## Question 5:

| Answer/Working | Marks | Guidance |
|---|---|---|
| $x^2 + x(k-2) + (k-2)(=0)$ | **M1** | Equate and move terms to one side of equation |
| $(k-2)^2 - 4(k-2)(> 0)$ | **M1** | Apply $b^2 - 4ac\ (>0)$. Allow $\geq$ at this stage |
| $(k-2)(k-6)(> 0)$ | **DM1** | |
| $k < 2$ or $k > 6$ (condone $\leq,\ \geq$) | **A2** | Attempt to factorise or solve or find 2 solutions. **SCA1** for $2, 6$ seen with wrong inequalities |
| Allow $\{-\infty, 2\} \cup \{6, \infty\}$ etc. | **[5]** | |

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5 Find the set of values of $k$ for which the line $y = 2 x - k$ meets the curve $y = x ^ { 2 } + k x - 2$ at two distinct points.

\hfill \mbox{\textit{CAIE P1 2014 Q5 [5]}}
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