CAIE P1 2021 March — Question 4 5 marks

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2021
SessionMarch
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicInequalities
TypeLine-curve intersection conditions
DifficultyModerate -0.3 This is a standard discriminant problem requiring students to set the equations equal, form a quadratic, and apply b²-4ac > 0. It's slightly easier than average because it's a routine technique with straightforward algebra and no conceptual surprises, though the presence of parameter k requires careful manipulation.
Spec1.02c Simultaneous equations: two variables by elimination and substitution1.02d Quadratic functions: graphs and discriminant conditions
4 A line has equation \(y = 3 x + k\) and a curve has equation \(y = x ^ { 2 } + k x + 6\), where \(k\) is a constant. Find the set of values of \(k\) for which the line and curve have two distinct points of intersection.
Question 4:
AnswerMarks Guidance
AnswerMarks Guidance
\(x^2 + kx + 6 = 3x + k\) leading to \(x^2 + x(k-3) + (6-k)[= 0]\)M1 Eliminate \(y\) and form 3-term quadratic
\((k-3)^2 - 4(6-k) [> 0]\)M1 OE. Apply \(b^2 - 4ac\)
\(k^2 - 2k - 15 [> 0]\)A1 Form 3-term quadratic
\((k+3)(k-5)[> 0]\)A1 Or \(k = -3, 5\) from use of formula or completing the square
\(k < -3,\ k > 5\)A1 FT Any correct alternative notation, do not allow \(\leqslant, \geqslant\). FT for *their* outside regions 
## Question 4:

| Answer | Marks | Guidance |
|--------|-------|----------|
| $x^2 + kx + 6 = 3x + k$ leading to $x^2 + x(k-3) + (6-k)[= 0]$ | M1 | Eliminate $y$ and form 3-term quadratic |
| $(k-3)^2 - 4(6-k) [> 0]$ | M1 | OE. Apply $b^2 - 4ac$ |
| $k^2 - 2k - 15 [> 0]$ | A1 | Form 3-term quadratic |
| $(k+3)(k-5)[> 0]$ | A1 | Or $k = -3, 5$ from use of formula or completing the square |
| $k < -3,\ k > 5$ | A1 FT | Any correct alternative notation, do not allow $\leqslant, \geqslant$. FT for *their* outside regions |

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4 A line has equation $y = 3 x + k$ and a curve has equation $y = x ^ { 2 } + k x + 6$, where $k$ is a constant. Find the set of values of $k$ for which the line and curve have two distinct points of intersection.\\

\hfill \mbox{\textit{CAIE P1 2021 Q4 [5]}}
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Q1 5 Q2 4 Q3 4 Q4 5 Q5 6 Q6 7 Q7 8 Q8 7 Q9 9 Q10 8 Q11 12