CAIE P1 2007 November — Question 1 3 marks

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2007
SessionNovember
Marks3
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicInequalities
TypeLine-curve intersection conditions
DifficultyModerate -0.3 This is a standard discriminant problem requiring students to set up a quadratic equation from the intersection condition and apply Δ < 0. While it involves multiple steps (substitution, rearrangement, discriminant calculation), it's a routine textbook exercise with no novel insight required, making it slightly easier than average.
Spec1.02d Quadratic functions: graphs and discriminant conditions1.02q Use intersection points: of graphs to solve equations
1 Determine the set of values of the constant \(k\) for which the line \(y = 4 x + k\) does not intersect the curve \(y = x ^ { 2 }\).
AnswerMarks Guidance
\(y = 4x + k\) and \(y = x^2\) → \(x^2 - 4x - k = 0\)M1 Complete elimination of \(x\) or \(y\)
\(b^2 - 4ac < 0\) → \(16+4k < 0\) → \(k < -4\)M1, A1 Any use of \(b^2 - 4ac\) (=0, >0 etc) Co - condone \(\leq\)
"2x=4⟹x=2" M1 "y=4 ⟹ k=-4" M1⟹ A1[3] 
$y = 4x + k$ and $y = x^2$ → $x^2 - 4x - k = 0$ | M1 | Complete elimination of $x$ or $y$
$b^2 - 4ac < 0$ → $16+4k < 0$ → $k < -4$ | M1, A1 | Any use of $b^2 - 4ac$ (=0, >0 etc) Co - condone $\leq$
"2x=4⟹x=2" M1 "y=4 ⟹ k=-4" M1⟹ A1 | [3] |
1 Determine the set of values of the constant $k$ for which the line $y = 4 x + k$ does not intersect the curve $y = x ^ { 2 }$.

\hfill \mbox{\textit{CAIE P1 2007 Q1 [3]}}
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