CAIE P1 2011 June — Question 2 5 marks

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2011
SessionJune
Marks5
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Mark schemeDownload PDF ↗
TopicInequalities
TypeLine-curve intersection conditions
DifficultyStandard +0.3 This is a standard discriminant problem requiring students to set the equations equal, form a quadratic, and 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 conceptual complications.
Spec1.02d Quadratic functions: graphs and discriminant conditions1.02g Inequalities: linear and quadratic in single variable
2 Find the set of values of \(m\) for which the line \(y = m x + 4\) intersects the curve \(y = 3 x ^ { 2 } - 4 x + 7\) at two distinct points.
\(y = mx + 4\) and \(y = 3x^2 - 4x + 7\)
AnswerMarks Guidance
Equate \(\rightarrow 3x^2 - (4 + m)x + 3 = 0\)M1 Eliminates \(y\) (or \(x\)) completely
Uses \(b^2 - 4ac \rightarrow (4 + m)^2 - 36\)M1 Any use of \(b^2 - 4ac\)
Solution of quadratic \(m = 2\) or \(-10\)DM1 A1 Method shown. Correct end-values
Set of values \(m > 2\) or \(m < -10\)A1 co
[5] 
$y = mx + 4$ and $y = 3x^2 - 4x + 7$

Equate $\rightarrow 3x^2 - (4 + m)x + 3 = 0$ | M1 | Eliminates $y$ (or $x$) completely
Uses $b^2 - 4ac \rightarrow (4 + m)^2 - 36$ | M1 | Any use of $b^2 - 4ac$
Solution of quadratic $m = 2$ or $-10$ | DM1 A1 | Method shown. Correct end-values
Set of values $m > 2$ or $m < -10$ | A1 | co
| [5] |
2 Find the set of values of $m$ for which the line $y = m x + 4$ intersects the curve $y = 3 x ^ { 2 } - 4 x + 7$ at two distinct points.

\hfill \mbox{\textit{CAIE P1 2011 Q2 [5]}}
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Q1 5 Q2 5 Q3 5 Q4 7 Q5 7 Q6 8 Q7 8 Q8 7 Q9 11 Q10 12