CAIE P1 2020 November — Question 4 5 marks

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2020
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 the equations equal, form a quadratic, and apply b²-4ac > 0 for two distinct roots. While it involves algebraic manipulation and understanding of intersection conditions, it follows a well-practiced procedure 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
4 A curve has equation \(y = 3 x ^ { 2 } - 4 x + 4\) and a straight line has equation \(y = m x + m - 1\), where \(m\) is a constant. Find the set of values of \(m\) for which the curve and the line have two distinct points of intersection.
Question 4:
AnswerMarks Guidance
\(3x^2 - 4x + 4 = mx + m - 1 \rightarrow 3x^2 - (4+m)x + (5-m)(= 0)\)M1 3-term quadratic
\(b^2 - 4ac = (4+m)^2 - 4\times3\times(5-m)\)M1 Find \(b^2 - 4ac\) for *their* quadratic
\(m^2 + 20m - 44\)A1
\((m+22)(m-2)\)A1 Or use of formula or completing square. This step must be seen
\(m > 2\), \(m < -22\)A1 Allow \(x > 2\), \(x < -22\) 
## Question 4:

| $3x^2 - 4x + 4 = mx + m - 1 \rightarrow 3x^2 - (4+m)x + (5-m)(= 0)$ | M1 | 3-term quadratic |
|---|---|---|
| $b^2 - 4ac = (4+m)^2 - 4\times3\times(5-m)$ | M1 | Find $b^2 - 4ac$ for *their* quadratic |
| $m^2 + 20m - 44$ | A1 | |
| $(m+22)(m-2)$ | A1 | Or use of formula or completing square. This step must be seen |
| $m > 2$, $m < -22$ | A1 | Allow $x > 2$, $x < -22$ |

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4 A curve has equation $y = 3 x ^ { 2 } - 4 x + 4$ and a straight line has equation $y = m x + m - 1$, where $m$ is a constant.

Find the set of values of $m$ for which the curve and the line have two distinct points of intersection.\\

\hfill \mbox{\textit{CAIE P1 2020 Q4 [5]}}
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