CAIE P1 2020 June — Question 1 4 marks

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2020
SessionJune
Marks4
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. It's slightly easier than average as it's a routine technique with straightforward algebra and no conceptual complications.
Spec1.02c Simultaneous equations: two variables by elimination and substitution1.02d Quadratic functions: graphs and discriminant conditions1.02q Use intersection points: of graphs to solve equations
1 Find the set of values of \(m\) for which the line with equation \(y = m x + 1\) and the curve with equation \(y = 3 x ^ { 2 } + 2 x + 4\) intersect at two distinct points.
Question 1:
AnswerMarks Guidance
AnswerMark Guidance
\(3x^2 + 2x + 4 = mx + 1 \rightarrow 3x^2 + x(2-m) + 3 = 0\)B1
\((2-m)^2 - 36\)M1 SOI
\((m+4)(m-8) \neq 0\) or \(2-m \geq 6\) and \(2-m \leq -6\)A1 OE
\(m < -4,\ m > 8\)A1 WWW
Alternative method for Question 1:
AnswerMarks Guidance
AnswerMark Guidance
\(\frac{dy}{dx} = 6x + 2 \rightarrow m = 6x + 2 \rightarrow 3x^2 + 2x + 4 = (6x+2)x + 1\)M1
\(x = \pm 1\)A1
\(m = \pm 6 + 2 \rightarrow m = 8\) or \(-4\)A1
\(m < -4,\ m > 8\)A1 WWW
Total: 4 marks
**Question 1:**

| Answer | Mark | Guidance |
|--------|------|----------|
| $3x^2 + 2x + 4 = mx + 1 \rightarrow 3x^2 + x(2-m) + 3 = 0$ | B1 | |
| $(2-m)^2 - 36$ | M1 | SOI |
| $(m+4)(m-8) \neq 0$ **or** $2-m \geq 6$ and $2-m \leq -6$ | A1 | OE |
| $m < -4,\ m > 8$ | A1 | WWW |

**Alternative method for Question 1:**

| Answer | Mark | Guidance |
|--------|------|----------|
| $\frac{dy}{dx} = 6x + 2 \rightarrow m = 6x + 2 \rightarrow 3x^2 + 2x + 4 = (6x+2)x + 1$ | M1 | |
| $x = \pm 1$ | A1 | |
| $m = \pm 6 + 2 \rightarrow m = 8$ or $-4$ | A1 | |
| $m < -4,\ m > 8$ | A1 | WWW |

**Total: 4 marks**

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1 Find the set of values of $m$ for which the line with equation $y = m x + 1$ and the curve with equation $y = 3 x ^ { 2 } + 2 x + 4$ intersect at two distinct points.\\

\hfill \mbox{\textit{CAIE P1 2020 Q1 [4]}}
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