CAIE P1 2022 March — Question 2 5 marks

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2022
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. While it involves algebraic manipulation and understanding of intersection conditions, it's a routine textbook exercise with a well-established method, making it slightly easier than average.
Spec1.02c Simultaneous equations: two variables by elimination and substitution1.02d Quadratic functions: graphs and discriminant conditions
2 A curve has equation \(y = x ^ { 2 } + 2 c x + 4\) and a straight line has equation \(y = 4 x + c\), where \(c\) is a constant. Find the set of values of \(c\) for which the curve and line intersect at two distinct points.
Question 2:
AnswerMarks Guidance
AnswerMarks Guidance
\(x^2 + 2cx + 4 = 4x + c\) leading to \(x^2 + 2cx - 4x + 4 - c\ [=0]\)\*M1 Equate \(y\)s and move terms to one side of equation.
\(b^2 - 4ac = (2c-4)^2 - 4(4-c)\)DM1 Use of discriminant with *their* correct coefficients.
\(\left[4c^2 - 16c + 16 - 16 + 4c =\right] 4c^2 - 12c\)A1
\(b^2 - 4ac > 0\) leading to \((4)c(c-3) > 0\)M1 Correctly apply '\(> 0\)' considering both regions.
\(c < 0,\ c > 3\)A1 Must be in terms of \(c\). SC B1 instead of M1A1 for \(c \leqslant 0,\ c \geqslant 3\)
5 
**Question 2:**

| Answer | Marks | Guidance |
|--------|-------|----------|
| $x^2 + 2cx + 4 = 4x + c$ leading to $x^2 + 2cx - 4x + 4 - c\ [=0]$ | \*M1 | Equate $y$s and move terms to one side of equation. |
| $b^2 - 4ac = (2c-4)^2 - 4(4-c)$ | DM1 | Use of discriminant with *their* correct coefficients. |
| $\left[4c^2 - 16c + 16 - 16 + 4c =\right] 4c^2 - 12c$ | A1 | |
| $b^2 - 4ac > 0$ leading to $(4)c(c-3) > 0$ | M1 | Correctly apply '$> 0$' considering both regions. |
| $c < 0,\ c > 3$ | A1 | Must be in terms of $c$. **SC B1** instead of M1A1 for $c \leqslant 0,\ c \geqslant 3$ |
| | **5** | |
2 A curve has equation $y = x ^ { 2 } + 2 c x + 4$ and a straight line has equation $y = 4 x + c$, where $c$ is a constant. Find the set of values of $c$ for which the curve and line intersect at two distinct points.\\

\hfill \mbox{\textit{CAIE P1 2022 Q2 [5]}}
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