Standard +0.3 This is a straightforward discriminant problem requiring students to set equations equal, form a quadratic, and analyze when solutions exist. While it involves a parameter c, the algebraic manipulation is routine and the discriminant method is a standard technique taught explicitly for intersection problems. Slightly above average due to the parameter handling and need to interpret the result correctly.
2 A line has equation \(y = 2 c x + 3\) and a curve has equation \(y = c x ^ { 2 } + 3 x - c\), where \(c\) is a constant.
Showing all necessary working, determine which of the following statements is correct.
A The line and curve intersect only for a particular set of values of \(c\).
B The line and curve intersect for all values of \(c\).
C The line and curve do not intersect for any values of \(c\).
\(cx^2 + 3x - c = 2cx + 3\) leading to \(cx^2 + (3-2c)x - (c+3)\ [=0]\)
M1
Forming a 3-term quadratic, all terms on one side
\(b^2 - 4ac = (3-2c)^2 + 4c(c+3)\)
M1
2nd M1 for \(b^2 - 4ac\) correct for *their* \(a\), \(b\), \(c\), i.e. no sign errors
\(= 8c^2 + 9\)
A1
\(> 0\) [for all values of \(c\)] leading to B [Intersects for all values of \(c\)]
A1
WWW
**Question 2:**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $cx^2 + 3x - c = 2cx + 3$ leading to $cx^2 + (3-2c)x - (c+3)\ [=0]$ | M1 | Forming a 3-term quadratic, all terms on one side |
| $b^2 - 4ac = (3-2c)^2 + 4c(c+3)$ | M1 | 2nd M1 for $b^2 - 4ac$ correct for *their* $a$, $b$, $c$, i.e. no sign errors |
| $= 8c^2 + 9$ | A1 | |
| $> 0$ [for all values of $c$] leading to B [Intersects for all values of $c$] | A1 | WWW |
2 A line has equation $y = 2 c x + 3$ and a curve has equation $y = c x ^ { 2 } + 3 x - c$, where $c$ is a constant.\\
Showing all necessary working, determine which of the following statements is correct.\\
A The line and curve intersect only for a particular set of values of $c$.\\
B The line and curve intersect for all values of $c$.\\
C The line and curve do not intersect for any values of $c$.\\
\hfill \mbox{\textit{CAIE P1 2023 Q2 [4]}}