| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2011 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Completing the square and sketching |
| Type | Find quadratic from roots/properties |
| Difficulty | Moderate -0.8 Part (i) is straightforward application of sum and product of roots formulas (p = -2, q = -15). Part (ii) requires recognizing that equal roots means discriminant = 0, giving a routine calculation. Both parts are standard textbook exercises with no problem-solving insight required, making this easier than average but not trivial. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown4.05a Roots and coefficients: symmetric functions |
| Answer | Marks | Guidance |
|---|---|---|
| \(\to p = -2, q = -15\) (any other method ok) | M1 A1 [2] | Must be \((x + 3)\) and \((x - 5)\). co |
| Answer | Marks | Guidance |
|---|---|---|
| \(r = 16\) | M1 DM1 A1 [3] | Any use of "\(b^2 - 4ac\)" \(c\) must include both \(q\) and \(r\). co |
**(i)** $x^2 + px + q = (x + 3)(x - 5)$
$\to p = -2, q = -15$ (any other method ok) | M1 A1 [2] | Must be $(x + 3)$ and $(x - 5)$. co
**(ii)** $x^2 + px + q + r = 0$
Use of "$b^2 - 4ac$"
Uses $a, b$ and $c$ correctly
$r = 16$ | M1 DM1 A1 [3] | Any use of "$b^2 - 4ac$" $c$ must include both $q$ and $r$. co
**or** $= (x + k)^2 \to 2k = p$ (M1) $k^2 = q + r$ (M1) $\to k = -1 \to r = 16$ (A1)The equation $x^2 + px + q = 0$, where $p$ and $q$ are constants, has roots $-3$ and $5$.
\begin{enumerate}[label=(\roman*)]
\item Find the values of $p$ and $q$. [2]
\item Using these values of $p$ and $q$, find the value of the constant $r$ for which the equation $x^2 + px + q + r = 0$ has equal roots. [3]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2011 Q3 [5]}}