| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2022 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Solve p(exponential) = 0 |
| Difficulty | Standard +0.3 Part (a) is a standard factor theorem application requiring substitution of x=-1/2 to find a, then factorisation - routine A-level work. Part (b) adds a straightforward substitution (x=e^(4y)) and logarithm step to solve for y. This is a typical textbook exercise combining two standard techniques with no novel insight required, making it slightly easier than average overall. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.06g Equations with exponentials: solve a^x = b |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Substitute \(x = -\frac{1}{2}\), equate to zero and attempt solution | M1 | |
| Obtain \(a = 6\) | A1 | |
| Divide by \(2x+1\) at least as far as the \(x\) term | M1 | or use of identity or by inspection |
| Obtain \(3x^2 + 10x - 8\) | A1 | |
| Conclude \((2x+1)(3x-2)(x+4)\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Apply logarithms and use power law for \(e^{4y} = k\) where \(k > 0\) | M1 | Using *their* answer to (a) |
| Obtain \(y = -0.101\) and no other answers | A1 | or greater accuracy |
## Question 4(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Substitute $x = -\frac{1}{2}$, equate to zero and attempt solution | M1 | |
| Obtain $a = 6$ | A1 | |
| Divide by $2x+1$ at least as far as the $x$ term | M1 | or use of identity or by inspection |
| Obtain $3x^2 + 10x - 8$ | A1 | |
| Conclude $(2x+1)(3x-2)(x+4)$ | A1 | |
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## Question 4(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Apply logarithms and use power law for $e^{4y} = k$ where $k > 0$ | M1 | Using *their* answer to **(a)** |
| Obtain $y = -0.101$ and no other answers | A1 | or greater accuracy |
---4 The polynomial $\mathrm { p } ( x )$ is defined by
$$\mathrm { p } ( x ) = a x ^ { 3 } + 23 x ^ { 2 } - a x - 8$$
where $a$ is a constant. It is given that $( 2 x + 1 )$ is a factor of $\mathrm { p } ( x )$.
\begin{enumerate}[label=(\alph*)]
\item Find the value of $a$ and hence factorise $\mathrm { p } ( x )$ completely.
\item Hence solve the equation $\mathrm { p } \left( \mathrm { e } ^ { 4 y } \right) = 0$, giving your answer correct to 3 significant figures.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2022 Q4 [7]}}