| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2022 |
| Session | June |
| 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 routine application of factor theorem to find 'a' and then factorisation - standard textbook exercise. Part (b) requires substitution x=e^(3y) and recognising that e^(3y) must equal the one positive root, then solving for y using logarithms. Straightforward multi-step question with no novel insight required, slightly above average due to the exponential substitution component. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.06a Exponential function: a^x and e^x graphs and properties1.06g Equations with exponentials: solve a^x = b |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Substitute \(x = 4\), equate to zero and attempt solution | M1 | |
| Obtain \(a = -7\) | A1 | |
| Divide by \(x-4\) at least as far as the \(x\) term | M1 | or use of identity or by inspection |
| Obtain \(2x^2 + x + 1\) and conclude \((x-4)(2x^2+x+1)\) | A1 | |
| Alternative method: Divide by \(x-4\) at least as far as the \(x\) term | M1 | |
| Equate the remainder to zero | M1 | |
| Obtain \(a = -7\) | A1 | |
| Obtain \(2x^2+x+1\) and conclude \((x-4)(2x^2+x+1)\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Apply logarithms and use power law for \(e^{3y} = 4\) | M1 | |
| Obtain \(\frac{1}{3}\ln 4\) or exact equivalent | A1 | |
| Use discriminant \([= 1 - 8 = -7]\) or equivalent to show no other roots | B1 | AG – necessary detail needed |
## Question 5(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Substitute $x = 4$, equate to zero and attempt solution | M1 | |
| Obtain $a = -7$ | A1 | |
| Divide by $x-4$ at least as far as the $x$ term | M1 | or use of identity or by inspection |
| Obtain $2x^2 + x + 1$ and conclude $(x-4)(2x^2+x+1)$ | A1 | |
| **Alternative method:** Divide by $x-4$ at least as far as the $x$ term | M1 | |
| Equate the remainder to zero | M1 | |
| Obtain $a = -7$ | A1 | |
| Obtain $2x^2+x+1$ and conclude $(x-4)(2x^2+x+1)$ | A1 | |
---
## Question 5(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Apply logarithms and use power law for $e^{3y} = 4$ | M1 | |
| Obtain $\frac{1}{3}\ln 4$ or exact equivalent | A1 | |
| Use discriminant $[= 1 - 8 = -7]$ or equivalent to show no other roots | B1 | AG – necessary detail needed |
---5 The polynomial $\mathrm { p } ( x )$ is defined by
$$\mathrm { p } ( x ) = 2 x ^ { 3 } + a x ^ { 2 } - 3 x - 4$$
where $a$ is a constant. It is given that ( $x - 4$ ) is a factor of $\mathrm { p } ( x )$.
\begin{enumerate}[label=(\alph*)]
\item Find the value of $a$ and hence factorise $\mathrm { p } ( x )$.
\item Show that the equation $\mathrm { p } \left( \mathrm { e } ^ { 3 y } \right) = 0$ has only one real root and find its exact value.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2022 Q5 [7]}}