| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2021 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Solve p(exponential) = 0 |
| Difficulty | Standard +0.3 Part (a) uses the factor theorem directly to find a constant—straightforward substitution. Part (b) is routine factorisation after finding a. Part (c) requires solving e^y + e^(-y) = k for various k values, which involves hyperbolic substitution or quadratic manipulation—a modest extension but still a standard technique. Overall slightly easier than average due to the guided structure. |
| 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 \(x = 3\), equate to zero and attempt solution | M1 | |
| Obtain \(a = 6\) | A1 | |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Divide by \(x - 3\) at least as far as the \(x\) term | M1 | Or use of identity or by inspection |
| Obtain \(6x^2 + 7x + 2\) | A1 | |
| Conclude \((x-3)(3x+2)(2x+1)\) | A1 | |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Equate \(e^y + e^{-y}\) to positive value resulting from part (b) | B1 | Condone if seen with equating to negative values as well. |
| Multiply by \(e^y\) and use quadratic formula | M1 | |
| Obtain \(e^y = \dfrac{3 \pm \sqrt{5}}{2}\) | A1 | or exact equivalents |
| Obtain \(\ln\dfrac{3 \pm \sqrt{5}}{2}\) | A1 | or exact equivalents and no other values which are undefined. |
| Total: 4 |
## Question 7(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Substitute $x = 3$, equate to zero and attempt solution | M1 | |
| Obtain $a = 6$ | A1 | |
| **Total: 2** | | |
## Question 7(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Divide by $x - 3$ at least as far as the $x$ term | M1 | Or use of identity or by inspection |
| Obtain $6x^2 + 7x + 2$ | A1 | |
| Conclude $(x-3)(3x+2)(2x+1)$ | A1 | |
| **Total: 3** | | |
## Question 7(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| Equate $e^y + e^{-y}$ to positive value resulting from part **(b)** | B1 | Condone if seen with equating to negative values as well. |
| Multiply by $e^y$ and use quadratic formula | M1 | |
| Obtain $e^y = \dfrac{3 \pm \sqrt{5}}{2}$ | A1 | or exact equivalents |
| Obtain $\ln\dfrac{3 \pm \sqrt{5}}{2}$ | A1 | or exact equivalents and no other values which are undefined. |
| **Total: 4** | | |7 The polynomial $\mathrm { p } ( x )$ is defined by
$$\mathrm { p } ( x ) = a x ^ { 3 } - 11 x ^ { 2 } - 19 x - a$$
where $a$ is a constant. It is given that $( x - 3 )$ is a factor of $\mathrm { p } ( x )$.
\begin{enumerate}[label=(\alph*)]
\item Find the value of $a$.
\item When $a$ has this value, factorise $\mathrm { p } ( x )$ completely.
\item Hence find the exact values of $y$ that satisfy the equation $\mathrm { p } \left( \mathrm { e } ^ { y } + \mathrm { e } ^ { - y } \right) = 0$.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2021 Q7 [9]}}