| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2021 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polynomial Division & Manipulation |
| Type | Solving Equations via Substitution After Division |
| Difficulty | Standard +0.3 Part (a) is routine polynomial division with a quadratic divisor. Part (b) requires recognizing that the factorization follows from part (a) by adjusting the constant. Part (c) is a standard substitution (let u = e^(-3y)) that reduces to the factorized form. This is a well-structured multi-part question requiring several techniques but following a clear guided path with no novel insights needed—slightly easier than average. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.06g Equations with exponentials: solve a^x = b |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Carry out division at least as far as \(x^2 + kx\) or equivalent | M1 | OE, e.g. comparing coefficients with coefficient of \(x^2\) equal to 1 and attempt at a second coefficient |
| Obtain quotient \(x^2 + 4x + 12\) | A1 | |
| Confirm remainder is 7 | A1 | AG |
| Total | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Include \((x-2)^2\) as a factor | M1 | Must be a product of factors only. SC B1 for \((x^2 - 4x + 4)(x^2 + 4x + 12)\) |
| Conclude \((x-2)^2(x^2 + 4x + 12)\) | A1 | isw any attempt to factorise the quotient |
| Total | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Apply logarithms and use power law for \(e^{-3y} = k\) where \(k > 0\) | M1 | |
| Obtain \(y = -\frac{1}{3}\ln 2\), \(\frac{1}{3}\ln\frac{1}{2}\) | A1 | Or exact equivalent. Must be simplified e.g. not \(\ln\frac{6}{3}\). ISW extra solutions but A0 if undefined solutions are included |
| Total | 2 |
## Question 5(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Carry out division at least as far as $x^2 + kx$ or equivalent | M1 | OE, e.g. comparing coefficients with coefficient of $x^2$ equal to 1 and attempt at a second coefficient |
| Obtain quotient $x^2 + 4x + 12$ | A1 | |
| Confirm remainder is 7 | A1 | AG |
| **Total** | **3** | |
---
## Question 5(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Include $(x-2)^2$ as a factor | M1 | Must be a product of factors only. **SC B1** for $(x^2 - 4x + 4)(x^2 + 4x + 12)$ |
| Conclude $(x-2)^2(x^2 + 4x + 12)$ | A1 | isw any attempt to factorise the quotient |
| **Total** | **2** | |
---
## Question 5(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| Apply logarithms and use power law for $e^{-3y} = k$ where $k > 0$ | M1 | |
| Obtain $y = -\frac{1}{3}\ln 2$, $\frac{1}{3}\ln\frac{1}{2}$ | A1 | Or exact equivalent. Must be simplified e.g. not $\ln\frac{6}{3}$. ISW extra solutions but A0 if undefined solutions are included |
| **Total** | **2** | |
---5
\begin{enumerate}[label=(\alph*)]
\item Find the quotient when $x ^ { 4 } - 32 x + 55$ is divided by $( x - 2 ) ^ { 2 }$ and show that the remainder is 7 .
\item Factorise $x ^ { 4 } - 32 x + 48$.
\item Hence solve the equation $\mathrm { e } ^ { - 12 y } - 32 \mathrm { e } ^ { - 3 y } + 48 = 0$, giving your answer in an exact form.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2021 Q5 [7]}}