| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2021 |
| Session | March |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | One factor, one non-zero remainder |
| Difficulty | Moderate -0.3 Part (a) is a standard application of factor and remainder theorems requiring solving two simultaneous equations from p(-2)=0 and p(3)=5. Part (b) adds a mild substitution step with exponentials but remains routine. This is slightly easier than average due to the straightforward algebraic manipulation and lack of conceptual complexity. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.06g Equations with exponentials: solve a^x = b |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Substitute \(x=-2\) and equate to zero | *M1 | |
| Substitute \(x=3\) and equate to 5 | *M1 | |
| Obtain \(-8-2a+b=0\) and \(27+3a+b=5\) or equivalents | A1 | |
| Solve a pair of relevant linear simultaneous equations for \(a\) or \(b\) | DM1 | Dependent on at least one M mark |
| Obtain \(a=-6\) and \(b=-4\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Attempt division by \(x+2\) at least as far as \(x^2+kx\) | M1 | |
| Obtain \(x^2-2x-2\) | A1 | |
| Obtain (at least) the positive root \(\frac{2+\sqrt{12}}{2}\) or exact equivalent | A1 | |
| Equate \(e^{2y}\) to positive root, apply logarithms and use power law | M1 | |
| Obtain \(\frac{1}{2}\ln\left(\frac{2+\sqrt{12}}{2}\right)\) or \(\frac{1}{2}\ln(1+\sqrt{3})\) or exact equivalent | A1 |
## Question 6(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Substitute $x=-2$ and equate to zero | *M1 | |
| Substitute $x=3$ and equate to 5 | *M1 | |
| Obtain $-8-2a+b=0$ and $27+3a+b=5$ or equivalents | A1 | |
| Solve a pair of relevant linear simultaneous equations for $a$ or $b$ | DM1 | Dependent on at least one M mark |
| Obtain $a=-6$ and $b=-4$ | A1 | |
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## Question 6(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Attempt division by $x+2$ at least as far as $x^2+kx$ | M1 | |
| Obtain $x^2-2x-2$ | A1 | |
| Obtain (at least) the positive root $\frac{2+\sqrt{12}}{2}$ or exact equivalent | A1 | |
| Equate $e^{2y}$ to positive root, apply logarithms and use power law | M1 | |
| Obtain $\frac{1}{2}\ln\left(\frac{2+\sqrt{12}}{2}\right)$ or $\frac{1}{2}\ln(1+\sqrt{3})$ or exact equivalent | A1 | |6 The polynomial $\mathrm { p } ( x )$ is defined by
$$\mathrm { p } ( x ) = x ^ { 3 } + a x + b$$
where $a$ and $b$ are constants. It is given that $( x + 2 )$ is a factor of $\mathrm { p } ( x )$ and that the remainder is 5 when $\mathrm { p } ( x )$ is divided by $( x - 3 )$.
\begin{enumerate}[label=(\alph*)]
\item Find the values of $a$ and $b$.
\item Hence find the exact root of the equation $\mathrm { p } \left( \mathrm { e } ^ { 2 y } \right) = 0$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2021 Q6 [10]}}