| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2024 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Verify, factorise, solve with substitution |
| Difficulty | Standard +0.3 This is a straightforward multi-part question requiring standard techniques: using the factor theorem to find a constant (substitution and solving linear equation), factorising a cubic (by inspection or division), then solving a trigonometric equation by substitution. All steps are routine A-level procedures with no novel insight required, making it slightly easier than average. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Substitute \(x = -2\), equate to zero and attempt solution | M1 | |
| Obtain \(a = 4\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Divide by \(x+2\) at least as far as \(k_1x^2 + k_2x\) | M1 | |
| Obtain \(4x^2 - 12x + 9\) | A1 | |
| Obtain \((x+2)(2x-3)^2\) or equivalent with integer coefficients only | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Equate \(\sin^2\theta\) to appropriate value from factorised form and attempt solution | M1 | Using *their* \(\frac{2}{3}\) |
| Obtain \(54.7\) | A1 | Or greater accuracy |
| Obtain \(-54.7\) | A1 | Or greater accuracy. No others in \(-90° < \theta < 90°\) |
## Question 4(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Substitute $x = -2$, equate to zero and attempt solution | M1 | |
| Obtain $a = 4$ | A1 | |
---
## Question 4(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Divide by $x+2$ at least as far as $k_1x^2 + k_2x$ | M1 | |
| Obtain $4x^2 - 12x + 9$ | A1 | |
| Obtain $(x+2)(2x-3)^2$ or equivalent with integer coefficients only | A1 | |
---
## Question 4(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| Equate $\sin^2\theta$ to appropriate value from factorised form and attempt solution | M1 | Using *their* $\frac{2}{3}$ |
| Obtain $54.7$ | A1 | Or greater accuracy |
| Obtain $-54.7$ | A1 | Or greater accuracy. No others in $-90° < \theta < 90°$ |
---4 The polynomial $\mathrm { p } ( x )$ is defined by
$$\mathrm { p } ( x ) = a x ^ { 3 } - a x ^ { 2 } - 15 x + 18$$
where $a$ is a constant. It is given that ( $x + 2$ ) is a factor of $\mathrm { p } ( x )$.
\begin{enumerate}[label=(\alph*)]
\item Find the value of $a$.
\item Hence factorise $\mathrm { p } ( x )$ completely.\\
\begin{center}
\end{center}
\item Solve the equation $\mathrm { p } \left( \operatorname { cosec } ^ { 2 } \theta \right) = 0$ for $- 90 ^ { \circ } < \theta < 90 ^ { \circ }$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2024 Q4 [8]}}