| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2024 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Find constant then factorise |
| Difficulty | Moderate -0.8 This is a straightforward application of the factor theorem requiring substitution of x=-2 to find a, followed by routine polynomial division or inspection to complete the factorisation. Both parts use standard techniques with no problem-solving insight needed, making it easier than average but not trivial due to the algebraic manipulation required. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| 4(a) | Substitute x=−2, equate to zero and attempt solution | M1 |
| Obtain a=4 | A1 |
| Answer | Marks |
|---|---|
| 4(b) | Divide by x+2at least as far as k x2 +k x |
| 1 2 | M1 |
| Obtain 4x2 −12x+9 | A1 |
| Obtain(x+2)(2x−3)2 or equivalent with integer coefficients only | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
| Answer | Marks | Guidance |
|---|---|---|
| 4(c) | Equate sin2 to appropriate value from factorised form and attempt solution | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| Obtain 54.7 | A1 | Or greater accuracy. |
| Obtain –54.7 | A1 | Or greater accuracy. |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 4:
--- 4(a) ---
4(a) | Substitute x=−2, equate to zero and attempt solution | M1
Obtain a=4 | A1
2
--- 4(b) ---
4(b) | Divide by x+2at least as far as k x2 +k x
1 2 | M1
Obtain 4x2 −12x+9 | A1
Obtain(x+2)(2x−3)2 or equivalent with integer coefficients only | A1
3
Question | Answer | Marks | Guidance
--- 4(c) ---
4(c) | Equate sin2 to appropriate value from factorised form and attempt solution | M1 | 2
Usingtheir .
3
Obtain 54.7 | A1 | Or greater accuracy.
Obtain –54.7 | A1 | Or greater accuracy.
No others in −9090.
3
Question | Answer | Marks | GuidanceThe polynomial $\text{p}(x)$ is defined by
$$\text{p}(x) = ax^3 - ax^2 - 15x + 18,$$
where $a$ is a constant. It is given that $(x + 2)$ is a factor of $\text{p}(x)$.
\begin{enumerate}[label=(\alph*)]
\item Find the value of $a$. [2]
\item Hence factorise $\text{p}(x)$ completely. [3]
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2024 Q4 [5]}}