| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2024 |
| Session | March |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Verify, factorise, solve with substitution |
| Difficulty | Moderate -0.3 This is a straightforward application of the factor theorem requiring students to substitute x=1/2 to find a, then perform polynomial division or comparison of coefficients to complete the factorization. Part (b) adds a minor twist by requiring substitution of cos θ and solving basic trigonometric equations. The question is slightly easier than average as it's a standard textbook exercise with clear signposting and routine techniques, though the two-part structure and trig element prevent it from being significantly below average. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| 3(a) | Substitute x=1, equate to zero and attempt solution | |
| 2 | M1 | |
| Obtain a=31 | A1 | |
| Divide by 2x−1 at least as far as 3x2 +mx | M1 | Or equivalent (for example by inspection, …). |
| Obtain 3x2 +17x+10 | A1 | |
| Obtain (2x−1)(3x+2)(x+5) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| 3(b) | Attempt solution of sin=k where k is valid constant from answer to part (a) | M1 |
| Obtain −11.5 | A1 | Or greater accuracy. |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 3:
--- 3(a) ---
3(a) | Substitute x=1, equate to zero and attempt solution
2 | M1
Obtain a=31 | A1
Divide by 2x−1 at least as far as 3x2 +mx | M1 | Or equivalent (for example by inspection, …).
Obtain 3x2 +17x+10 | A1
Obtain (2x−1)(3x+2)(x+5) | A1
5
--- 3(b) ---
3(b) | Attempt solution of sin=k where k is valid constant from answer to part (a) | M1
Obtain −11.5 | A1 | Or greater accuracy.
2
Question | Answer | Marks | GuidanceThe polynomial $\mathrm{p}(x)$ is defined by
$$\mathrm{p}(x) = 6x^3 + ax^2 + 3x - 10,$$
where $a$ is a constant. It is given that $(2x - 1)$ is a factor of $\mathrm{p}(x)$.
\begin{enumerate}[label=(\alph*)]
\item Find the value of $a$ and hence factorise $\mathrm{p}(x)$ completely. [5]
\item Solve the equation $\mathrm{p}(\cos\theta) = 0$ for $-90° < \theta < 90°$. [2]
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2024 Q3 [7]}}