| Exam Board | SPS |
|---|---|
| Module | SPS SM (SPS SM) |
| Year | 2025 |
| Session | February |
| Marks | 6 |
| Topic | Tangents, normals and gradients |
| Type | Prove constraint relationship |
| Difficulty | Moderate -0.3 Part (a) is straightforward algebraic expansion requiring students to match coefficients, a routine skill. Part (b) asks students to prove K is composite by using the factorization from (a), which is guided and requires only basic reasoning about when factors are distinct (n>2 ensures n+1 and n²-n+1 are both >1 and unequal). This is easier than average A-level proof questions as the factorization is provided and the argument is relatively direct. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem4.01a Mathematical induction: construct proofs |
\begin{enumerate}[label=(\alph*)]
\item The number $K$ is defined by $K = n^3 + 1$, where $n$ is an integer greater than $2$.
Given that $n^3 + 1 = (n + 1) (n^2 + bn + c)$, find the constants $b$ and $c$. [1]
\item Prove that $K$ has at least two distinct factors other than $1$ and $K$. [5]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM 2025 Q4 [6]}}