| Exam Board | SPS |
|---|---|
| Module | SPS SM (SPS SM) |
| Year | 2025 |
| Session | October |
| Marks | 4 |
| Topic | Discriminant and conditions for roots |
| Type | Prove/show always positive |
| Difficulty | Moderate -0.8 This question involves straightforward application of the discriminant formula and the quadratic formula with no algebraic complications. Part (a) requires showing Δ = k² + 4k² = 5k² ≥ 0, and part (b) is direct substitution into the quadratic formula. Both parts are routine A-level techniques with minimal problem-solving required, making this easier than average but not trivial since it involves algebraic manipulation with a parameter. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.02f Solve quadratic equations: including in a function of unknown |
\begin{enumerate}[label=(\alph*)]
\item Show that the equation $x^2 + kx - k^2 = 0$ has real roots for all real values of $k$. [2]
\item Show that the roots of the equation $x^2 + kx - k^2 = 0$ are $\left(\frac{-1 \pm \sqrt{5}}{2}\right)k$. [2]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM 2025 Q9 [4]}}