| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2021 |
| Session | June |
| Marks | 8 |
| Topic | Indefinite & Definite Integrals |
| Type | Integration with given constant |
| Difficulty | Standard +0.3 This is a straightforward integration and algebraic manipulation question. Part (a) requires integrating two standard functions and substituting limits—routine C3/C4 work. Part (b) involves solving a quadratic in √k, which is a standard substitution technique. The question is slightly easier than average as it's methodical with clear steps and no conceptual challenges beyond basic integration and algebraic manipulation. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.08d Evaluate definite integrals: between limits |
Given that $k$ is a positive constant and $\int_1^k \left(\frac{5}{2\sqrt{x}} + 3\right)dx = 4$
\begin{enumerate}[label=(\alph*)]
\item show that $3k + 5\sqrt{k} - 12 = 0$ [4]
\item Hence, using algebra, find any values of $k$ such that
$$\int_1^k \left(\frac{5}{2\sqrt{x}} + 3\right)dx = 4$$
[4]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2021 Q7 [8]}}