| Exam Board | AQA |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2007 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Definite integral with trigonometric functions |
| Difficulty | Standard +0.3 This is a structured multi-part question that guides students through standard techniques: (a) is direct recall, (b) is routine quotient rule application, (c) is a straightforward identity proof using basic trig definitions, and (d) requires combining previous parts to integrate—all standard C3 material with clear scaffolding. Slightly above average due to the multi-step nature and need to connect parts, but no novel insight required. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.07q Product and quotient rules: differentiation1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08d Evaluate definite integrals: between limits |
8
\begin{enumerate}[label=(\alph*)]
\item Write down $\int \sec ^ { 2 } x \mathrm {~d} x$.
\item Given that $y = \frac { \cos x } { \sin x }$, use the quotient rule to show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = - \operatorname { cosec } ^ { 2 } x$.
\item Prove the identity $( \tan x + \cot x ) ^ { 2 } = \sec ^ { 2 } x + \operatorname { cosec } ^ { 2 } x$.
\item Hence find $\int _ { 0.5 } ^ { 1 } ( \tan x + \cot x ) ^ { 2 } \mathrm {~d} x$, giving your answer to two significant figures.
\end{enumerate}
\hfill \mbox{\textit{AQA C3 2007 Q8 [12]}}