| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2013 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differentiation from First Principles |
| Type | Stationary point via first principles |
| Difficulty | Standard +0.3 Part (a) is a straightforward application of first principles differentiation using the chord gradient formula, requiring only substitution and simplification to show the gradient is 2h-1, then taking the limit as h→0. Part (b) is a standard improper integral evaluation. Both parts are routine FP1 exercises requiring careful algebra but no novel insight or complex problem-solving. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.07a Derivative as gradient: of tangent to curve1.08a Fundamental theorem of calculus: integration as reverse of differentiation4.08c Improper integrals: infinite limits or discontinuous integrands |
**(a) A curve has equation $y = 2x^2 - 5x$.**
The point $P$ on the curve has coordinates $(1, -3)$.
The point $Q$ on the curve has $x$-coordinate $1 + h$.
(i) Show that the gradient of the line $PQ$ is $2h - 1$. **(3 marks)**
(ii) Explain how the result of part (a)(i) can be used to show that the tangent to the curve at the point $P$ is parallel to the line $x + y = 0$. **(2 marks)**
**(b) For the improper integral**
$$\int_1^{\infty} \frac{x - 4}{2x^2 - 5x} \, dx$$
**either show that the integral has a finite value and state its value, or explain why the integral does not have a finite value. (3 marks)**5
\begin{enumerate}[label=(\alph*)]
\item A curve has equation $y = 2 x ^ { 2 } - 5 x$.\\
The point $P$ on the curve has coordinates $( 1 , - 3 )$.\\
The point $Q$ on the curve has $x$-coordinate $1 + h$.
\begin{enumerate}[label=(\roman*)]
\item Show that the gradient of the line $P Q$ is $2 h - 1$.
\item Explain how the result of part (a)(i) can be used to show that the tangent to the curve at the point $P$ is parallel to the line $x + y = 0$.
\end{enumerate}\item For the improper integral $\int _ { 1 } ^ { \infty } x ^ { - 4 } \left( 2 x ^ { 2 } - 5 x \right) \mathrm { d } x$, either show that the integral has a finite value and state its value, or explain why the integral does not have a finite value.
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2013 Q5 [8]}}