| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2024 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differentiation from First Principles |
| Type | Chord gradient with h (algebraic) |
| Difficulty | Moderate -0.8 This is a straightforward first principles differentiation question requiring students to find the gradient of a chord and take the limit as h→0. The algebra is simple (quadratic function), and the conceptual step of recognizing the limit process is standard textbook material for this topic. Easier than average due to minimal algebraic complexity and being a direct application of a core technique. |
| Spec | 1.07a Derivative as gradient: of tangent to curve |
| Answer | Marks | Guidance |
|---|---|---|
| 3(a) | f(2+h)= 2(2+h)2−3 | |
| | B1 | SOI |
| Answer | Marks | Guidance |
|---|---|---|
| | M1 | ( ) |
| Answer | Marks |
|---|---|
| 2h+8 or 2(h+4) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| 3(b) | h→0, or chord [AB] → tangent [at A] | B1 |
| 8 | B1FT | Could come from anywhere except wrong working. |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 3:
--- 3(a) ---
3(a) | f(2+h)= 2(2+h)2−3
| B1 | SOI
( )
2(2+h)2 −3 − 5 2h2 +8h
=
(2+h)−2 h
| M1 | ( )
their 2(2+h)2 −3 −their5
can be implied by the
(2+h)−2
simplified expression or the correct answer.
Their 5 must come from 2(2)2 −3.
2h+8 or 2(h+4) | A1
3
--- 3(b) ---
3(b) | h→0, or chord [AB] → tangent [at A] | B1 | Either of these statements or any sight of h = 0.
8 | B1FT | Could come from anywhere except wrong working.
Either correct or FT their linear expression from (a).
2
Question | Answer | Marks | GuidanceThe equation of a curve is $y = 2x^2 - 3$. Two points $A$ and $B$ with $x$-coordinates 2 and $(2 + h)$ respectively lie on the curve.
\begin{enumerate}[label=(\alph*)]
\item Find and simplify an expression for the gradient of the chord $AB$ in terms of $h$. [3]
\item Explain how the gradient of the curve at the point $A$ can be deduced from the answer to part (a), and state the value of this gradient. [2]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2024 Q3 [5]}}