| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2021 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differentiation from First Principles |
| Type | Numerical gradient deduction |
| Difficulty | Easy -1.2 This is a guided numerical investigation of differentiation from first principles requiring only substitution into the curve equation, calculation of chord gradients using the two-point formula, and observation of a pattern. No algebraic manipulation of limits or formal derivative work is needed—students simply compute numerical values and state that the gradients approach 2. |
| Spec | 1.07a Derivative as gradient: of tangent to curve1.07b Gradient as rate of change: dy/dx notation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(1.2679\) | B1 | AWRT. ISW if correct answer seen. \(3-\sqrt{3}\) scores B0 |
| 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(1.7321\) | B1 | AWRT. ISW if correct answer seen |
| 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Sight of 2 or 2.0000 or two in reference to the gradient | *B1 | |
| This is because the gradient at \(E\) is the limit of the gradients of the chords as the \(x\)-value tends to 3 or \(\delta x\) tends to 0 | DB1 | Allow: it gets nearer/approaches/tends/almost/approximately 2 |
| 2 |
## Question 3(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $1.2679$ | B1 | AWRT. ISW if correct answer seen. $3-\sqrt{3}$ scores B0 |
| | **1** | |
## Question 3(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $1.7321$ | B1 | AWRT. ISW if correct answer seen |
| | **1** | |
## Question 3(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Sight of 2 or 2.0000 or two in reference to the gradient | *B1 | |
| This is because the gradient at $E$ is the limit of the gradients of the chords as the $x$-value tends to 3 or $\delta x$ tends to 0 | DB1 | Allow: it gets nearer/approaches/tends/almost/approximately 2 |
| | **2** | |3 The equation of a curve is $y = ( x - 3 ) \sqrt { x + 1 } + 3$. The following points lie on the curve. Non-exact values are rounded to 4 decimal places.
$$A ( 2 , k ) \quad B ( 2.9,2.8025 ) \quad C ( 2.99,2.9800 ) \quad D ( 2.999,2.9980 ) \quad E ( 3,3 )$$
\begin{enumerate}[label=(\alph*)]
\item Find $k$, giving your answer correct to 4 decimal places.
\item Find the gradient of $A E$, giving your answer correct to 4 decimal places.\\
The gradients of $B E , C E$ and $D E$, rounded to 4 decimal places, are 1.9748, 1.9975 and 1.9997 respectively.
\item State, giving a reason for your answer, what the values of the four gradients suggest about the gradient of the curve at the point $E$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2021 Q3 [4]}}