| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2024 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differentiating Transcendental Functions |
| Type | Find normal line equation |
| Difficulty | Standard +0.3 This is a straightforward multi-part question testing standard A-level calculus techniques: finding a normal line (gradient calculation), integration with fractional powers, and analyzing whether a function is increasing/decreasing using the discriminant. All parts follow routine procedures with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07m Tangents and normals: gradient and equations1.07n Stationary points: find maxima, minima using derivatives1.07o Increasing/decreasing: functions using sign of dy/dx1.08b Integrate x^n: where n != -1 and sums1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| 10(a) | −18 | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| 18 | M1 | Use of m m =−1from f(x) with x=1. |
| Answer | Marks | Guidance |
|---|---|---|
| x−1 18 | A1 | OE |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
| Answer | Marks |
|---|---|
| 10(b) | |
| Answer | Marks | Guidance |
|---|---|---|
| | B1B1 | B1 for each unsimplified {}. |
| Answer | Marks | Guidance |
|---|---|---|
| 0=3 ( 2(1)−3 ) 3 −6(1) 3 +c 0=3−6+c | M1 | Use of x=1 and y=0 in their integrated f(x), defined as an |
| Answer | Marks | Guidance |
|---|---|---|
| f (x) or y= 3(2x−3) 3 −6x3 +3 | A1 | Only condone c = 3 as their final answer if all coefficients have |
| Answer | Marks |
|---|---|
| 10(c) | b2 −4ac=1282 −4125192 and stating “< 0” |
| Answer | Marks | Guidance |
|---|---|---|
| OR sketch of the given quadratic and stating positive. | M1* | b2 −4ac=−79616 can be accepted in place of working. |
| No turning points [in the original function.] | DM1 | |
| Decreasing because f(any positive x value)0 | A1 | WWW |
Question 10:
--- 10(a) ---
10(a) | −18 | B1 | SOI
1
18 | M1 | Use of m m =−1from f(x) with x=1.
1 2
y − 0 1
=
x−1 18 | A1 | OE
ISW
3
Question | Answer | Marks | Guidance
--- 10(b) ---
10(b) |
4 1 1 5 1
f(x)= 8(2x−3) 3. . −10x3. +c
2 4 5
3 3
4 5
3(2x−3) 3 −6x3 +c
| B1B1 | B1 for each unsimplified {}.
Can be implied by equivalent simplified or partly simplified
versions.
4 5
0=3 ( 2(1)−3 ) 3 −6(1) 3 +c 0=3−6+c | M1 | Use of x=1 and y=0 in their integrated f(x), defined as an
expression with at least one correct power, which must contain
+ c.
4 5
f (x) or y= 3(2x−3) 3 −6x3 +3 | A1 | Only condone c = 3 as their final answer if all coefficients have
previously been simplified in a correct statement.
4
--- 10(c) ---
10(c) | b2 −4ac=1282 −4125192 and stating “< 0”
OR use of the quadratic formula and stating “No solutions”
OR completing the square for the given quadratic and stating positive or > 0.
OR sketch of the given quadratic and stating positive. | M1* | b2 −4ac=−79616 can be accepted in place of working.
No turning points [in the original function.] | DM1
Decreasing because f(any positive x value)0 | A1 | WWW
e.g. f'(1)=−18.
3A function f with domain $x > 0$ is such that $\mathrm{f}'(x) = 8(2x - 3)^{\frac{1}{3}} - 10x^{\frac{3}{5}}$. It is given that the curve with equation $y = \mathrm{f}(x)$ passes through the point $(1, 0)$.
\begin{enumerate}[label=(\alph*)]
\item Find the equation of the normal to the curve at the point $(1, 0)$. [3]
\item Find f$(x)$. [4]
\end{enumerate}
It is given that the equation $\mathrm{f}'(x) = 0$ can be expressed in the form
$$125x^2 - 128x + 192 = 0.$$
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Determine, making your reasoning clear, whether f is an increasing function, a decreasing function or neither. [3]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2024 Q10 [10]}}