| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2024 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indefinite & Definite Integrals |
| Type | Find curve from gradient |
| Difficulty | Moderate -0.8 This is a straightforward integration question requiring basic techniques: (a) solving a simple equation with fractional powers, and (b) integrating polynomial and root terms then finding the constant using a boundary condition. Both parts are routine textbook exercises with no problem-solving insight required, making it easier than average but not trivial due to the algebraic manipulation needed. |
| Spec | 1.08b Integrate x^n: where n != -1 and sums1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08d Evaluate definite integrals: between limits |
| Answer | Marks | Guidance |
|---|---|---|
| 5(a) | Attempt correct process for solving 3-term quadratic equation in x | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| Obtain at least 2 x −3=0 or equivalent | A1 | Ignore 4 x + 3 = 0. |
| Answer | Marks | Guidance |
|---|---|---|
| 4 16 | A1 | SC B1 if no method shown for solving the 3-term |
| Answer | Marks |
|---|---|
| 2 4 | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| 4 16 | A1 | SC B1 if no method shown for solving the 3-term |
| Answer | Marks | Guidance |
|---|---|---|
| 4 16 | A1 | SC B1 if no method shown for solving the 3-term |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
| Answer | Marks |
|---|---|
| 5(b) | 3 |
| Answer | Marks |
|---|---|
| 1 2 3 1 2 3 | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| Obtain correct 2x2 −2x2 +x or equivalent | A1 | Allow unsimplified. |
| Substitute x=4 andy=11 to attempt value of c | M1 | Dependent on at least 2 correct terms involving x. |
| Answer | Marks | Guidance |
|---|---|---|
| Obtain y=2x2 −2x2 +x−9 | A1 | Must be simplified. |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 5:
--- 5(a) ---
5(a) | Attempt correct process for solving 3-term quadratic equation in x | M1 | Accept 8y2 – 6y – 9 → (2y – 3)(4y + 3), if y = √x
specified.
Obtain at least 2 x −3=0 or equivalent | A1 | Ignore 4 x + 3 = 0.
3
SC B1 for x = with no method shown for
2
solving the 3-term quadratic.
9 9
Conclude x= ignore
4 16 | A1 | SC B1 if no method shown for solving the 3-term
quadratic.
Alternative Method for Q5(a)
9 81
3 x = 4x – → 16x2 – 45x + o.e and attempt correct process to solve
2 4 | M1
9 9
Obtain x = or
4 16 | A1 | SC B1 if no method shown for solving the 3-term
quadratic.
9 9
x = ignore
4 16 | A1 | SC B1 if no method shown for solving the 3-term
quadratic.
3
Question | Answer | Marks | Guidance
--- 5(b) ---
5(b) | 3
Integrate to obtain form k x2 +k x2 +k x where kk k 0
1 2 3 1 2 3 | M1
3
Obtain correct 2x2 −2x2 +x or equivalent | A1 | Allow unsimplified.
Substitute x=4 andy=11 to attempt value of c | M1 | Dependent on at least 2 correct terms involving x.
3
Obtain y=2x2 −2x2 +x−9 | A1 | Must be simplified.
Allow ‘f(x) =’.
Allow y missing if y appears previously.
4
Question | Answer | Marks | GuidanceThe equation of a curve is such that $\frac{dy}{dx} = 4x - 3\sqrt{x} + 1$.
\begin{enumerate}[label=(\alph*)]
\item Find the $x$-coordinate of the point on the curve at which the gradient is $\frac{11}{2}$. [3]
\item Given that the curve passes through the point $(4, 11)$, find the equation of the curve. [4]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2024 Q5 [7]}}