| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2022 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas Between Curves |
| Type | Two Curves Intersection Area |
| Difficulty | Standard +0.3 Part (a) is a standard area-between-curves integration requiring subtraction of two simple functions. Part (b) involves finding derivatives and using the angle-between-tangents formula, which is routine A-level content. Both parts require multiple steps but use well-practiced techniques with no novel insight needed. Slightly easier than average due to straightforward setup and given intersection points. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07m Tangents and normals: gradient and equations1.08f Area between two curves: using integration |
| Answer | Marks | Guidance |
|---|---|---|
| \(\pm\int\left(2x^{1/2}+1\right) - \left(\frac{1}{2}x^2 - x + 1\right)dx \left[= \pm\int 2x^{1/2} - \frac{1}{2}x^2 + x\, dx\right]\) | *M1 | |
| \(\pm\left(\frac{4x^{3/2}}{3} + x - \left(\frac{x^3}{6} - \frac{x^2}{2} + x\right)\right)\) or \(\pm\left(\frac{4x^{3/2}}{3} - \frac{x^3}{6} + \frac{x^2}{2}\right)\) | B2, 1, 0 | OE Coefficients may be unsimplified |
| \(\pm\left(\frac{32}{3} - \frac{32}{3} + 8\right)\) or \(\pm\left(\frac{44}{3} - 0 - \frac{20}{3} + 0\right)\) | DM1 | \(\pm(F(4) - F(0))\) using *their* integral(s) |
| \(= 8\) | A1 | Depends on all previous marks. If *M1 B2 DM0* and limits stated, SC B1 for \(+8\) |
| Answer | Marks | Guidance |
|---|---|---|
| Upper curve: \(\frac{dy}{dx} = x^{-\frac{1}{2}}\). Lower curve: \(\frac{dy}{dx} = x-1\) | M1 A1 | Attempt at differentiating one function. A1 if both correct |
| At \(x=4\): gradient of upper curve \(= \frac{1}{2}\), gradient of lower curve \(= 3\) | M1 | Evaluate two gradients using \(x=4\) |
| \(\alpha = \tan^{-1}3 - \tan^{-1}\frac{1}{2}\ [= 71.57 - 26.57]\) | M1 | Use inverse tan to find angles then subtract. OR find equations of both tangents then Pythagoras using a point on each e.g. on axes. OR cosine rule using intercepts or proportion |
| \([\alpha =]\ 45°\) | A1 | AWRT |
## Question 10:
### Part 10(a):
| $\pm\int\left(2x^{1/2}+1\right) - \left(\frac{1}{2}x^2 - x + 1\right)dx \left[= \pm\int 2x^{1/2} - \frac{1}{2}x^2 + x\, dx\right]$ | *M1 | |
|---|---|---|
| $\pm\left(\frac{4x^{3/2}}{3} + x - \left(\frac{x^3}{6} - \frac{x^2}{2} + x\right)\right)$ or $\pm\left(\frac{4x^{3/2}}{3} - \frac{x^3}{6} + \frac{x^2}{2}\right)$ | B2, 1, 0 | OE Coefficients may be unsimplified |
| $\pm\left(\frac{32}{3} - \frac{32}{3} + 8\right)$ or $\pm\left(\frac{44}{3} - 0 - \frac{20}{3} + 0\right)$ | DM1 | $\pm(F(4) - F(0))$ using *their* integral(s) |
| $= 8$ | A1 | Depends on all previous marks. If *M1 B2 DM0* **and** limits stated, **SC B1** for $+8$ |
### Part 10(b):
| Upper curve: $\frac{dy}{dx} = x^{-\frac{1}{2}}$. Lower curve: $\frac{dy}{dx} = x-1$ | M1 A1 | Attempt at differentiating one function. A1 if both correct |
|---|---|---|
| At $x=4$: gradient of upper curve $= \frac{1}{2}$, gradient of lower curve $= 3$ | M1 | Evaluate two gradients using $x=4$ |
| $\alpha = \tan^{-1}3 - \tan^{-1}\frac{1}{2}\ [= 71.57 - 26.57]$ | M1 | Use inverse tan to find angles then subtract. **OR** find equations of both tangents then Pythagoras using a point on each e.g. on axes. **OR** cosine rule using intercepts or proportion |
| $[\alpha =]\ 45°$ | A1 | AWRT |
---10\\
\includegraphics[max width=\textwidth, alt={}, center]{0f59214c-df46-46a6-ae9e-b9c8f104e159-16_942_933_262_605}
Curves with equations $y = 2 x ^ { \frac { 1 } { 2 } } + 1$ and $y = \frac { 1 } { 2 } x ^ { 2 } - x + 1$ intersect at $A ( 0,1 )$ and $B ( 4,5 )$, as shown in the diagram.
\begin{enumerate}[label=(\alph*)]
\item Find the area of the region between the two curves.\\
The acute angle between the two tangents at $B$ is denoted by $\alpha ^ { \circ }$, and the scales on the axes are the same.
\item Find $\alpha$.\\
\includegraphics[max width=\textwidth, alt={}, center]{0f59214c-df46-46a6-ae9e-b9c8f104e159-18_951_725_267_703}
The diagram shows the circle with equation $x ^ { 2 } + y ^ { 2 } = 20$. Tangents touching the circle at points $B$ and $C$ pass through the point $A ( 0,10 )$.\\
(a) By letting the equation of a tangent be $y = m x + 10$, find the two possible values of $m$.\\
(b) Find the coordinates of $B$ and $C$.\\
The point $D$ is where the circle crosses the positive $x$-axis.
\item Find angle $B D C$ in degrees.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2022 Q10 [10]}}