| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2017 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chain Rule |
| Type | Find stationary points and nature |
| Difficulty | Moderate -0.3 This is a straightforward stationary points question requiring standard differentiation of y = 8√x - 2x, solving dy/dx = 0, finding the second derivative, and analyzing intersections with horizontal lines. All techniques are routine for P1 level with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.07d Second derivatives: d^2y/dx^2 notation1.07i Differentiate x^n: for rational n and sums1.07n Stationary points: find maxima, minima using derivatives1.07o Increasing/decreasing: functions using sign of dy/dx |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{dy}{dx} = 4x^{-\frac{1}{2}} - 2\) | B1 | Accept unsimplified |
| \(= 0\) when \(\sqrt{x} = 2\) | ||
| \(x=4,\ y=8\) | B1B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{d^2y}{dx^2} = -2x^{-\frac{3}{2}}\) | B1FT | FT providing \(-\)ve power of \(x\) |
| \(\left(\frac{d^2y}{dx^2} = -\frac{1}{4}\right) \rightarrow\) Maximum | B1 | Correct \(\frac{d^2y}{dx^2}\) and \(x=4\) in (i) are required. Followed by "\( < 0\) or negative" is sufficient but \(\frac{d^2y}{dx^2}\) must be correct if evaluated |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| *EITHER:* Recognises a quadratic in \(\sqrt{x}\) | (M1 | E.g. \(\sqrt{x}=u \rightarrow 2u^2-8u+6=0\) |
| \(1\) and \(3\) as solutions to this equation | A1 | |
| \(\rightarrow x=9,\ x=1\) | A1 | |
| *OR:* Rearranges then squares | (M1 | \(\sqrt{x}\) needs to be isolated before squaring both sides |
| \(\rightarrow x^2-10x+9=0\) oe | A1 | |
| \(\rightarrow x=9,\ x=1\) | A1 | Both correct by trial and improvement gets 3/3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(k > 8\) | B1 |
## Question 9(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{dy}{dx} = 4x^{-\frac{1}{2}} - 2$ | B1 | Accept unsimplified |
| $= 0$ when $\sqrt{x} = 2$ | | |
| $x=4,\ y=8$ | B1B1 | |
---
## Question 9(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{d^2y}{dx^2} = -2x^{-\frac{3}{2}}$ | B1FT | FT providing $-$ve power of $x$ |
| $\left(\frac{d^2y}{dx^2} = -\frac{1}{4}\right) \rightarrow$ Maximum | B1 | Correct $\frac{d^2y}{dx^2}$ and $x=4$ in (i) are required. Followed by "$ < 0$ or negative" is sufficient but $\frac{d^2y}{dx^2}$ must be correct if evaluated |
---
## Question 9(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| *EITHER:* Recognises a quadratic in $\sqrt{x}$ | (M1 | E.g. $\sqrt{x}=u \rightarrow 2u^2-8u+6=0$ |
| $1$ and $3$ as solutions to this equation | A1 | |
| $\rightarrow x=9,\ x=1$ | A1 | |
| *OR:* Rearranges then squares | (M1 | $\sqrt{x}$ needs to be isolated before squaring both sides |
| $\rightarrow x^2-10x+9=0$ oe | A1 | |
| $\rightarrow x=9,\ x=1$ | A1 | Both correct by trial and improvement gets 3/3 |
---
## Question 9(iv):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $k > 8$ | B1 | |
---9 The equation of a curve is $y = 8 \sqrt { } x - 2 x$.\\
(i) Find the coordinates of the stationary point of the curve.\\
(ii) Find an expression for $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ and hence, or otherwise, determine the nature of the stationary point.\\
(iii) Find the values of $x$ at which the line $y = 6$ meets the curve.\\
(iv) State the set of values of $k$ for which the line $y = k$ does not meet the curve.\\
\hfill \mbox{\textit{CAIE P1 2017 Q9 [9]}}