| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2022 |
| Session | March |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Product & Quotient Rules |
| Type | Determine increasing/decreasing intervals |
| Difficulty | Standard +0.3 This is a straightforward application of the quotient/chain rule to find derivatives, followed by standard stationary point analysis. Part (a) requires finding dy/dx and setting it to zero, part (b) applies this to a specific case with second derivative test, and part (c) checks the sign of g'(x). All steps are routine A-level techniques with no novel insight required, making it slightly easier than average. |
| Spec | 1.07i 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 |
| \(\frac{dy}{dx} = \{-k(3x-k)^{-2}\}\{\times 3\}\{+3\}\) | B2, 1, 0 | |
| \(\frac{-3k}{(3x-k)^2} + 3 = 0\) leading to \((3)(3x-k)^2 = (3)k\), leading to \(3x - k = [\pm]\sqrt{k}\) | M1 | Set \(\frac{dy}{dx} = 0\) and remove the denominator |
| \(x = \frac{k \pm \sqrt{k}}{3}\) | A1 | OE |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(a = \frac{4 \pm \sqrt{4}}{3}\) leading to \(a = 2\) | B1 | Substitute \(x = a\) when \(k = 4\). Allow \(x = 2\). |
| \(f''(x) = f'\left[-12(3x-4)^{-2} + 3\right] = 72(3x-4)^{-3}\) | B1 | Allow \(18k(3x-k)^{-3}\) |
| \(> 0\) (or 9) when \(x = 2 \rightarrow\) minimum | B1 FT | FT on *their* \(x = 2\), providing their \(x \geq \frac{3}{2}\) and \(f''(x)\) is correct |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Substitute \(k = -1\) leading to \(g'(x) = \frac{3}{(3x+1)^2} + 3\) | M1 | Condone one error. |
| \(g'(x) > 0\) or \(g'(x)\) always positive, hence g is an increasing function | A1 | WWW. A0 if conclusion depends on substitution of values into \(g'(x)\). |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(x = \frac{k \pm \sqrt{k}}{3}\) when \(k = -1\) has no solutions, so g is increasing or decreasing | M1 | Allow the statement 'no turning points' for increasing or decreasing |
| Show \(g'(x)\) is positive for any value of \(x\), hence g is an increasing function | A1 | Or show \(g(b) > g(a)\) for \(b > a \rightarrow\) g is an increasing function |
## Question 11(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{dy}{dx} = \{-k(3x-k)^{-2}\}\{\times 3\}\{+3\}$ | B2, 1, 0 | |
| $\frac{-3k}{(3x-k)^2} + 3 = 0$ leading to $(3)(3x-k)^2 = (3)k$, leading to $3x - k = [\pm]\sqrt{k}$ | M1 | Set $\frac{dy}{dx} = 0$ and remove the denominator |
| $x = \frac{k \pm \sqrt{k}}{3}$ | A1 | OE |
---
## Question 11(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $a = \frac{4 \pm \sqrt{4}}{3}$ leading to $a = 2$ | B1 | Substitute $x = a$ when $k = 4$. Allow $x = 2$. |
| $f''(x) = f'\left[-12(3x-4)^{-2} + 3\right] = 72(3x-4)^{-3}$ | B1 | Allow $18k(3x-k)^{-3}$ |
| $> 0$ (or 9) when $x = 2 \rightarrow$ minimum | B1 FT | FT on *their* $x = 2$, providing their $x \geq \frac{3}{2}$ and $f''(x)$ is correct |
---
## Question 11(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Substitute $k = -1$ leading to $g'(x) = \frac{3}{(3x+1)^2} + 3$ | M1 | Condone one error. |
| $g'(x) > 0$ or $g'(x)$ always positive, hence g is an increasing function | A1 | WWW. A0 if conclusion depends on substitution of values into $g'(x)$. |
**Alternative method for 11(c):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x = \frac{k \pm \sqrt{k}}{3}$ when $k = -1$ has no solutions, so g is increasing or decreasing | M1 | Allow the statement 'no turning points' for increasing or decreasing |
| Show $g'(x)$ is positive for any value of $x$, hence g is an increasing function | A1 | Or show $g(b) > g(a)$ for $b > a \rightarrow$ g is an increasing function |11 It is given that a curve has equation $y = k ( 3 x - k ) ^ { - 1 } + 3 x$, where $k$ is a constant.
\begin{enumerate}[label=(\alph*)]
\item Find, in terms of $k$, the values of $x$ at which there is a stationary point.\\
The function f has a stationary value at $x = a$ and is defined by
$$f ( x ) = 4 ( 3 x - 4 ) ^ { - 1 } + 3 x \quad \text { for } x \geqslant \frac { 3 } { 2 }$$
\item Find the value of $a$ and determine the nature of the stationary value.
\item The function g is defined by $\mathrm { g } ( x ) = - ( 3 x + 1 ) ^ { - 1 } + 3 x$ for $x \geqslant 0$.
Determine, making your reasoning clear, whether $g$ is an increasing function, a decreasing function or neither.\\
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 Q11 [9]}}