| Exam Board | OCR MEI |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2011 |
| Session | June |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Basic factored form sketching |
| Difficulty | Standard +0.8 This is an extended investigation question from FP2 requiring multiple sketches, calculus optimization (finding maxima via differentiation of a product), pattern recognition across cases, and conjectures about limiting behavior. While individual parts are accessible, the multi-part nature, need for systematic analysis, and requirement to synthesize observations across different parameter values elevates this above standard A-level questions but remains within typical Further Maths scope. |
| Spec | 1.02m Graphs of functions: difference between plotting and sketching1.02n Sketch curves: simple equations including polynomials1.02w Graph transformations: simple transformations of f(x)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/dx1.08d Evaluate definite integrals: between limits1.08e Area between curve and x-axis: using definite integrals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| (A) \(m=1, n=1\): Negative parabola from \((0,0)\) to \((1,0)\), symmetrical about \(x=0.5\) | G1 | |
| (B) \(m=2, n=2\): Bell-shape from \((0,0)\) to \((1,0)\), symmetrical about \(x=0.5\); flat ends, and obviously different to (A) | G1 | |
| (C) \(m=2, n=4\): Skewed curve from \((0,0)\) to \((1,0)\), maximum to left of \(x=0.5\) | G1 | |
| (D) \(m=4, n=2\): Skewed curve from \((0,0)\) to \((1,0)\), maximum to right of \(x=0.5\) | G1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| When \(m=n\), the curve is symmetrical | B1 | |
| Exchanging \(m\) and \(n\) reflects the curve | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| If \(m>n\), the maximum is to the right of \(x=0.5\) | B1 | o.e. Give B1B0 if idea is correct but vaguely expressed |
| As \(m\) increases relative to \(n\), the maximum point moves further to the right | B1 | |
| \(y = x^m(1-x)^n \Rightarrow \frac{dy}{dx} = mx^{m-1}(1-x)^n - nx^m(1-x)^{n-1} = x^{m-1}(1-x)^{n-1}[m(1-x)-nx]\) | M1, A1 | Using product rule; any correct form |
| \(\frac{dy}{dx}=0 \Rightarrow\) maximum at \(x=\frac{m}{m+n}\) | M1, A1 | Setting derivative \(= 0\) and solving to find a value of \(x\) other than 0 or 1 |
| Answer | Marks |
|---|---|
| \(y'(0) = 0\) provided \(m > 1\) | B1 |
| \(y'(1) = 0\) provided \(n > 1\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| For large \(m\) and \(n\), the curve approaches the \(x\)-axis | B1 | Comment on shape |
| \(\Rightarrow \int_0^1 x^m(1-x)^n dx \to 0\) as \(m, n \to \infty\) | B1 | Independent |
| Answer | Marks | Guidance |
|---|---|---|
| [Graph showing curve near \(y = 1\) across \([0,1]\), dropping steeply at endpoints] | M1 | Evidence of investigation s.o.i. |
| The curve tends to \(y = 1\) | A1 | Accept "three sides of (unit) square" |
# Question 5:
## Part (i)
| Answer | Mark | Guidance |
|--------|------|----------|
| **(A)** $m=1, n=1$: Negative parabola from $(0,0)$ to $(1,0)$, symmetrical about $x=0.5$ | G1 | |
| **(B)** $m=2, n=2$: Bell-shape from $(0,0)$ to $(1,0)$, symmetrical about $x=0.5$; flat ends, and obviously different to (A) | G1 | |
| **(C)** $m=2, n=4$: Skewed curve from $(0,0)$ to $(1,0)$, maximum to left of $x=0.5$ | G1 | |
| **(D)** $m=4, n=2$: Skewed curve from $(0,0)$ to $(1,0)$, maximum to right of $x=0.5$ | G1 | |
## Part (ii)
| Answer | Mark | Guidance |
|--------|------|----------|
| When $m=n$, the curve is symmetrical | B1 | |
| Exchanging $m$ and $n$ reflects the curve | B1 | |
## Part (iii)
| Answer | Mark | Guidance |
|--------|------|----------|
| If $m>n$, the maximum is to the right of $x=0.5$ | B1 | o.e. Give B1B0 if idea is correct but vaguely expressed |
| As $m$ increases relative to $n$, the maximum point moves further to the right | B1 | |
| $y = x^m(1-x)^n \Rightarrow \frac{dy}{dx} = mx^{m-1}(1-x)^n - nx^m(1-x)^{n-1} = x^{m-1}(1-x)^{n-1}[m(1-x)-nx]$ | M1, A1 | Using product rule; any correct form |
| $\frac{dy}{dx}=0 \Rightarrow$ maximum at $x=\frac{m}{m+n}$ | M1, A1 | Setting derivative $= 0$ and solving to find a value of $x$ other than 0 or 1 |
## Question (iv):
$y'(0) = 0$ provided $m > 1$ | B1 |
$y'(1) = 0$ provided $n > 1$ | B1 |
**Total: 2**
---
## Question (v):
For large $m$ and $n$, the curve approaches the $x$-axis | B1 | Comment on shape
$\Rightarrow \int_0^1 x^m(1-x)^n dx \to 0$ as $m, n \to \infty$ | B1 | Independent
**Total: 2**
---
## Question (vi):
e.g. $m = 0.01$, $n = 0.01$
[Graph showing curve near $y = 1$ across $[0,1]$, dropping steeply at endpoints] | M1 | Evidence of investigation s.o.i.
The curve tends to $y = 1$ | A1 | Accept "three sides of (unit) square"
**Total: 2**
**Section Total: 18**5 In this question, you are required to investigate the curve with equation
$$y = x ^ { m } ( 1 - x ) ^ { n } , \quad 0 \leqslant x \leqslant 1 ,$$
for various positive values of $m$ and $n$.
\begin{enumerate}[label=(\roman*)]
\item On separate diagrams, sketch the curve in each of the following cases.\\
(A) $m = 1 , n = 1$,\\
(B) $m = 2 , n = 2$,\\
(C) $m = 2 , n = 4$,\\
(D) $m = 4 , n = 2$.
\item What feature does the curve have when $m = n$ ?
What is the effect on the curve of interchanging $m$ and $n$ when $m \neq n$ ?
\item Describe how the $x$-coordinate of the maximum on the curve varies as $m$ and $n$ vary. Use calculus to determine the $x$-coordinate of the maximum.
\item Find the condition on $m$ for the gradient to be zero when $x = 0$. State a corresponding result for the gradient to be zero when $x = 1$.
\item Use your calculator to investigate the shape of the curve for large values of $m$ and $n$. Hence conjecture what happens to the value of the integral $\int _ { 0 } ^ { 1 } x ^ { m } ( 1 - x ) ^ { n } \mathrm {~d} x$ as $m$ and $n$ tend to infinity.
\item Use your calculator to investigate the shape of the curve for small values of $m$ and $n$. Hence conjecture what happens to the shape of the curve as $m$ and $n$ tend to zero.
}{www.ocr.org.uk}) after the live examination series.\\
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\end{enumerate}
\hfill \mbox{\textit{OCR MEI FP2 2011 Q5 [18]}}