CAIE FP2 2014 June — Question 3

Exam BoardCAIE
ModuleFP2 (Further Pure Mathematics 2)
Year2014
SessionJune
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicMoments
TypeParticle suspended by strings
DifficultyStandard +0.8 This is a statics problem involving a particle in equilibrium under multiple forces (weight, two string tensions at angles). It requires resolving forces in two directions, applying equilibrium conditions, and likely involves trigonometric manipulation. The two-part structure with part (ii) asking to show a specific relationship suggests moderate algebraic complexity beyond routine mechanics problems.
Spec1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta3.03a Force: vector nature and diagrams
3 hours
Additional Materials:
Answer Booklet/Paper
Graph Paper
List of Formulae (MF10) \section*{READ THESE INSTRUCTIONS FIRST} If you have been given an Answer Booklet, follow the instructions on the front cover of the Booklet.
Write your Centre number, candidate number and name on all the work you hand in.
Write in dark blue or black pen.
You may use an HB pencil for any diagrams or graphs.
Do not use staples, paper clips, glue or correction fluid.
DO NOT WRITE IN ANY BARCODES. Answer all the questions.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question.
Where a numerical value is necessary, take the acceleration due to gravity to be \(10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
The use of a calculator is expected, where appropriate.
Results obtained solely from a graphic calculator, without supporting working or reasoning, will not receive credit.
You are reminded of the need for clear presentation in your answers.
At the end of the examination, fasten all your work securely together.
[0pt] The number of marks is given in brackets [ ] at the end of each question or part question.
Question 3:
Part (i)
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(\frac{1}{2}mv^2 = \frac{1}{2}mu^2 - mg(a - a\cos\theta)\)B1 Conservation of energy
\(T - mg\cos\theta = \frac{mv^2}{a}\)B1 \(F=ma\) radially
\(mu = J\)B1 Relate \(u\) to impulse \(J\)
\(T = \frac{J^2}{ma} - mg(2 - 3\cos\theta)\) A.G.M1 A1 Eliminate \(u\) and \(v\) to find \(T\)
Part (ii)(a)
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(v^2 = ga(2\cos\theta - 1)\) \(k=1\)
\([T = mg(3\cos\theta - 1)]\)
\(v=0\) [and \(T>0\)] when \(\cos\theta = \frac{1}{2}\)M1 Investigate \(v\) [and \(T\)] for \(k=1\)
so \(P\) oscillates (A.E.F.)A1 S.R. Award B1 for correct result based only on \(T\)
Part (ii)(b)
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(T = mg(3\cos\theta + 4)\) \(k=6\)
\([v^2 = ga(2\cos\theta + 4)]\)
\(T > 0\) for e.g. \(\theta = \pi\) and \(v > 0\)M1 Investigate \(v\) and \(T\) for \(k=6\)
so \(P\) does full circle (A.E.F.)A1 S.R. Award B1 for correct result based only on \(T\) 
# Question 3:

## Part (i)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{1}{2}mv^2 = \frac{1}{2}mu^2 - mg(a - a\cos\theta)$ | B1 | Conservation of energy |
| $T - mg\cos\theta = \frac{mv^2}{a}$ | B1 | $F=ma$ radially |
| $mu = J$ | B1 | Relate $u$ to impulse $J$ |
| $T = \frac{J^2}{ma} - mg(2 - 3\cos\theta)$ **A.G.** | M1 A1 | Eliminate $u$ and $v$ to find $T$ |

## Part (ii)(a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $v^2 = ga(2\cos\theta - 1)$ | | $k=1$ |
| $[T = mg(3\cos\theta - 1)]$ | | |
| $v=0$ [and $T>0$] when $\cos\theta = \frac{1}{2}$ | M1 | Investigate $v$ [and $T$] for $k=1$ |
| so $P$ oscillates (A.E.F.) | A1 | S.R. Award B1 for correct result based only on $T$ |

## Part (ii)(b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $T = mg(3\cos\theta + 4)$ | | $k=6$ |
| $[v^2 = ga(2\cos\theta + 4)]$ | | |
| $T > 0$ for e.g. $\theta = \pi$ and $v > 0$ | M1 | Investigate $v$ and $T$ for $k=6$ |
| so $P$ does full circle (A.E.F.) | A1 | S.R. Award B1 for correct result based only on $T$ |
3 hours\\
Additional Materials:\\
Answer Booklet/Paper\\
Graph Paper\\
List of Formulae (MF10)

\section*{READ THESE INSTRUCTIONS FIRST}
If you have been given an Answer Booklet, follow the instructions on the front cover of the Booklet.\\
Write your Centre number, candidate number and name on all the work you hand in.\\
Write in dark blue or black pen.\\
You may use an HB pencil for any diagrams or graphs.\\
Do not use staples, paper clips, glue or correction fluid.\\
DO NOT WRITE IN ANY BARCODES.

Answer all the questions.\\
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question.\\
Where a numerical value is necessary, take the acceleration due to gravity to be $10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$.\\
The use of a calculator is expected, where appropriate.\\
Results obtained solely from a graphic calculator, without supporting working or reasoning, will not receive credit.\\
You are reminded of the need for clear presentation in your answers.\\
At the end of the examination, fasten all your work securely together.\\[0pt]
The number of marks is given in brackets [ ] at the end of each question or part question.

\hfill \mbox{\textit{CAIE FP2 2014 Q3}}
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