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author | Karl Berry <karl@freefriends.org> | 2014-05-05 22:02:58 +0000 |
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committer | Karl Berry <karl@freefriends.org> | 2014-05-05 22:02:58 +0000 |
commit | c2e26b130cd3c16ddba1e50bd65f8ff334ba8acc (patch) | |
tree | b1e3190b569092239516018cca8b73c918accb9f /Master/texmf-dist/doc/latex/exam-n/sample | |
parent | e6a0a4076d4c6d8eb715a0350a499642c68164df (diff) |
exam-n (5may14)
git-svn-id: svn://tug.org/texlive/trunk@33862 c570f23f-e606-0410-a88d-b1316a301751
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diff --git a/Master/texmf-dist/doc/latex/exam-n/sample/Makefile b/Master/texmf-dist/doc/latex/exam-n/sample/Makefile new file mode 100644 index 00000000000..ff34e72ae46 --- /dev/null +++ b/Master/texmf-dist/doc/latex/exam-n/sample/Makefile @@ -0,0 +1,16 @@ +all: sample_exam.pdf + +sample_exam.pdf: numerical1-solution.pdf + +sample_exam_solution.pdf: sample_exam_solution.tex + +sample_exam_solution.tex: sample_exam.tex + rm -f $@ + sed '1s/documentclass/documentclass[showsolutions]/' sample_exam.tex >sample_exam_solution.tex + +%.pdf: %.tex + TEXINPUTS=..: pdflatex $< + if grep 'Rerun to get cross' ${<:.tex=.log}; then TEXINPUTS=..: pdflatex $<; else :; fi + +clean: + rm -f sample_exam*.pdf sample_exam_solution.tex *.log *.aux diff --git a/Master/texmf-dist/doc/latex/exam-n/sample/cosmo1.tex b/Master/texmf-dist/doc/latex/exam-n/sample/cosmo1.tex new file mode 100644 index 00000000000..cd9ae51974f --- /dev/null +++ b/Master/texmf-dist/doc/latex/exam-n/sample/cosmo1.tex @@ -0,0 +1,91 @@ +\documentclass[compose]{exam-n} + +\usepackage{graphics} + +\begin{document} + +\begin{question}{20} \author{Andrew Davies} +% Have a blank line here, to check that the question number remains +% nicely lined up, even if there are lines between the environment +% opening and the text. + +First, \emph{admire} the restful picture of a spiral in Fig.\ \ref{f:spiral}, +included as a graphic. Fully zenned up? Then let us begin\dots. +\begin{figure} +\ifbigfont + \includegraphics[width=\textwidth]{spiral} +\else + \includegraphics{spiral} +\fi +\caption{\label{f:spiral}A spiral} +\end{figure} + +\part Show that, under the action of gravity alone, the scale size +of the Universe varies according to +\begin{equation*} +\ddot{R}=-\frac{4\pi G \rho_0}{3R^2} +\end{equation*} +\partmarks*{4} +and that, consequently, +\begin{equation*} +\dot{R}^2=-\frac{8\pi G \rho_0}{3R}=-K. +\end{equation*} +\partmarks*{3} + +Express $K$ in terms of the present values of the Hubble constant +$H_0$ and of the density parameter $\Omega_0$. +\partmarks{3} +\begin{solution} +This can be solved by \emph{remembering} the solution +\partmarks{3} +\end{solution} + +\part In the early Universe, the relation between time and +temperature has the form +\begin{equation*} +t=\sqrt{\frac{3c^2}{16\pi G g_{\rm eff}a}}\frac{1}{T^2}, +\end{equation*} +where $a$ is the radiation constant. Discuss the assumptions +leading to this equation, but do not carry out the mathematical +derivation. Discuss the meaning of the factor $g_{\rm eff}$ , and +find its value just before and after annihilation of electrons and +positrons. +\partmarks{6} +\begin{solution} +Before, well, geee; after\dots kazamm! +\end{solution} + +\part +Explain how the present-day neutron/proton ratio was established +by particle interactions in the Early Universe. How is the ratio +of deuterium to helium relevant to the nature of dark matter? It is +\emph{crucially vital} to note +that Table~\ref{t:dullness} is of absolutely no relevance to this question. +\begin{table} +\begin{tabular}{r|l} +Column 1&and row 1\\ +More content&in row 2 +\end{tabular} +\caption{\label{t:dullness}A remarkably dull table} +\end{table} +\partmarks{4} +\begin{solution} +Explanations are superfluous; all that is, is. +\begin{table} +\begin{tabular}{r|l} +First rows&are premier\\ +subsequent rows&are of secondary interest +\end{tabular} +\caption{\label{t:dullnessII}A table o'erbrimming with otioseness} +\end{table} +In addition, Table~\ref{t:dullnessII} adds nothing to the discussion, +adds nothing to our understanding of our place in the cosmos, but it +\emph{does} contribute slightly to the heat-death of the universe (can +you work out how many deuterium nuclei decayed during the typing of +this table?). +\end{solution} +\begin{questiondata} +Hubble's law: $v=H_0 D$ +\end{questiondata} +\end{question} +\end{document} diff --git a/Master/texmf-dist/doc/latex/exam-n/sample/cosmo2.tex b/Master/texmf-dist/doc/latex/exam-n/sample/cosmo2.tex new file mode 100644 index 00000000000..bb6c512b9dd --- /dev/null +++ b/Master/texmf-dist/doc/latex/exam-n/sample/cosmo2.tex @@ -0,0 +1,60 @@ +\documentclass[compose]{exam-n} +\begin{document} + +\begin{question}{30} \comment{by John Brown} +Show by considering the Newtonian rules of vector and velocity +addition that in Newtonian cosmology the cosmological principle +demands Hubble's Law $v_r\propto r$.\partmarks{10} + +Prove that, in Euclidean geometry, the number $N(F)$ of objects +of identical luminosity $L$, and of space density $n(r)$ at +distance $r$, observed with radiation flux $\ge F$ is (neglecting +other selection and redshift effects) +\begin{equation*} +N(F)=4\pi\int_0^{(\frac{L}{4\pi F})^{1/2}} n(r) r^2\ddd r. +\end{equation*} +\partmarks*{5} + +Use this to show that for $n=n_1=$constant at $r<r_1$ and +$n=n_2=$constant at $r>r_1$, +\begin{equation*} +N(F) = N_1\left(\frac{F}{F_1}\right)^{-3/2}\qquad \text{for +$F>F_1$}, +\end{equation*} +and +\ifbigfont + \begin{multline*} + N(F) = + N_1\left\{1+\frac{n_2}{n_1}\left[\left(\frac{F}{F_1}\right)^{-3/2}-1\right] + \right\}\\\text{for $F<F_1$}, + \end{multline*} +\else + \begin{equation*} + N(F) = + N_1\left\{1+\frac{n_2}{n_1}\left[\left(\frac{F}{F_1}\right)^{-3/2}-1\right] + \right\} \qquad \text{for $F<F_1$}, + \end{equation*} +\fi +where $F_1=L/4\pi r_1^2$, $N_1=N(F_1)=\frac{4}{3}\pi r_1^3 n_1$. +\partmarks{9} + +Reduce these two expressions to the result for a completely +uniform density universe with $n_1=n_2=n_0$. +\partmarks{3} + +% An itemized list followed by partmarks* +Sketch how $n(F)$ would look in universes which are +\begin{itemize} +\item flat, +\item open, +\item and closed. +\end{itemize} +\partmarks*{3} + +\begin{solution} +A sufficiently heavy weight will reduce expressions to completely +uniform sheets of paper if it is placed on top of them.\partmarks3 +In a flat universe, $n(F)$ will look like n(F).\partmarks*3 +\end{solution} +\end{question} +\end{document} diff --git a/Master/texmf-dist/doc/latex/exam-n/sample/cosmo3.tex b/Master/texmf-dist/doc/latex/exam-n/sample/cosmo3.tex new file mode 100644 index 00000000000..b7b74c01077 --- /dev/null +++ b/Master/texmf-dist/doc/latex/exam-n/sample/cosmo3.tex @@ -0,0 +1,50 @@ +\documentclass[compose]{exam-n} +\begin{document} + +\begin{question}{30} \comment{by Andrew Davies} + +The Friedmann equations are written, in a standard notation, +\begin{gather*} +H^2 = \frac{8\pi G\rho}{3}-\frac{kc^2}{R^2}+\frac{\Lambda}{3},\\ +\Diffl {}t (\rho c^2R^3)=-p\Diffl {R^3}t, +\end{gather*} +Discuss briefly the meaning of each of $H$, $\rho$ , $k$ and +$\Lambda$. \partmarks{4} + +Suppose the Universe consists of a single substance with equation +of state $p=w\rho c^2$, where $ w=$constant. Consider the +following cases, with $k=\Lambda=0$: + +\part For $w = 0$, find the relation between $R$ and $\rho$. Hence +show that $H=\frac{2}{3t}$. What is the physical interpretation of +this case? +\partmarks{8} + +\part In the case $w=-1$ , show that $H = $constant and $R = A \exp(Ht)$, +with $A$ constant. +\partmarks{4} + +\part Explain how the case, $w=-1$, $k=\Lambda=0$, $\rho=0$ is +equivalent to an empty, flat, Universe with a non-zero $\Lambda$. +\partmarks{2} + +\part Consider a model Universe which contained matter with equation +of state with $w = 0$ for $0 < t < t_0$, but which changes to +$W=0$ for $t\ge t_0$ without any discontinuity in $H(t)$. +Regarding this second stage as driven by a non-zero $\Lambda$ what +is the value of $\Lambda$ if $t_0 = 10^{24}$\units{\mu s}? Define the +dimensionless deceleration parameter, $q$, and find its value +before and after $t_0$. +\shout{Shout it loud: I'm a geek and I'm proud} +\partmarks{8} + +Note: that's +\[ +t_0=10^{24}\units{\mu s}\qquad\text{with a letter mu: $\mu$}. +\] + +\part To what extent does this idealized model resemble the currently +accepted picture of the development of our Universe? +\partmarks{4} +\end{question} +\end{document} diff --git a/Master/texmf-dist/doc/latex/exam-n/sample/dynamical1.tex b/Master/texmf-dist/doc/latex/exam-n/sample/dynamical1.tex new file mode 100644 index 00000000000..e3a51a803bb --- /dev/null +++ b/Master/texmf-dist/doc/latex/exam-n/sample/dynamical1.tex @@ -0,0 +1,96 @@ +\documentclass[compose]{exam-n} + +\begin{document} + +\begin{question}{20} \author{John Brown and Declan Diver} +\part An earth satellite in a highly eccentric orbit of (constant) +perigee distance $q$ undergoes a targential velocity impulse +$-\Delta V$ at each perigee passage. By considering the mean rate +of change of velocity at perigee, show that the mean rate of +change of the semi-major axis $a$ ($\gg q$) satisfies +\begin{equation*} +\frac{1}{a^2} \Diffl at = +\left(\frac{8}{GMq}\right)^{1/2}\frac{\Delta V}{T}, +\end{equation*} +where $M$ is the Earth's mass and $T$ the orbital period. +\partmarks{3} +\begin{questiondata} +You may assume $\displaystyle v^2(r)=GM\left(\frac{2}{r}-\frac{1}{a}\right)$. +\end{questiondata} + +Using $T=2\pi(a^3/GM)^{1/2}$ show that with $a_0=a(0)$, (where +$a(t)$ is the semimajor axis at time $t$) +\begin{equation*} +\frac{a(t)}{a_0}=\left[1-\frac{t\Delta V}{2^{1/2}\pi +a_0(1-e_0)^{1/2}}\right]^2 +\end{equation*} +\partmarks*{2} +and +\begin{equation*} +\frac{T(t)}{T_0}=\left[1-\frac{t\Delta V}{2^{1/2}\pi +a_0(1-e_0)^{1/2}}\right]^3 +\end{equation*} +\partmarks*{1} +and the eccentricity satisfies (with $e_0=e(0)$) +\begin{equation*} +e(t)=1-\frac{1-e_0}{\left[1-\frac{t\Delta V}{2^{1/2}\pi +a_0(1-e_0)^{1/2}}\right]^2}. +\end{equation*} +\partmarks*{2} + +Show that, once the orbit is circular, its radius decays +exponentially with time on timescale $m_0/2\dot{m}$ where $m_0$ +is the satellite mass and $\dot{m}$ the mass of atmosphere +`stopped' by it per second. \partmarks{2} + +\part What is meant by (a) the sphere of influence of a star, and +(b) the passage distance? +\partmarks{2} + +Consider a system of $N$ identical stars, each of mass $m$. + +\part Given that the change $\delta u$ in the speed of one such star +due to the cumulative effect over time $t$ of many gravitational +encounters with other stars in the system can be approximated by +\begin{equation*} +(\delta u)^2 \propto [\nu tm^2\log(p_{\rm max}/p_{\rm +min})]/\bar{u}, +\end{equation*} +where $\bar{u}$ is the rms mutual speed, $\nu$ is the stellar +number density, and $p_{\rm max, min}$ are the maximum, minimum +passage distances for the system, show that this leads to a natural +time $T$ for the system, where +\begin{equation*} +T\propto\frac{\bar{u}u^2}{m^2\nu\log N}. +\end{equation*} +\partmarks*{5} + +\begin{questiondata} +You may assume that the sphere of influence radius of a star is +approximated by $(m/M)^{2/5}R$ where $R$ and $M$ are the radius +and mass of the whole system respectively. +\end{questiondata} + +\part Deduce that $T$ is the disintegration timescale for the system, +by showing that a star with initial speed $u_0$ in a stable circular +orbit reaches escape speed after time $T$. +\partmarks{3} + +Dummy text, to lengthen the question to the extent that it spreads across three pages. +Dummy text, to lengthen the question to the extent that it spreads across three pages. +Dummy text, to lengthen the question to the extent that it spreads across three pages. +Dummy text, to lengthen the question to the extent that it spreads across three pages. +Dummy text, to lengthen the question to the extent that it spreads across three pages. +Dummy text, to lengthen the question to the extent that it spreads across three pages. +Dummy text, to lengthen the question to the extent that it spreads across three pages. +Dummy text, to lengthen the question to the extent that it spreads across three pages. +Dummy text, to lengthen the question to the extent that it spreads across three pages. +Dummy text, to lengthen the question to the extent that it spreads across three pages. +Dummy text, to lengthen the question to the extent that it spreads across three pages. +Dummy text, to lengthen the question to the extent that it spreads across three pages. +Dummy text, to lengthen the question to the extent that it spreads across three pages. +Dummy text, to lengthen the question to the extent that it spreads across three pages. +Dummy text, to lengthen the question to the extent that it spreads across three pages. +Dummy text, to lengthen the question to the extent that it spreads across three pages. +\end{question} +\end{document} diff --git a/Master/texmf-dist/doc/latex/exam-n/sample/dynamical2.tex b/Master/texmf-dist/doc/latex/exam-n/sample/dynamical2.tex new file mode 100644 index 00000000000..517ae849205 --- /dev/null +++ b/Master/texmf-dist/doc/latex/exam-n/sample/dynamical2.tex @@ -0,0 +1,24 @@ +\documentclass[final]{exam-n} +\begin{document} + +\begin{question}{30} \comment{by John Brown} +Give the equations of motion for $i=1,\ldots, N$ particles of +masses $m_i$ and positions $r_i(t)$ under the action of mutual +gravity alone in an arbitrary inertial frame. +\partmarks{4} + +Use these to derive the following conservation laws of the system: + +\part Constancy of linear momentum -- i.e., centre of mass fixed in a +suitable inertial frame. \partmarks{4} + \part Constancy of angular momentum. \partmarks{6} + \part Constancy of total energy. \partmarks{8} + +How many integrals of motion exist in total? +\partmarks{2} + +Derive the moment of inertia of the system and demonstrate its +relevance to criteria for escape of particles from the system. +\partmarks{6} +\end{question} +\end{document} diff --git a/Master/texmf-dist/doc/latex/exam-n/sample/dynamical3.tex b/Master/texmf-dist/doc/latex/exam-n/sample/dynamical3.tex new file mode 100644 index 00000000000..abc51017ead --- /dev/null +++ b/Master/texmf-dist/doc/latex/exam-n/sample/dynamical3.tex @@ -0,0 +1,59 @@ +\documentclass[compose]{exam-n} +\begin{document} + +\begin{question}{30} \comment{by Declan Diver} +For a system of $N$ objects, each having mass $m_i$ and position +vector $\mathbf{R}_i$ with respect to a fixed co-ordinate system, +use the moment of inertia +\[ +I=\sum_{i=1}^N m_i R_i^2 +\] +to deduce the virial theorem in the forms +\[ +\ddot{I}=4E_k+2E_G=2E_k+2E +\] +where $E_k$ and $E_G$ are respectively the total kinetic and +gravitational potential energy, and $E$ is the total energy of +the system. +\partmarks{8} + +Given the inequality +\ifbigfont + \begin{multline*} + \left(\sum_{i=1}^N + a_i^2\right) \left(\sum_{i=1}^N b_i^2\right) \\ +\ge \left(\sum_{i=1}^N \mathbf{a}_i\cdot\mathbf{b}_i\right)^2 \\ ++ \left(\sum_{i=1}^N \mathbf{a}_i\times\mathbf{b}_i\right)^2 + \end{multline*} +\else + \begin{equation*} + \left(\sum_{i=1}^N + a_i^2\right) \left(\sum_{i=1}^N b_i^2\right) \ge \left(\sum_{i=1}^N + \mathbf{a}_i\cdot\mathbf{b}_i\right)^2 + \left(\sum_{i=1}^N + \mathbf{a}_i\times\mathbf{b}_i\right)^2 + \end{equation*} +\fi +for arbitrary vectors $\mathbf{a}_i$, $\mathbf{b}_i$, +$i=1,\ldots,N$, deduce the following relationship for the $N$-body +system +\begin{equation*} +\frac{1}{4}\dot{I}^2+J^2\le 2IE_k, +\end{equation*} +where $\mathbf{J}$ is the total angular momentum of the system. +\partmarks{8} + +Assuming the system is isolated, use the virial theorem to deduce +further the generalised Sundman inequality +\begin{equation*} +\frac{\dot{\sigma}}{\dot{\rho}}\ge 0, +\end{equation*} +in which $\rho^2=I$ and +$\displaystyle\sigma=\rho\dot{\rho}^2+\frac{J^2}{\rho}-2\rho E $. +\partmarks{8} + +Why does this inequality preclude the possibility of an +$N$-fold collision for a system with finite angular momentum? +\partmarks{6} + +\end{question} +\end{document} diff --git a/Master/texmf-dist/doc/latex/exam-n/sample/excos1.tex b/Master/texmf-dist/doc/latex/exam-n/sample/excos1.tex new file mode 100644 index 00000000000..34fa6966ec4 --- /dev/null +++ b/Master/texmf-dist/doc/latex/exam-n/sample/excos1.tex @@ -0,0 +1,46 @@ +\documentclass[compose]{exam-n} + +\multiplechoiceanswers{4} + +\begin{document} + +\begin{mcq} +In 1908, where was there an airburst `impact'? +\answer Tunguska +\item Arizona +\item Off the Mexican coast +\item Egypt +\begin{solution} +The evidence for this is a dirty big hole in the ground in Siberia. +\end{solution} +\end{mcq} + +\begin{mcq} +The fossil record suggests that mass extinction events occur once every how many years? +\item 2.6 Billion Years +\item 260 Million Years +\item 26 Million Years % not marked as correct +\item 26 Thousand Years +\end{mcq} + +\begin{mcq} +The habitable zone of our Solar system extends over what distances from the Sun? +\answer 0.6--1.5 AU +\item 6--15 AU +\item 60--150 AU +\item 600--1500 AU +% excess item, and a second \answer... +\answer From the little bear's bed all the way through to daddy bear's + bed. This is known as the `Goldilocks zone'. +\end{mcq} + +\begin{mcq} +If the temperature of the Sun were to increase by 10\%, how would the +position of the solar habitable zone change? +\item It would move closer to the Sun. +\answer It would move further from the Sun. +\item It would move to Stornoway. +%\item There would no longer be a habitable zone. +\end{mcq} + +\end{document} diff --git a/Master/texmf-dist/doc/latex/exam-n/sample/numerical1-solution.pdf b/Master/texmf-dist/doc/latex/exam-n/sample/numerical1-solution.pdf Binary files differnew file mode 100644 index 00000000000..5da7ebd7f71 --- /dev/null +++ b/Master/texmf-dist/doc/latex/exam-n/sample/numerical1-solution.pdf diff --git a/Master/texmf-dist/doc/latex/exam-n/sample/numerical1-solution.tex b/Master/texmf-dist/doc/latex/exam-n/sample/numerical1-solution.tex new file mode 100644 index 00000000000..3c9b175984b --- /dev/null +++ b/Master/texmf-dist/doc/latex/exam-n/sample/numerical1-solution.tex @@ -0,0 +1,18 @@ +\documentclass{article} +\usepackage{color} +\begin{document} +\color{red} +\noindent This page and the following two should appear on separate pages (as +opposed to superimposed on each other), and disappear when the +noshowsolutions option is present. + +\fontsize{72}{72}\selectfont +\centering +Numerical 1 solution, page one +\newpage +\null\vskip\baselineskip +Numerical 1 solution, page two +\newpage +\null\vskip2\baselineskip +Numerical 1 solution, page three +\end{document} diff --git a/Master/texmf-dist/doc/latex/exam-n/sample/numerical1.tex b/Master/texmf-dist/doc/latex/exam-n/sample/numerical1.tex new file mode 100644 index 00000000000..98157ebf683 --- /dev/null +++ b/Master/texmf-dist/doc/latex/exam-n/sample/numerical1.tex @@ -0,0 +1,58 @@ +\documentclass[compose]{exam-n} +\begin{document} + +\begin{question}{20} \author{Graham Woan} + +\part The recently-launched \emph{Swift} Gamma Ray Burst telescope is +expected to detect about 200 bursts of gamma rays during its +2-year lifespan. Explain why the Poisson distribution, +\begin{equation*} +P(n|\lambda)=\exp(-\lambda)\lambda^n /n! +\end{equation*} +is appropriate to describe the probability of detecting $n$ +bursts, and carefully explain the significance of the parameter +$\lambda$. Table~\ref{t:excess} has absolutely nothing to do with +this question, and its presence here is proof positive of the existence of +aliens who wish to do us typographical harm.\partmarks{4} + +% This table is here to check that the table in the _solution_ to the +% previous question doesn't cause the table counter to be +% incremented. This should be Table 2, whether or not solutions are +% being shown. +\begin{table} +\begin{centering} +\begin{tabular}{r|l} +\hline +left&right\\ +\hline +\end{tabular} +\caption{\label{t:excess}This is a table} +\end{centering} +\end{table} + +Given the above, estimate the probability that \emph{Swift} will +detect more than three bursts on any particular calendar day. +% Fill up the line, so that we check that (default) \partmarks doesn't +% include \@partmarksspace, since (default) \partmarks is now fully in +% the margin. +Blah. Blah. Blaah. Fill the line.\partmarks{6} + +\part Explain how Bayesian inference uses the observed number of +bursts to infer the true burst rate at the sensitivity limit of +\emph{Swift}, and explain the significance of the posterior +probability distribution for $\lambda$. \partmarks{5} +\begin{solution} +\includepdf[pages=-]{numerical1-solution.pdf} +\end{solution} + +Assuming that the posterior, $p$, for $\lambda$ can be +approximated as a gaussian, show that, quite generally, the +uncertainty in $\lambda$ inferred from \emph{Swift} will be +\begin{equation*} +\sigma \simeq \left( -\frac{\partial^2\ln p}{\partial +\lambda^2}\Big|_{\lambda_0} \right)^{-1/2}, +\end{equation*} +where $\lambda_0$ is the most probable value of $\lambda$. +\partmarks{5} +\end{question} +\end{document} diff --git a/Master/texmf-dist/doc/latex/exam-n/sample/numerical2.tex b/Master/texmf-dist/doc/latex/exam-n/sample/numerical2.tex new file mode 100644 index 00000000000..b47bed21815 --- /dev/null +++ b/Master/texmf-dist/doc/latex/exam-n/sample/numerical2.tex @@ -0,0 +1,41 @@ +\documentclass[compose]{exam-n} +\begin{document} + +\begin{question}{30} \comment{by Graham Woan} +Two variables, $A$ and $B$, have a joint Gaussian probability +distribution function (pdf) with a negative correlation +coefficient. Sketch the form of this function as a contour plot +in the $AB$ plane, and use it to distinguish between the most +probable joint values of $(A,B)$ and the most probable value of +$A$ given (a different) $B$. \partmarks{5} + +Explain what is meant by \emph{marginalisation} in Bayesian +inference and how it can be interpreted in terms the above plot. +\partmarks{5} + +Doppler observations of stars with extrasolar planets give us data +on $m\sin i$ of the planet, where $m$ is the planet's mass and +$i$ the angle between the normal to the planetary orbit and the +line of sight to Earth (i.e. the orbital inclination), which can +take a value between 0 and $\pi/2$ . + +Assuming that planets can orbit stars in any plane, show that the +probability distribution for $i$ is $p(i) = \sin i$. +\partmarks{5} + +A paper reports a value for $m\sin i$ of $x$, subject to a Gaussian +error of variance $\sigma^2$. Assuming the mass has a uniform +prior, show that the posterior probability distribution for the mass +of the planet is +\begin{equation*} +p(m|x)\propto\int_0^1\exp\left[-\frac{\left(x-m\sqrt{1-\mu^2}\right)^2}{2\sigma^2}\right] +\ddd \mu, +\end{equation*} +where $\mu=\cos i$. +\partmarks{9} + +Determine the corresponding expression for the posterior pdf of +$\mu$, and explain how both are normalised. +\partmarks{6} +\end{question} +\end{document} diff --git a/Master/texmf-dist/doc/latex/exam-n/sample/numerical3.tex b/Master/texmf-dist/doc/latex/exam-n/sample/numerical3.tex new file mode 100644 index 00000000000..554de298e9d --- /dev/null +++ b/Master/texmf-dist/doc/latex/exam-n/sample/numerical3.tex @@ -0,0 +1,42 @@ +\documentclass[compose]{exam-n} +\begin{document} + +\begin{question}{30} \comment{by Graham Woan} +Distinguish between frequentist and Bayesian definitions of +probability, and explain carefully how parameter estimation is +performed in each regime.\partmarks{10} + +A square ccd with $M\times M$ pixels takes a dark frame for +calibration purposes, registering a small number of electrons in +each pixel from thermal noise. The probability of there being $n_i$ +electrons in the $i$th pixel follows a Poisson distribution, i.e. +\begin{equation*} + P(n_i|\lambda) = \exp(-\lambda)\lambda^{n_i}/n_i!, +\end{equation*} +where $\lambda$ is the same constant for all pixels. Show that the +expectation value of is $\langle n_i \rangle = \lambda$. +\partmarks{5} [You may assume the relation $\sum_0^\infty \frac{x^n}{n!}=\exp(x)$.] + +Show similarly that +\begin{equation*} + \langle n_i(n_i-1) \rangle = \lambda^2. +\end{equation*} +and hence, or otherwise, that the variance of $n_i$ is also +$\lambda$. +\partmarks{5} + +The pixels values are summed in columns. Show that these sums, +$S_j$, will be drawn from a parent probability distribution that is +approximately +\begin{equation*} p(S_j|\lambda)=\frac{1}{\sqrt{2\pi +M\lambda}}\exp\left[-\frac{(S_j-M\lambda)^2}{2M\lambda}\right], +\end{equation*} +clearly stating any theorems you use. +\partmarks{5} + +Given the set of $M$ values $\{S_j\}$, and interpreting the above +as a Bayesian likelihood, express the posterior probability for +$\lambda$, justifying any assumptions you make. +\partmarks{5} +\end{question} +\end{document} diff --git a/Master/texmf-dist/doc/latex/exam-n/sample/sample_exam.pdf b/Master/texmf-dist/doc/latex/exam-n/sample/sample_exam.pdf Binary files differnew file mode 100644 index 00000000000..89072959f30 --- /dev/null +++ b/Master/texmf-dist/doc/latex/exam-n/sample/sample_exam.pdf diff --git a/Master/texmf-dist/doc/latex/exam-n/sample/sample_exam.tex b/Master/texmf-dist/doc/latex/exam-n/sample/sample_exam.tex new file mode 100644 index 00000000000..b8a3eb4a3d2 --- /dev/null +++ b/Master/texmf-dist/doc/latex/exam-n/sample/sample_exam.tex @@ -0,0 +1,64 @@ +\documentclass{exam-n} +%\documentclass[bigfont,sansserif]{exam-n} +%\documentclass[mtpro2,showsolutions]{exam-n} +%\documentclass[mtpro2,showsolutions,sansserif]{exam-n} +%\documentclass[draft,showsolutions,sansserif]{exam-n} + +% Note: this sample question dates from an earlier phase, in which +% `paper n' would contain a bundle of questions from various courses. +% That's why the \coursetitle doesn't match the broad range of +% subjects being 'examined'. +% +% Several of the questions are gibberish. +% +% This collection of questions has in part turned into an informal +% regression test for the class file. + +\examdate{Wednesday, 23 May 2007} +\examtime{9:30 -- 10:45 (1 course)\\9:30 -- 12:00 (2 courses)} + +\exambanner{Examination for the degrees of \BSc\, + \MSci\ and \MA\ on the honours standard } + +\degreedescriptions{Physics 3\\Chemical Physics 3\\Physics with + Astrophysics 3\\Theoretical Physics 3M\\Joint Physics 3} +\schoolcoursecode{P304D and P304H} +\universitycoursecode{PHYS3031 and PHYS4025} +\coursetitle{Quantum Mechanics} + +\usepackage{pdfpages} + +\rubric{Candidates should answer Questions 1 and 2 (10~marks each), + \linebreak\textbf{and either} Question 3 \textbf{or} Question 4 (30~marks). + \linebreak The content of this sample exam derives from real + questions, but the result is in many cases test gibberish.} +%\norubric + +\numquestions{13} + +\usepackage{graphicx} % for graphic in dynamical1 + +\begin{document} +\maketitle + +\section{I} +\includequestion{cosmo1} +\includequestion{numerical1} +\includequestion{dynamical1} + +\section{II} +\includequestion{cosmo2} +\subsection{Cosmology question number 3} +\includequestion{cosmo3} +\includequestion{excos1} % four multiple-choice questions + +\section{III} +% Override the question number, to test/demo this +\includequestion[99]{numerical2} +\includequestion{numerical3} + +\section{IV} +\includequestion{dynamical2} +\includequestion{dynamical3} + +\end{document} diff --git a/Master/texmf-dist/doc/latex/exam-n/sample/sample_exam_solution.pdf b/Master/texmf-dist/doc/latex/exam-n/sample/sample_exam_solution.pdf Binary files differnew file mode 100644 index 00000000000..327cdf20faa --- /dev/null +++ b/Master/texmf-dist/doc/latex/exam-n/sample/sample_exam_solution.pdf diff --git a/Master/texmf-dist/doc/latex/exam-n/sample/sample_exam_solution.tex b/Master/texmf-dist/doc/latex/exam-n/sample/sample_exam_solution.tex new file mode 100644 index 00000000000..098c453572b --- /dev/null +++ b/Master/texmf-dist/doc/latex/exam-n/sample/sample_exam_solution.tex @@ -0,0 +1,64 @@ +\documentclass[showsolutions]{exam-n} +%\documentclass[bigfont,sansserif]{exam-n} +%\documentclass[mtpro2,showsolutions]{exam-n} +%\documentclass[mtpro2,showsolutions,sansserif]{exam-n} +%\documentclass[draft,showsolutions,sansserif]{exam-n} + +% Note: this sample question dates from an earlier phase, in which +% `paper n' would contain a bundle of questions from various courses. +% That's why the \coursetitle doesn't match the broad range of +% subjects being 'examined'. +% +% Several of the questions are gibberish. +% +% This collection of questions has in part turned into an informal +% regression test for the class file. + +\examdate{Wednesday, 23 May 2007} +\examtime{9:30 -- 10:45 (1 course)\\9:30 -- 12:00 (2 courses)} + +\exambanner{Examination for the degrees of \BSc\, + \MSci\ and \MA\ on the honours standard } + +\degreedescriptions{Physics 3\\Chemical Physics 3\\Physics with + Astrophysics 3\\Theoretical Physics 3M\\Joint Physics 3} +\schoolcoursecode{P304D and P304H} +\universitycoursecode{PHYS3031 and PHYS4025} +\coursetitle{Quantum Mechanics} + +\usepackage{pdfpages} + +\rubric{Candidates should answer Questions 1 and 2 (10~marks each), + \linebreak\textbf{and either} Question 3 \textbf{or} Question 4 (30~marks). + \linebreak The content of this sample exam derives from real + questions, but the result is in many cases test gibberish.} +%\norubric + +\numquestions{13} + +\usepackage{graphicx} % for graphic in dynamical1 + +\begin{document} +\maketitle + +\section{I} +\includequestion{cosmo1} +\includequestion{numerical1} +\includequestion{dynamical1} + +\section{II} +\includequestion{cosmo2} +\subsection{Cosmology question number 3} +\includequestion{cosmo3} +\includequestion{excos1} % four multiple-choice questions + +\section{III} +% Override the question number, to test/demo this +\includequestion[99]{numerical2} +\includequestion{numerical3} + +\section{IV} +\includequestion{dynamical2} +\includequestion{dynamical3} + +\end{document} diff --git a/Master/texmf-dist/doc/latex/exam-n/sample/sample_mcq.tex b/Master/texmf-dist/doc/latex/exam-n/sample/sample_mcq.tex new file mode 100644 index 00000000000..268417ff7d0 --- /dev/null +++ b/Master/texmf-dist/doc/latex/exam-n/sample/sample_mcq.tex @@ -0,0 +1,87 @@ +\documentclass[draft]{exam-n} + +\examdate{Today} +\examtime{Now! -- \emph{you're late!}} + +\exambanner{Multiple-choice exam} + +\degreedescriptions{Various degrees} +\schoolcoursecode{MCQ123} +\universitycoursecode{AssDes101} +\coursetitle{Assessment Design Strategies} + +\rubric{Candidates should choose answers randomly where they feel + unable to speculate wildly.} + +\multiplechoiceanswers{4} + +\begin{document} +\maketitle + +\begin{mcq} +Which is the first answer in this list? +\answer Answer one +\item Answer two +\item Answer three +\item Answer four +\end{mcq} + +\begin{mcq} +How many answers are in this list? +\item one +\item two +\item three +\answer four +\end{mcq} + +\begin{mcq} +Which is the incorrect answer in this list? +\item Correct answer +\item Another correct answer +\answer The wrong answer +\item A final correct answer +\end{mcq} + +\begin{mcq} +What is the nature of existence? +\item All good things come to those who wait +\item The universe makes sense +\answer Error is inevitable +\item Om +\end{mcq} + +\begin{mcq} +How many roads must a man walk down? +\item Three +\answer Oooh, I wouldn't walk down there -- it's terribly windy. +\item Can I introduce you to the notion of the tuning fork? +\item Third left, then second right, and the nature of existence is + available from the leprechaun in the little lane just after the bend + in the road. +\end{mcq} + +\begin{mcq} +How many questions are there in this test? +\item Six +\answer Eight +\item Ten +\item Twelve +\end{mcq} + +\begin{mcq} +Is this a reasonable question? +\item no +\item yes +\item is this an answer? +\answer if you say so +\end{mcq} + +\begin{mcq} +What is the most important thing about examinations? +\item They are an accurate measure of understanding +\item They are repeatable and reliable +\answer They are nicely formatted +\item They are stress-free occasions which everyone enjoys. +\end{mcq} + +\end{document} diff --git a/Master/texmf-dist/doc/latex/exam-n/sample/spiral.eps b/Master/texmf-dist/doc/latex/exam-n/sample/spiral.eps new file mode 100644 index 00000000000..fcf91bb97d1 --- /dev/null +++ b/Master/texmf-dist/doc/latex/exam-n/sample/spiral.eps @@ -0,0 +1,22 @@ +%!PS-Adobe-3.0 EPSF-3.0 +% Shrinking squares, cgl Feb 97 +% Adapted from http://www.ntg.nl/maps/pdf/19_12.pdf +% this version... +%%Creator: norman@astro.gla.ac.uk +%%BoundingBox: 0 0 200 125 +/b 0.6180339887 def % (\sqrt5-1)/2 +/x 200 def +/y x b mul def +/square { + 0 y lineto y y lineto y 0 lineto + y 0 y 180 90 arcn y y translate +} def +0.35 setlinewidth +16 { + 0 0 moveto square -90 rotate + /x y def /y x b mul def +} repeat +stroke + +%showpage + diff --git a/Master/texmf-dist/doc/latex/exam-n/sample/spiral.pdf b/Master/texmf-dist/doc/latex/exam-n/sample/spiral.pdf Binary files differnew file mode 100644 index 00000000000..0119b77d38b --- /dev/null +++ b/Master/texmf-dist/doc/latex/exam-n/sample/spiral.pdf |