Mailinglist Archive: opensuse-edu (43 mails)
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Re: LaTeX
- From: Roger Whittaker <roger@xxxxxxxxxx>
- Date: Tue, 4 Apr 2000 13:02:52 +0000 (UTC)
- Message-id: <Pine.LNX.4.10.10004041354540.21467-302000@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx>
Here are the examples.
A standard install will almost certainly have TeX and LaTeX installed.
To check, just type
latex
at the prompt. If it's there you'll get
This is TeX, Version 3.14159 (Web2C 7.3.1)
**
or similar.
When you do latex to a LaTeX source (.tex) file, it makes it into a .dvi
file. The DVI viewer can view that. You can further process it with
dvips to make a postscript file (.ps) and if you want with ps2pdf to make
a .pdf file (acrobat reader format)
Play with the examples I sent. I have better ones, but they're at home (I
don't do much `A'-level maths teaching here at SuSE...)
I hope you dont mind, but I'll forward this to the list also.
On Tue, 4 Apr 2000, Peter Rutherford wrote:
> Dear Roger
>
> I am re-sending the following reply to your message about LaTeX; my first effort was "bounced" back to me!
>
> Re. LaTeX
>
> I would love to see such examples and look forward to receiving them.
>
> I have performed a "standard" installation on my old 486. Will LaTeX be
> there as a matter of routine and, if so, how do I get to it?
>
> Is the DVI viewer in the KDE related to TeX in some way?
>
> Thanks
>
> Peter
>
--
Roger Whittaker
SuSE Linux Ltd
The Kinetic Centre
Theobald Street
Borehamwood
Herts
WD6 4PJ
----------------------
020 8387 1482
----------------------
roger@xxxxxxxxxxxxxxxx
----------------------
\documentclass[10 pt, a4paper, fleqn]{article} % fleqn aligns equations left
\setlength{\textwidth}{14.66 cm} % width of actual text
\setlength{\textheight}{24 cm} % height of actual text
\setlength{\oddsidemargin}{0.46 cm} % left margin minus one inch
% In practice this seems to give equal margins with just about the right text width.
\setlength{\topmargin}{0 cm}
\setlength{\headheight}{0 cm}
\pagestyle{empty} % avoids page numbers
\setlength{\mathindent}{0 cm} % no indent for maths
\setlength{\parindent}{0 cm} % no indent for paragraphs
\setlength{\parskip}{11pt plus 1pt minus 1pt} % gap between paragraphs
\setlength{\marginparwidth}{0.8 cm} % width of margin paragraph
\usepackage{times}
\begin{document}
\section*{S6 PURE MATHEMATICS \hfill 22-10-98} % \hfill gets date right aligned
\bfseries Half-term homework \hfill \underline{Show all working clearly}\\
\normalfont
To be handed in on Monday 2-11-98
\vspace{-0.2 cm}
\newcounter{qno}
% if we don't declare the counter it won't work.
\begin{list}{ \arabic{qno})}{\usecounter {qno} \setlength{\leftmargin}{0 cm}}
% setlength{\leftmargin} gets the question text exactly aligned to the main left margin. Otherwise by default it is indented.
\setlength{\labelwidth}{1 cm}
\setlength{\labelsep}{0.5 cm}
\setlength{\itemsep}{0.2 ex plus 0.2 ex}
\newcommand{\marks}[1]{\marginpar{\hfill {\bfseries \small{#1}}}}
\item The first term of a geometric series is \(2500\) and the fifth term is \(4\). Given that the common ratio is positive, calculate\\
a) the seventh term of the series\\
b) the sum to infinity of the series. \marks{[5]}
\item \[\mbox{i) Find }\sum_{r=1}^{100}\frac{2r}{3}\]
ii) Find the sum to infinity of \( \frac{7}{10}+\frac{7}{100}+\frac{7}{1000}+\ldots\), giving your answer as a fraction.\\ \vspace{-0.4 cm}
\[\mbox{Hence or otherwise, find the value of }\sum_{r=1}^{\infty}\frac{k}{10^r} \mbox{ in terms of } k\] \vspace{-0.8 cm} \marks{[6]} \vspace{0.8 cm}
\item An arithmetic series has first term \(24\) and common difference \(-3\).\\
a) Prove that the sum \(S_n\) of the first \(n\) terms of this series is given by \(2S_n = 3n(17-n)\)\\
b) Given that \(S_n=105\) find the possible values of \(n\). \marks{[8]}
\item The third and sixth terms of a geometric series are \(4\) and \(-\frac{1}{2}\) respectively.\\
a) Find the first term of this series.\\
b) Find, to three significant figures, the sum of the first \(10\) terms of this series. \marks{[7]}
\item \[\mbox{Calculate the values of a) }\sum_{r=1}^{100}r \mbox{ , \quad b) }\sum_{r=1}^{50}2r \] \vspace{-0.2 cm}
\[\mbox{Hence or otherwise, evaluate the sum } \sum_{r=1}^{50}(2r-1)\] \vspace{ -0.8 cm} \marks{[5]} \vspace{0.8 cm}
\item The points \emph{A}, \emph{B} and \emph{C} have coordinates \((-3, 4)\), \((-1, 9)\) and \((7, 8)\) respectively. \\
a) Calculate the gradient of \emph{AB}.\\
b) Find an equation of the line which passes through \emph{C} and is parallel to \emph{AB}. \marks{[4]}
\item a) Show that the triangle \emph{PQR} with vertices \emph{P}\((-1, 7)\), \emph{Q}\((8, 19)\) and \emph{R}\((-8, 31)\) is right-angled.\\
b) Find an equation of the line \emph{PQ}.\\
c) Calculate the area of \(\Delta \)\emph{PQR}. \marks{[6]}
\item The first four terms of an arithmetic progression are \(2\), \(a-b\), \(2a+b+7\) and \(a-3b\) respectively, where \(a\) and \(b\) are constants. Find \(a\) and \(b\) and hence determine the sum of the first \(30\) terms of the progression. \marks{[5]}
\item A line \(\ell\) passes through the point \emph{P}\((3, k)\), where \(k\) is a constant, and is parallel to the line joining the points \emph{A}\((1, 4)\) and \emph{B}\((k, 5)\).\\
a) Find, in terms of \(k\), an equation of \(\ell\).\\
b) Show that \(\ell\) passes through the point whose coordinates are \((k+2, k+1)\).\\
c) Given that the angle between the line \(\ell\) and the positive \(x\) axis is \(45^\circ\), find the possible values of \(k\). \marks{[8]}
\item a) Find, in terms of \(x\) and \(y\), the distance between the point \emph{P}\((x, y)\) and the point \emph{M}\((3, 7)\).\\
b) In the same way, find the distance between \emph{P}\((x, y)\) and the point \emph{N}\((7, 1)\).\\
c) Use your answers to form and simplify an equation for the path of \emph{P} if it moves so that the distances \emph{PM} and \emph{PN} are equal. \marks{[8]}
\item \(\ell_1\) is the line \(2x+y=5\) and \(\ell_2\) is the line \(x+4y=6\).\\
a) Calculate the coordinates of the point of intersection of \(\ell_1\) and \(\ell_2\).\\
b) Calculate in degrees to one decimal place, the angle between \(\ell_1\) and \(\ell_2\).\\
\emph{(A sketch diagram might be useful.)} \marks{[6]}
\end {list}
\end{document}\documentclass[11 pt, a4paper, fleqn]{article} % fleqn aligns equations left
\setlength{\textwidth}{14.66 cm} % width of actual text
\setlength{\textheight}{24 cm} % height of actual text
\setlength{\oddsidemargin}{0.46 cm} % left margin minus one inch
% In practice this seems to give equal margins with just about the right text width.
\setlength{\topmargin}{0 cm}
\setlength{\headheight}{0 cm}
\pagestyle{empty} % avoids page numbers
\setlength{\mathindent}{0 cm} % no indent for maths
\setlength{\parindent}{0 cm} % no indent for paragraphs
\setlength{\parskip}{10pt plus 1pt minus 1pt} % gap between paragraphs
\setlength{\marginparwidth}{0.8 cm} % width of margin paragraph
\usepackage{times} % nice text fonts
\usepackage{multicol}
%\usepackage{amsmath} % easier equation alignment etc
%\usepackage{amsfonts} % to get extra symbols
%\usepackage{graphicx} if you want to include *.eps pictures
% USE \begin{quest} ... \end{quest} for main questions list.
% USE \begin{subq}... \end{subq} for alphabetically listed subquestions.
\begin{document}
\section*{S6 PURE MATHEMATICS \hfill 2-2-99} % \hfill gets date right aligned
\bfseries Trigonometry \hfill \underline{Show all working clearly}
\normalfont
\newcommand{\marks}[1]{\marginpar{\hfill {\bfseries \small{#1}}}} %defining marks command
\newcounter{qno}% if we don't declare the counter it won't work.
\newenvironment{quest} % main question list environment
{
\begin{list}{\arabic{qno})}
{
\usecounter {qno}
\setlength{\leftmargin}{0 cm} % align to left margin
\setlength{\labelwidth}{1 cm}
\setlength{\labelsep}{0.5 cm}
\setlength{\itemsep}{0.5 ex plus 0.2 ex}
}
}
{\end{list}
} % end of definition
\newcounter{qpart} % if we don't declare the counter it won't work.
\newenvironment{subq} % defining subquestion environment
{
\begin{list}{\alph{qpart}\hfill)} % alphabetic counter
{
\usecounter {qpart}
\setlength{\itemindent}{1.65 em}
\setlength{\labelwidth}{0.8 em}
\setlength{\labelsep}{0.8 em}
\setlength{\leftmargin}{0 em}
\setlength{\itemsep}{0.2 ex plus 0.1 ex} %these lengths look right at the moment
}
}
{\end{list}
} % end of definition
\begin{quest}
\vspace{-0.2 cm}
\item In each case express the trigonometrical ratio in terms of an acute angle.\\
(Examples: \(\sin 200^\circ=-\sin 20^\circ\), \quad \(\cos 300^\circ=\cos 60^\circ.\))
\begin{multicols}{4}
\begin{subq}
\item \(\sin 250^\circ\)
\item \(\cos 250^\circ\)
\item \(\tan 250^\circ\)
\item \(\sin 132^\circ\)
\item \(\cos 132^\circ\)
\item \(\tan 132^\circ\)
\item \(\sin (-75^\circ)\)
\item \(\cos (-75^\circ)\)
\item \(\tan (-75^\circ)\)
\item \(\sin 346^\circ\)
\item \(\cos 346^\circ\)
\item \(\tan 346^\circ\)
\end{subq}
\end{multicols}
\marks{[24]}
\item Express the following angles in radians:
\begin{subq}
\begin{multicols}{4}
\item \(300^\circ\)
\item \(60^\circ\)
\item \(120^\circ\)
\item \(315^\circ\)
\item \(225^\circ\)
\item \(30^\circ\)
\item \(600^\circ\)
\item \(15^\circ\)
\item \(330^\circ\)
\item \(720^\circ\)
\item \(135^\circ\)
\item \(270^\circ\)
\end{multicols}
\end{subq}
\marks{[12]}
\item Express the following angles in degrees:
\begin{subq}
\begin{multicols}{4}
\item \(2 \pi\)
\item \(\frac{\pi}{4}\)
\item \(\frac{3 \pi}{4}\)
\item \(\frac{5 \pi}{6}\)
\item \(\frac{7 \pi}{6}\)
\item \(\frac{5 \pi}{3}\)
\item \(\frac{5 \pi}{4}\)
\item \(\frac{3 \pi}{2}\)
\item \(\frac{2 \pi}{3}\)
\item \(\frac{5 \pi}{3}\)
\item \(\frac{2 \pi}{9}\)
\item \(\frac{\pi}{12}\)
\end{multicols}
\end{subq}
\marks{[12]}
\item Find all solutions between \(0^\circ\) and \(360^\circ\) of the following equations:
\begin{multicols}{4}
\begin{subq}
\item \(\sin \theta = \frac{1}{2}\)
\item \(\sin \theta = 0\)
\item \(\cos \theta = -\frac{1}{2}\)
\item \(\tan \theta = \sqrt{3}\)
\item \(\sin 2\theta = -\frac {1}{\sqrt{2}}\)
\item \(\cos 3\theta = 0\)
\item \(\sin^2\theta = \frac{3}{4}\)
\item \(\tan \theta = 1\)
\item \(\tan 2\theta = -\sqrt{3}\)
\item \(\sin \theta = -1\)
\item \(\sin 5\theta = \frac{1}{2}\)
\item \(\cos 5\theta = -1\)
\end{subq}
\end{multicols}
\marks{[24]}
\item Draw neat, clearly labelled sketch graphs of the following for \(0^\circ \leq x \leq 360^\circ\) :
\begin{subq}
\item \(y=\cos 2x\)
\item \(y=5+\sin x\)
\item \(y=\sin(x-45^\circ)\)
\item \(y=1+\cos 3x\)
\end{subq}
\marks{[12]}
\item
\begin{subq}
\item Make a table of values and draw an accurate graph of \(y=\sin^2 x\).
\item Draw a clear labelled sketch of the graph of \(y=1-\cos 2x\).
\item Draw an intellegent conclusion.
\end{subq}
\marks{[12]}
\item Prove that for all angles \(\theta\), \(\sin^2 \theta + \cos^2 \theta = 1\). \marks{[4]}
\item Given that \(\cos \phi = \frac {5}{13}\) and \(\phi\) is acute, find the exact values of \(\sin \phi\) and \(\tan \phi\). \marks{[4]}
\item Given that \(\sin \psi = \frac {2}{7}\) and \(\psi\) is obtuse, find, as surds the values of \(\cos \psi\) and \(\tan \psi\). \marks{[4]}
\end{quest}
\end{document}
A standard install will almost certainly have TeX and LaTeX installed.
To check, just type
latex
at the prompt. If it's there you'll get
This is TeX, Version 3.14159 (Web2C 7.3.1)
**
or similar.
When you do latex to a LaTeX source (.tex) file, it makes it into a .dvi
file. The DVI viewer can view that. You can further process it with
dvips to make a postscript file (.ps) and if you want with ps2pdf to make
a .pdf file (acrobat reader format)
Play with the examples I sent. I have better ones, but they're at home (I
don't do much `A'-level maths teaching here at SuSE...)
I hope you dont mind, but I'll forward this to the list also.
On Tue, 4 Apr 2000, Peter Rutherford wrote:
> Dear Roger
>
> I am re-sending the following reply to your message about LaTeX; my first effort was "bounced" back to me!
>
> Re. LaTeX
>
> I would love to see such examples and look forward to receiving them.
>
> I have performed a "standard" installation on my old 486. Will LaTeX be
> there as a matter of routine and, if so, how do I get to it?
>
> Is the DVI viewer in the KDE related to TeX in some way?
>
> Thanks
>
> Peter
>
--
Roger Whittaker
SuSE Linux Ltd
The Kinetic Centre
Theobald Street
Borehamwood
Herts
WD6 4PJ
----------------------
020 8387 1482
----------------------
roger@xxxxxxxxxxxxxxxx
----------------------
\documentclass[10 pt, a4paper, fleqn]{article} % fleqn aligns equations left
\setlength{\textwidth}{14.66 cm} % width of actual text
\setlength{\textheight}{24 cm} % height of actual text
\setlength{\oddsidemargin}{0.46 cm} % left margin minus one inch
% In practice this seems to give equal margins with just about the right text width.
\setlength{\topmargin}{0 cm}
\setlength{\headheight}{0 cm}
\pagestyle{empty} % avoids page numbers
\setlength{\mathindent}{0 cm} % no indent for maths
\setlength{\parindent}{0 cm} % no indent for paragraphs
\setlength{\parskip}{11pt plus 1pt minus 1pt} % gap between paragraphs
\setlength{\marginparwidth}{0.8 cm} % width of margin paragraph
\usepackage{times}
\begin{document}
\section*{S6 PURE MATHEMATICS \hfill 22-10-98} % \hfill gets date right aligned
\bfseries Half-term homework \hfill \underline{Show all working clearly}\\
\normalfont
To be handed in on Monday 2-11-98
\vspace{-0.2 cm}
\newcounter{qno}
% if we don't declare the counter it won't work.
\begin{list}{ \arabic{qno})}{\usecounter {qno} \setlength{\leftmargin}{0 cm}}
% setlength{\leftmargin} gets the question text exactly aligned to the main left margin. Otherwise by default it is indented.
\setlength{\labelwidth}{1 cm}
\setlength{\labelsep}{0.5 cm}
\setlength{\itemsep}{0.2 ex plus 0.2 ex}
\newcommand{\marks}[1]{\marginpar{\hfill {\bfseries \small{#1}}}}
\item The first term of a geometric series is \(2500\) and the fifth term is \(4\). Given that the common ratio is positive, calculate\\
a) the seventh term of the series\\
b) the sum to infinity of the series. \marks{[5]}
\item \[\mbox{i) Find }\sum_{r=1}^{100}\frac{2r}{3}\]
ii) Find the sum to infinity of \( \frac{7}{10}+\frac{7}{100}+\frac{7}{1000}+\ldots\), giving your answer as a fraction.\\ \vspace{-0.4 cm}
\[\mbox{Hence or otherwise, find the value of }\sum_{r=1}^{\infty}\frac{k}{10^r} \mbox{ in terms of } k\] \vspace{-0.8 cm} \marks{[6]} \vspace{0.8 cm}
\item An arithmetic series has first term \(24\) and common difference \(-3\).\\
a) Prove that the sum \(S_n\) of the first \(n\) terms of this series is given by \(2S_n = 3n(17-n)\)\\
b) Given that \(S_n=105\) find the possible values of \(n\). \marks{[8]}
\item The third and sixth terms of a geometric series are \(4\) and \(-\frac{1}{2}\) respectively.\\
a) Find the first term of this series.\\
b) Find, to three significant figures, the sum of the first \(10\) terms of this series. \marks{[7]}
\item \[\mbox{Calculate the values of a) }\sum_{r=1}^{100}r \mbox{ , \quad b) }\sum_{r=1}^{50}2r \] \vspace{-0.2 cm}
\[\mbox{Hence or otherwise, evaluate the sum } \sum_{r=1}^{50}(2r-1)\] \vspace{ -0.8 cm} \marks{[5]} \vspace{0.8 cm}
\item The points \emph{A}, \emph{B} and \emph{C} have coordinates \((-3, 4)\), \((-1, 9)\) and \((7, 8)\) respectively. \\
a) Calculate the gradient of \emph{AB}.\\
b) Find an equation of the line which passes through \emph{C} and is parallel to \emph{AB}. \marks{[4]}
\item a) Show that the triangle \emph{PQR} with vertices \emph{P}\((-1, 7)\), \emph{Q}\((8, 19)\) and \emph{R}\((-8, 31)\) is right-angled.\\
b) Find an equation of the line \emph{PQ}.\\
c) Calculate the area of \(\Delta \)\emph{PQR}. \marks{[6]}
\item The first four terms of an arithmetic progression are \(2\), \(a-b\), \(2a+b+7\) and \(a-3b\) respectively, where \(a\) and \(b\) are constants. Find \(a\) and \(b\) and hence determine the sum of the first \(30\) terms of the progression. \marks{[5]}
\item A line \(\ell\) passes through the point \emph{P}\((3, k)\), where \(k\) is a constant, and is parallel to the line joining the points \emph{A}\((1, 4)\) and \emph{B}\((k, 5)\).\\
a) Find, in terms of \(k\), an equation of \(\ell\).\\
b) Show that \(\ell\) passes through the point whose coordinates are \((k+2, k+1)\).\\
c) Given that the angle between the line \(\ell\) and the positive \(x\) axis is \(45^\circ\), find the possible values of \(k\). \marks{[8]}
\item a) Find, in terms of \(x\) and \(y\), the distance between the point \emph{P}\((x, y)\) and the point \emph{M}\((3, 7)\).\\
b) In the same way, find the distance between \emph{P}\((x, y)\) and the point \emph{N}\((7, 1)\).\\
c) Use your answers to form and simplify an equation for the path of \emph{P} if it moves so that the distances \emph{PM} and \emph{PN} are equal. \marks{[8]}
\item \(\ell_1\) is the line \(2x+y=5\) and \(\ell_2\) is the line \(x+4y=6\).\\
a) Calculate the coordinates of the point of intersection of \(\ell_1\) and \(\ell_2\).\\
b) Calculate in degrees to one decimal place, the angle between \(\ell_1\) and \(\ell_2\).\\
\emph{(A sketch diagram might be useful.)} \marks{[6]}
\end {list}
\end{document}\documentclass[11 pt, a4paper, fleqn]{article} % fleqn aligns equations left
\setlength{\textwidth}{14.66 cm} % width of actual text
\setlength{\textheight}{24 cm} % height of actual text
\setlength{\oddsidemargin}{0.46 cm} % left margin minus one inch
% In practice this seems to give equal margins with just about the right text width.
\setlength{\topmargin}{0 cm}
\setlength{\headheight}{0 cm}
\pagestyle{empty} % avoids page numbers
\setlength{\mathindent}{0 cm} % no indent for maths
\setlength{\parindent}{0 cm} % no indent for paragraphs
\setlength{\parskip}{10pt plus 1pt minus 1pt} % gap between paragraphs
\setlength{\marginparwidth}{0.8 cm} % width of margin paragraph
\usepackage{times} % nice text fonts
\usepackage{multicol}
%\usepackage{amsmath} % easier equation alignment etc
%\usepackage{amsfonts} % to get extra symbols
%\usepackage{graphicx} if you want to include *.eps pictures
% USE \begin{quest} ... \end{quest} for main questions list.
% USE \begin{subq}... \end{subq} for alphabetically listed subquestions.
\begin{document}
\section*{S6 PURE MATHEMATICS \hfill 2-2-99} % \hfill gets date right aligned
\bfseries Trigonometry \hfill \underline{Show all working clearly}
\normalfont
\newcommand{\marks}[1]{\marginpar{\hfill {\bfseries \small{#1}}}} %defining marks command
\newcounter{qno}% if we don't declare the counter it won't work.
\newenvironment{quest} % main question list environment
{
\begin{list}{\arabic{qno})}
{
\usecounter {qno}
\setlength{\leftmargin}{0 cm} % align to left margin
\setlength{\labelwidth}{1 cm}
\setlength{\labelsep}{0.5 cm}
\setlength{\itemsep}{0.5 ex plus 0.2 ex}
}
}
{\end{list}
} % end of definition
\newcounter{qpart} % if we don't declare the counter it won't work.
\newenvironment{subq} % defining subquestion environment
{
\begin{list}{\alph{qpart}\hfill)} % alphabetic counter
{
\usecounter {qpart}
\setlength{\itemindent}{1.65 em}
\setlength{\labelwidth}{0.8 em}
\setlength{\labelsep}{0.8 em}
\setlength{\leftmargin}{0 em}
\setlength{\itemsep}{0.2 ex plus 0.1 ex} %these lengths look right at the moment
}
}
{\end{list}
} % end of definition
\begin{quest}
\vspace{-0.2 cm}
\item In each case express the trigonometrical ratio in terms of an acute angle.\\
(Examples: \(\sin 200^\circ=-\sin 20^\circ\), \quad \(\cos 300^\circ=\cos 60^\circ.\))
\begin{multicols}{4}
\begin{subq}
\item \(\sin 250^\circ\)
\item \(\cos 250^\circ\)
\item \(\tan 250^\circ\)
\item \(\sin 132^\circ\)
\item \(\cos 132^\circ\)
\item \(\tan 132^\circ\)
\item \(\sin (-75^\circ)\)
\item \(\cos (-75^\circ)\)
\item \(\tan (-75^\circ)\)
\item \(\sin 346^\circ\)
\item \(\cos 346^\circ\)
\item \(\tan 346^\circ\)
\end{subq}
\end{multicols}
\marks{[24]}
\item Express the following angles in radians:
\begin{subq}
\begin{multicols}{4}
\item \(300^\circ\)
\item \(60^\circ\)
\item \(120^\circ\)
\item \(315^\circ\)
\item \(225^\circ\)
\item \(30^\circ\)
\item \(600^\circ\)
\item \(15^\circ\)
\item \(330^\circ\)
\item \(720^\circ\)
\item \(135^\circ\)
\item \(270^\circ\)
\end{multicols}
\end{subq}
\marks{[12]}
\item Express the following angles in degrees:
\begin{subq}
\begin{multicols}{4}
\item \(2 \pi\)
\item \(\frac{\pi}{4}\)
\item \(\frac{3 \pi}{4}\)
\item \(\frac{5 \pi}{6}\)
\item \(\frac{7 \pi}{6}\)
\item \(\frac{5 \pi}{3}\)
\item \(\frac{5 \pi}{4}\)
\item \(\frac{3 \pi}{2}\)
\item \(\frac{2 \pi}{3}\)
\item \(\frac{5 \pi}{3}\)
\item \(\frac{2 \pi}{9}\)
\item \(\frac{\pi}{12}\)
\end{multicols}
\end{subq}
\marks{[12]}
\item Find all solutions between \(0^\circ\) and \(360^\circ\) of the following equations:
\begin{multicols}{4}
\begin{subq}
\item \(\sin \theta = \frac{1}{2}\)
\item \(\sin \theta = 0\)
\item \(\cos \theta = -\frac{1}{2}\)
\item \(\tan \theta = \sqrt{3}\)
\item \(\sin 2\theta = -\frac {1}{\sqrt{2}}\)
\item \(\cos 3\theta = 0\)
\item \(\sin^2\theta = \frac{3}{4}\)
\item \(\tan \theta = 1\)
\item \(\tan 2\theta = -\sqrt{3}\)
\item \(\sin \theta = -1\)
\item \(\sin 5\theta = \frac{1}{2}\)
\item \(\cos 5\theta = -1\)
\end{subq}
\end{multicols}
\marks{[24]}
\item Draw neat, clearly labelled sketch graphs of the following for \(0^\circ \leq x \leq 360^\circ\) :
\begin{subq}
\item \(y=\cos 2x\)
\item \(y=5+\sin x\)
\item \(y=\sin(x-45^\circ)\)
\item \(y=1+\cos 3x\)
\end{subq}
\marks{[12]}
\item
\begin{subq}
\item Make a table of values and draw an accurate graph of \(y=\sin^2 x\).
\item Draw a clear labelled sketch of the graph of \(y=1-\cos 2x\).
\item Draw an intellegent conclusion.
\end{subq}
\marks{[12]}
\item Prove that for all angles \(\theta\), \(\sin^2 \theta + \cos^2 \theta = 1\). \marks{[4]}
\item Given that \(\cos \phi = \frac {5}{13}\) and \(\phi\) is acute, find the exact values of \(\sin \phi\) and \(\tan \phi\). \marks{[4]}
\item Given that \(\sin \psi = \frac {2}{7}\) and \(\psi\) is obtuse, find, as surds the values of \(\cos \psi\) and \(\tan \psi\). \marks{[4]}
\end{quest}
\end{document}
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