summaryrefslogtreecommitdiff
path: root/macros/latex/contrib/webquiz/doc/examples/quiz-page.tex
blob: 9bc0740db1d99003109fb4e5a18e607a52bc6ccb (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
\documentclass[]{webquiz}
\BreadCrumbs{quizindex | title}
\DeclareMathOperator{\cis}{cis}
\newcommand{\R}{\mathbb R}
\newcommand{\C}{\mathbb C}
\usepackage{pst-all}

\title{Quiz 1: Numbers and sets}

\begin{document}

    \begin{question}
    Which of the following are correct ways of writing the set
    \[ A=\{x \in \mathbb{R} \mid -3 < x \leq -1 \text{ or } x\geq 0\} ?\]
    \begin{choice}[columns=2]
    \incorrect $(-3,\infty)$
    \feedback The interval $(-3,\infty)$ includes the real numbers
    between $-1$ and $0$, which do not belong to $A$.

    \incorrect $[-3,\infty)$
    \feedback The interval $[-3,\infty)$ includes $-3$, and
    the real numbers between $-1$ and $0$, which do not belong to $A$.

    \incorrect $[-3,-1]\cap[0,\infty)$
    \feedback The interval $[-3,-1]\cap[0,\infty)$ is the empty set
    $\emptyset$. As $A$ is not empty (for example, $A$ includes
    $-1$), this option cannot be correct.

    \incorrect $(-3,-1)\cup[0,\infty)$
    \feedback The interval $-1$ is not in $(-3,-1)\cup[0,\infty)$,
    but $-1$ is in $A$.

    \correct $(-3,-1]\cup[0,\infty)$

    \end{choice}
    \end{question}

%%%% 2
\begin{question}
What is another way of writing the set
\[B= \{x \in {\mathbb R}\ |\ |x-3|<2 \}\ \rm{?}\]
\begin{choice}[columns=2]
\incorrect $(2,3]$
\feedback For example, $4$ belongs to $B$ but is not in $(2,3]$.

\incorrect $[2,4]$
\feedback For example, $1.5$ belongs to $B$ but is not in $[2,4]$.

\correct $(1,5)$

\feedback $B$ is the set of all points whose distance from 3 on the
number line is less than 2. \\
The solution to $|x-3|<2$ is $1<x<5$.

\incorrect $[1,5]$
\feedback Neither $1$ nor $5$  belong to $B$, but both $1$ and $5$ belong to $[1,5]$.

\incorrect $[2,3)$
\feedback For example, $4$ belongs to $B$ but is not in $[2,3)$.

\end{choice}
\end{question}


%%%%%%%%%% 3
\begin{question}
If $A=\{7,8,9,10\}$ and $B=\{5,6,7,8\}$ then $(A\backslash
B)\cup(B\backslash A)$ is
\begin{choice}[columns=2]
\incorrect $\{5,6,7,8,9,10\}$

\correct $\{5,6,9,10\}$

\feedback $A\backslash B=\{9,10\}$ and $B\backslash A=\{5,6\}$ so
$(A\backslash B)\cup(B\backslash A)=\{5,6,9,10\}$.

\incorrect $\emptyset$, the empty set.

\incorrect $\{7,8\}$

\incorrect None of the above.

\end{choice}
\end{question}

%%%%%%%%%%%%%% 4
\begin{question}
The set \(\{0,1,\pm\sqrt{-1},\pi,12\}\) is a subset of
\begin{choice}[columns=2]
\incorrect $\mathbb N$
\feedback The number \(\pi\) is not a natural number.

\incorrect $\mathbb Z$
\feedback The number \(\pi\) is not an integer.

\incorrect $\mathbb Q$
\feedback The number \(\pi\) is not a rational number.

\incorrect $\mathbb R$
\feedback \(\sqrt{-1}\) is not real.

\correct $\mathbb C$
\feedback Since \(\pm\sqrt{-1}\)  denotes the two imaginary numbers  $i$ and $-i$, the given set cannot be
in any of the sets $\mathbb{N,Z,Q}\ \rm{or}\ \mathbb{R}$.  \\
Hence the
right answer must be ${\mathbb C}$ which contains all imaginary
numbers.

\end{choice}
\end{question}

%%%%%%%%%% 5
\begin{question}
Which of the following alternatives is the best feedback to `Solve
$x^{2}-3x+4=0$ over $\mathbb{C}$'.
\begin{choice}[columns=2]
\incorrect There are no real solutions.
\feedback As the question asks us to solve the equation over $\mathbb C$ (that is, to find all solutions belonging to the set of complex numbers), this is not the best feedback.

\incorrect $x=1,4$

\incorrect \(x=\dfrac{3\pm\sqrt{7}}{2}\)

\correct  \(x=\dfrac{3\pm i\sqrt{7}}{2}\)

\feedback Using the quadratic formula,
\(x =\dfrac{3\pm\sqrt{9-16}}{2} = \dfrac{3\pm\sqrt{-7}}{2}\).


\incorrect None of the above is correct.
\end{choice}
\end{question}

%%%%%%% 6
\begin{question}
If $z=9+3i$ and $w=2-i$ then $z+w$ equals
\begin{choice}[columns=2]
\incorrect $9-i$

\correct $11+2i$
\feedback $z+w=(9+3i)+(2-i)=(9+2)+(3-1)i=11+2i$.

\incorrect $6+3i$

\incorrect $8$

\incorrect None of the above

\end{choice}
\end{question}

%%%%%%% 7

\begin{question}
If $w=2-i$ then $\overline{w}$ equals
\begin{choice}[columns=2]
\incorrect $2-i$

\incorrect $2$

\correct $2+i$

\feedback
$\overline{w}=\overline{2-i}=2+i$.

\incorrect $-2+i$

\incorrect None of the above
\end{choice}
\end{question}

%%%%%%%%% 8
\begin{question}
If $p=9+3i$ and $q=2-i$ then $p\overline{q}$ equals
\begin{choice}[columns=2]

\correct $15+15i$

\feedback
$p\overline{q}=(9+3i)\overline{(2-i)}$ \\
$=(9+3i)(2+i)= (18-3)+(6+9)i$\\
$=15+15i$.

\incorrect $21+15i$

\incorrect $18+3i$

\incorrect $1-i$

\incorrect None of the above

\end{choice}
\end{question}


%%%%%%%%% 9
\begin{question}
If $z=9+3i$ and $w=2-i$ then $\dfrac{z}{w}$ equals
\begin{choice}[columns=2]
\incorrect $15+15i$

\incorrect $6+3i$

\incorrect $12+15i$

\incorrect $3-3i$

\correct None of the above

\feedback
\(\dfrac{z}{w}=\dfrac{9+3i}{2-i}
            =\dfrac{9+3i}{2-i}\times \dfrac{2+i}{2+i}
            =\dfrac{15+15i}{5}=3+3i\).
\end{choice}
\end{question}

%%%%%%%%%% 10
\begin{question}
The shaded region in the graph
\begin{center}\begin{pspicture}(-3,-1.5)(3,4)
\pscircle[linewidth=2pt,linestyle=dashed,fillcolor=blue,fillstyle=solid](1,1){2}
\psaxes[linecolor=red,linewidth=1pt,labels=none]{->}(0,0)(-1.5,-1.5)(3.5,3.5)
\rput(3.75,0){$x$}
\rput(0,3.85){$iy$}
\rput(3,-0.4){3}
\rput(-0.4,3){3$i$}
\psdots(1,1)
\end{pspicture}
\end{center}
corresponds to which set of complex numbers?
\begin{choice}[columns=2]
\correct\(\{z \in \C : |z-(i+1)|<2\}\)


\incorrect \(\{z \in \C : |z|-|1+i|<2\}\)
\feedback This set corresponds to the interior of a circle, centre the origin, radius $2+\sqrt 2$.


\incorrect \(\{z \in \C : \text{Re}(z+(i+1))<2 \}\)
\feedback This set corresponds to the open half plane containing all complex numbers $z=x+iy$ with  $x<1$.

\incorrect \(\{z \in \C : |z-2|<|i+1-2|\}\)
\feedback This set corresponds to the interior of a circle, centre $2$, radius $\sqrt 2$.

\incorrect None of the above.

\end{choice}
\end{question}

\end{document}