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<quiz>
  <question type="stack">
    <name>
      <text>23.2.6.1 fs ex1 v2</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[<p class="HELM_exercise"><span style="font-size: 14px;">Exercise</span><br>
</p>Write out the terms of the Fourier series for the periodic signal given by 
\[ f(t) = \begin{cases} 1 \hspace{1cm} 0 &lt; t &lt; \dfrac{\pi}{2} \\ 0 \hspace{1cm} \dfrac{\pi}{2} &lt; t &lt; 2\pi \end{cases} \] \[ f(t + 2\pi) = f(t)\] by filling in the boxes below. You should give the constant term \(\dfrac{a_0}{2}\) and each term corresponding to \( n = 1\) to \( n = 6\). When you enter an answer into a box, it will appear in the series below, which takes the form \[ \begin{align} f(t) = \dfrac{a_0}{2} &amp;+ a_1 \cos c_1 t + a_2 \cos c_2 t + a_3 \cos c_3 t + a_4 \cos c_4 t + a_5 \cos c_5 t + a_6 \cos c_6 t \\ &amp;+ b_1 \sin c_1 t + b_2 \sin c_2 t + b_3 \sin c_3 t + b_4 \sin c_4 t + b_5 \sin c_5 t + b_6 \sin c_6 t \end{align} \]<p><br></p><p>
\(\dfrac{a_0}{2} = \) [[input:ans1]][[feedback:prt1]] 
\(a_1 = \)&nbsp;[[input:ans2]][[feedback:prt2]]&nbsp; \(c_1 = \)&nbsp;[[input:ans3]][[feedback:prt3]] 
\(a_2 = \)&nbsp;[[input:ans4]][[feedback:prt4]]&nbsp;\(c_2 =\)&nbsp;[[input:ans5]] [[feedback:prt5]]
\( a_3 =\)&nbsp;[[input:ans6]][[feedback:prt6]]&nbsp; \(c_3 = \)&nbsp;[[input:ans7]][[feedback:prt7]] 
\(a_4 = \)&nbsp;[[input:ans8]][[feedback:prt8]]&nbsp; \(c_4 = \)&nbsp;[[input:ans9]][[feedback:prt9]] 
\(a_5 = \)&nbsp;[[input:ans10]][[feedback:prt10]]&nbsp; \(c_5 =\)&nbsp;[[input:ans11]][[feedback:prt11]] 
\(a_6 =\)&nbsp;[[input:ans12]][[feedback:prt12]] &nbsp;\(c_6 =\)&nbsp;[[input:ans13]][[feedback:prt13]] 
\(b_1 =\)&nbsp;[[input:ans14]][[feedback:prt14]]&nbsp;\(s_1 =\)&nbsp;[[input:ans15]][[feedback:prt15]] 
\(b_2 =\)&nbsp; [[input:ans16]][[feedback:prt16]]&nbsp; \(s_2 =\)&nbsp;[[input:ans17]][[feedback:prt17]]
\( b_3 =\)&nbsp;[[input:ans18]][[feedback:prt18]]&nbsp;\(s_3 =\)&nbsp;[[input:ans19]][[feedback:prt19]]
\(b_4 =\)&nbsp;[[input:ans20]][[feedback:prt20]]&nbsp; \(s_4 =\)&nbsp;[[input:ans21]][[feedback:prt21]] 
\(b_5 =\)&nbsp;[[input:ans22]][[feedback:prt22]]&nbsp; \(s_5 =\)&nbsp;[[input:ans23]][[feedback:prt23]] 
\(b_6 =\)&nbsp;[[input:ans24]][[feedback:prt24]]&nbsp; \(s_6 =\)&nbsp;[[input:ans25]][[feedback:prt25]]
</p><p>
\(f(t) =\) [[validation:ans1]] +&nbsp; [[validation:ans2]]\( \cos (&nbsp; \)&nbsp; [[validation:ans3]] \( t&nbsp; )\) + [[validation:ans4]]\( \cos ( \)&nbsp; [[validation:ans5]] \( t)&nbsp; \)&nbsp;+&nbsp; [[validation:ans6]]\( \cos ( \) [[validation:ans7]]\( t ) \) + [[validation:ans8]] \( \cos ( \)&nbsp; [[validation:ans9]] \( t ) \) +&nbsp; [[validation:ans10]] \( \cos ( \)&nbsp; [[validation:ans11]] \( t ) \) + [[validation:ans12]] \( \cos ( \)&nbsp; [[validation:ans13]] \( t ) \) + [[validation:ans14]]\( \sin ( \)&nbsp; [[validation:ans15]]\( t ) \) + [[validation:ans16]]\( \sin ( \)&nbsp; [[validation:ans17]] \( t ) \) + [[validation:ans18]]\( \sin ( \) [[validation:ans19]] \( t ) \) + [[validation:ans20]] \( \sin ( \)&nbsp; [[validation:ans21]] \( t ) \) + [[validation:ans22]] \( \sin ( \)&nbsp; [[validation:ans23]] \( t ) \) +&nbsp; [[validation:ans24]] \( \sin ( \)&nbsp; [[validation:ans25]]\( t ) \)</p><p></p><p></p>]]></text>
    </questiontext>
    <generalfeedback format="html">
      <text><![CDATA[Since \(T = 2\pi\) and since the function has a non-zero average value, the form of the Fourier series is
\[ \dfrac{a_0}{2} + \sum_{n=1}^\infty \left\{ a_n \cos (n t) + b_n \sin (n t) \right\} \]
where
\[
\begin{align}
a_n &amp;= \dfrac{1}{\pi} \displaystyle{ \int _{0}^{2\pi} }f(t) \cos (n t) \hspace{0.15cm}dt \hspace{1cm} n = 0, 1, 2, \dots \\
b_n &amp;= \dfrac{1}{\pi} \displaystyle{ \int _{0}^{2\pi} } f(t)  \sin (n t) \hspace{0.15cm}dt  \hspace{1cm} n =  1, 2, \dots
\end{align}
\]
Because the function is non-negative there will be a constant term. 
The constant term will be \(\dfrac{a_0}{2}\) where \( a_0 = \dfrac{1}{\pi} \displaystyle{ \int_{0}^{2\pi}} f(t) \hspace{0.15cm}dt  = \dfrac{1}{\pi}  \displaystyle{ \int_{0}^{\frac{\pi}{2} } } dt = \dfrac{1}{2} \).
<p></p>
For \(n = 1, 2, 3, \dots , \) we have
\[
\begin{align}
a_n &amp;=  \dfrac{1}{\pi} \displaystyle{ \int_{0}^{\frac{\pi}{2}}} \cos (nt) \hspace{0.15cm} dt = \dfrac{1}{\pi} \left[ \dfrac{1}{n} \sin (nt) \right] _0^{\frac{\pi}{2}} \\
&amp;= \dfrac{1}{n\pi} \sin \left( \dfrac{n\pi}{2} \right) \\
&amp;= 
\begin{cases}
\dfrac{1}{n\pi}(-1)^{n+1} \hspace{1cm} n = 1,3,5,7, \dots \\
0 \hspace{2cm} n = 2,4,6,8, \dots
\end{cases}
\end{align}
\]
Similarly for \(b_n\) we have
\[
b_n = 
\begin{cases}
\dfrac{1}{n\pi} \hspace{1cm} n = 1,3,5, \dots\\
\dfrac{2}{n\pi} \hspace{1cm} n = 2, 6, 10, \dots\\
0 \hspace{1.2cm} n = 4, 8, 12, \dots
\end{cases}
\]
<p></p>
Thus the series is given by
\[ 
\begin{align} f(t) &amp;= \dfrac{1}{4} + \dfrac{1}{\pi} \cos t - \dfrac{1}{3\pi} \cos 3t + \dfrac{1}{5\pi} \cos 5t + \dots \\
&amp;+ \dfrac{1}{\pi} \sin t + \dfrac{2}{2\pi} \sin 2t + \dfrac{1}{3\pi} \sin 3t + \dfrac{1}{5\pi} \sin 5t + \dfrac{2}{6\pi}\sin 6t + \dots
\end{align}
\]]]></text>
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ta2:1/pi;
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ta19:3;
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ta25:6;</text>
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      <name>prt7</name>
      <value>1.0000000</value>
      <autosimplify>1</autosimplify>
      <feedbackstyle>1</feedbackstyle>
      <feedbackvariables>
        <text/>
      </feedbackvariables>
      <node>
        <name>0</name>
        <description/>
        <answertest>AlgEquiv</answertest>
        <sans>ans7</sans>
        <tans>ta7</tans>
        <testoptions/>
        <quiet>0</quiet>
        <truescoremode>=</truescoremode>
        <truescore>1.0000000</truescore>
        <truepenalty/>
        <truenextnode>-1</truenextnode>
        <trueanswernote>prt7-1-T</trueanswernote>
        <truefeedback format="html">
          <text/>
        </truefeedback>
        <falsescoremode>=</falsescoremode>
        <falsescore>0.0000000</falsescore>
        <falsepenalty/>
        <falsenextnode>-1</falsenextnode>
        <falseanswernote>prt7-1-F</falseanswernote>
        <falsefeedback format="html">
          <text/>
        </falsefeedback>
      </node>
    </prt>
    <prt>
      <name>prt8</name>
      <value>1.0000000</value>
      <autosimplify>1</autosimplify>
      <feedbackstyle>1</feedbackstyle>
      <feedbackvariables>
        <text/>
      </feedbackvariables>
      <node>
        <name>0</name>
        <description/>
        <answertest>AlgEquiv</answertest>
        <sans>ans8</sans>
        <tans>ta8</tans>
        <testoptions/>
        <quiet>0</quiet>
        <truescoremode>=</truescoremode>
        <truescore>1.0000000</truescore>
        <truepenalty/>
        <truenextnode>-1</truenextnode>
        <trueanswernote>prt8-1-T</trueanswernote>
        <truefeedback format="html">
          <text/>
        </truefeedback>
        <falsescoremode>=</falsescoremode>
        <falsescore>0.0000000</falsescore>
        <falsepenalty/>
        <falsenextnode>-1</falsenextnode>
        <falseanswernote>prt8-1-F</falseanswernote>
        <falsefeedback format="html">
          <text/>
        </falsefeedback>
      </node>
    </prt>
    <prt>
      <name>prt9</name>
      <value>1.0000000</value>
      <autosimplify>1</autosimplify>
      <feedbackstyle>1</feedbackstyle>
      <feedbackvariables>
        <text/>
      </feedbackvariables>
      <node>
        <name>0</name>
        <description/>
        <answertest>AlgEquiv</answertest>
        <sans>ans9</sans>
        <tans>ta9</tans>
        <testoptions/>
        <quiet>0</quiet>
        <truescoremode>=</truescoremode>
        <truescore>1.0000000</truescore>
        <truepenalty/>
        <truenextnode>-1</truenextnode>
        <trueanswernote>prt9-1-T</trueanswernote>
        <truefeedback format="html">
          <text/>
        </truefeedback>
        <falsescoremode>=</falsescoremode>
        <falsescore>0.0000000</falsescore>
        <falsepenalty/>
        <falsenextnode>-1</falsenextnode>
        <falseanswernote>prt9-1-F</falseanswernote>
        <falsefeedback format="html">
          <text/>
        </falsefeedback>
      </node>
    </prt>
    <qtest>
      <testcase>1</testcase>
      <description/>
      <testinput>
        <name>ans1</name>
        <value>ta1</value>
      </testinput>
      <testinput>
        <name>ans10</name>
        <value>ta10</value>
      </testinput>
      <testinput>
        <name>ans11</name>
        <value>ta11</value>
      </testinput>
      <testinput>
        <name>ans12</name>
        <value>ta12</value>
      </testinput>
      <testinput>
        <name>ans13</name>
        <value>ta13</value>
      </testinput>
      <testinput>
        <name>ans14</name>
        <value>ta14</value>
      </testinput>
      <testinput>
        <name>ans15</name>
        <value>ta15</value>
      </testinput>
      <testinput>
        <name>ans16</name>
        <value>ta16</value>
      </testinput>
      <testinput>
        <name>ans17</name>
        <value>ta17</value>
      </testinput>
      <testinput>
        <name>ans18</name>
        <value>ta18</value>
      </testinput>
      <testinput>
        <name>ans19</name>
        <value>ta19</value>
      </testinput>
      <testinput>
        <name>ans2</name>
        <value>ta2</value>
      </testinput>
      <testinput>
        <name>ans20</name>
        <value>ta20</value>
      </testinput>
      <testinput>
        <name>ans21</name>
        <value>ta21</value>
      </testinput>
      <testinput>
        <name>ans22</name>
        <value>ta22</value>
      </testinput>
      <testinput>
        <name>ans23</name>
        <value>ta23</value>
      </testinput>
      <testinput>
        <name>ans24</name>
        <value>ta24</value>
      </testinput>
      <testinput>
        <name>ans25</name>
        <value>ta25</value>
      </testinput>
      <testinput>
        <name>ans3</name>
        <value>ta3</value>
      </testinput>
      <testinput>
        <name>ans4</name>
        <value>ta4</value>
      </testinput>
      <testinput>
        <name>ans5</name>
        <value>ta5</value>
      </testinput>
      <testinput>
        <name>ans6</name>
        <value>ta6</value>
      </testinput>
      <testinput>
        <name>ans7</name>
        <value>ta7</value>
      </testinput>
      <testinput>
        <name>ans8</name>
        <value>ta8</value>
      </testinput>
      <testinput>
        <name>ans9</name>
        <value>ta9</value>
      </testinput>
      <expected>
        <name>prt1</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt1-1-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt10</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt10-1-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt11</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt11-1-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt12</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt12-1-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt13</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt13-1-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt14</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt14-1-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt15</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt15-1-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt16</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt16-1-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt17</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt17-1-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt18</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt18-1-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt19</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt19-1-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt2</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt2-1-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt20</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt20-1-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt21</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt21-1-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt22</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt22-1-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt23</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt23-1-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt24</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt24-1-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt25</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt25-1-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt3</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt3-1-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt4</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt4-1-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt5</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt5-1-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt6</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt6-1-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt7</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt7-1-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt8</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt8-1-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt9</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt9-1-T</expectedanswernote>
      </expected>
    </qtest>
    <qtest>
      <testcase>2</testcase>
      <description/>
      <testinput>
        <name>ans1</name>
        <value>ta1+1</value>
      </testinput>
      <testinput>
        <name>ans10</name>
        <value>ta10+1</value>
      </testinput>
      <testinput>
        <name>ans11</name>
        <value>ta11+1</value>
      </testinput>
      <testinput>
        <name>ans12</name>
        <value>ta12+1</value>
      </testinput>
      <testinput>
        <name>ans13</name>
        <value>ta13+1</value>
      </testinput>
      <testinput>
        <name>ans14</name>
        <value>ta14+1</value>
      </testinput>
      <testinput>
        <name>ans15</name>
        <value>ta15+1</value>
      </testinput>
      <testinput>
        <name>ans16</name>
        <value>ta16+1</value>
      </testinput>
      <testinput>
        <name>ans17</name>
        <value>ta17+1</value>
      </testinput>
      <testinput>
        <name>ans18</name>
        <value>ta18+1</value>
      </testinput>
      <testinput>
        <name>ans19</name>
        <value>ta19+1</value>
      </testinput>
      <testinput>
        <name>ans2</name>
        <value>ta2+1</value>
      </testinput>
      <testinput>
        <name>ans20</name>
        <value>ta20+1</value>
      </testinput>
      <testinput>
        <name>ans21</name>
        <value>ta21+1</value>
      </testinput>
      <testinput>
        <name>ans22</name>
        <value>ta22+1</value>
      </testinput>
      <testinput>
        <name>ans23</name>
        <value>ta23+1</value>
      </testinput>
      <testinput>
        <name>ans24</name>
        <value>ta24+1</value>
      </testinput>
      <testinput>
        <name>ans25</name>
        <value>ta25+1</value>
      </testinput>
      <testinput>
        <name>ans3</name>
        <value>ta3+1</value>
      </testinput>
      <testinput>
        <name>ans4</name>
        <value>ta4+1</value>
      </testinput>
      <testinput>
        <name>ans5</name>
        <value>ta5+1</value>
      </testinput>
      <testinput>
        <name>ans6</name>
        <value>ta6+1</value>
      </testinput>
      <testinput>
        <name>ans7</name>
        <value>ta7+1</value>
      </testinput>
      <testinput>
        <name>ans8</name>
        <value>ta8+1</value>
      </testinput>
      <testinput>
        <name>ans9</name>
        <value>ta9+1</value>
      </testinput>
      <expected>
        <name>prt1</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt1-1-F</expectedanswernote>
      </expected>
      <expected>
        <name>prt10</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt10-1-F</expectedanswernote>
      </expected>
      <expected>
        <name>prt11</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt11-1-F</expectedanswernote>
      </expected>
      <expected>
        <name>prt12</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt12-1-F</expectedanswernote>
      </expected>
      <expected>
        <name>prt13</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt13-1-F</expectedanswernote>
      </expected>
      <expected>
        <name>prt14</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt14-1-F</expectedanswernote>
      </expected>
      <expected>
        <name>prt15</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt15-1-F</expectedanswernote>
      </expected>
      <expected>
        <name>prt16</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt16-1-F</expectedanswernote>
      </expected>
      <expected>
        <name>prt17</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt17-1-F</expectedanswernote>
      </expected>
      <expected>
        <name>prt18</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt18-1-F</expectedanswernote>
      </expected>
      <expected>
        <name>prt19</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt19-1-F</expectedanswernote>
      </expected>
      <expected>
        <name>prt2</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt2-1-F</expectedanswernote>
      </expected>
      <expected>
        <name>prt20</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt20-1-F</expectedanswernote>
      </expected>
      <expected>
        <name>prt21</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt21-1-F</expectedanswernote>
      </expected>
      <expected>
        <name>prt22</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt22-1-F</expectedanswernote>
      </expected>
      <expected>
        <name>prt23</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt23-1-F</expectedanswernote>
      </expected>
      <expected>
        <name>prt24</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt24-1-F</expectedanswernote>
      </expected>
      <expected>
        <name>prt25</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt25-1-F</expectedanswernote>
      </expected>
      <expected>
        <name>prt3</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt3-1-F</expectedanswernote>
      </expected>
      <expected>
        <name>prt4</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt4-1-F</expectedanswernote>
      </expected>
      <expected>
        <name>prt5</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt5-1-F</expectedanswernote>
      </expected>
      <expected>
        <name>prt6</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt6-1-F</expectedanswernote>
      </expected>
      <expected>
        <name>prt7</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt7-1-F</expectedanswernote>
      </expected>
      <expected>
        <name>prt8</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt8-1-F</expectedanswernote>
      </expected>
      <expected>
        <name>prt9</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt9-1-F</expectedanswernote>
      </expected>
    </qtest>
  </question>
</quiz>
