<?xml version="1.0" encoding="UTF-8"?>
<quiz>
  <question type="stack">
    <name>
      <text>Cramers_Rule_1</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[<p>Use Cramer's rule to find the y-coordinate of the solution of the system: <br /> <br /> \[{@eqns[1]@}\ =\ {@bmat[1,1]@} , \] <br /> \[ {@eqns[2]@}\ =\ {@bmat[2,1]@} , \] <br /> \[ {@eqns[3]@}\ =\ {@bmat[3,1]@} . \] <br /> <br /> y= [[input:ans1]] [[validation:ans1]]</p>]]></text>
    </questiontext>
    <generalfeedback format="html">
      <text><![CDATA[<p>Cramer's rule states that y = \(\frac{D_{y}}{D}\) where \(D\) is the determinant of the matrix and \(D_{y}\) is the determinant of the matrix with the y column replaced by the numbers on the right hand side of the equations. Therefore for the system <br /> \[{@eqns[1]@}\ =\ {@bmat[1,1]@} , \] \[ {@eqns[2]@}\ =\ {@bmat[2,1]@} , \] \[ {@eqns[3]@}\ =\ {@bmat[3,1]@} , \] <br /> \[ D = det{@Amat@}={@D@}\] <br /> \[D_{y} = det{@Amaty@}={@Dy@}\] <br /> and hence, \[y=\frac{D_{y}}{D}=\frac{@Dy@}{@D@} = {@Dy/D@}\]</p>]]></text>
    </generalfeedback>
    <defaultgrade>1</defaultgrade>
    <penalty>0</penalty>
    <hidden>0</hidden>
    <idnumber/>
    <stackversion>
      <text/>
    </stackversion>
    <questionvariables>
      <text>Au:matrix([rand_with_step(-1,1,2),ev(rand(7)-3),ev(rand(7)-3)],[0,rand_with_step(-1,1,2),ev(rand(7)-3)],[0,0,rand_with_step(-1,1,2)]);
 Bu:matrix([rand_with_step(-1,1,2),ev(rand(7)-3),ev(rand(7)-3)],[0,rand_with_step(-1,1,2),ev(rand(7)-3)],[0,0,rand_with_step(-1,1,2)]);
 Amat:ev(transpose(Au).Bu);
 bmat:matrix([rand(5)+1],[rand(5)+1],[rand(5)+1]);
 xyz:matrix([x],[y],[z]);
 eqns[1]:ev(Amat.xyz)[1,1];
 eqns[2]:ev(Amat.xyz)[2,1];
 eqns[3]:ev(Amat.xyz)[3,1];
 truesolm:ev(invert(Amat) . bmat);
 correctanswer:truesolm[2,1];
 D:determinant(Amat);
 Amaty:copymatrix(Amat);
 Amaty[1,2]:ev(bmat[1,1]);
 Amaty[2,2]:ev(bmat[2,1]);
 Amaty[3,2]:ev(bmat[3,1]);
 Dy:determinant(Amaty);</text>
    </questionvariables>
    <specificfeedback format="html">
      <text><![CDATA[<p>[[feedback:Result]]</p>]]></text>
    </specificfeedback>
    <questionnote format="moodle_auto_format">
      <text>\( {@eqns[1]@}\ =\ {@bmat[1,1]@} ,  {@eqns[2]@}\ =\ {@bmat[2,1]@} ,{@eqns[3]@}\ =\ {@bmat[3,1]@} \) gives y= {@correctanswer@}</text>
    </questionnote>
    <questiondescription format="moodle_auto_format">
      <text/>
    </questiondescription>
    <questionsimplify>1</questionsimplify>
    <assumepositive>0</assumepositive>
    <assumereal>0</assumereal>
    <prtcorrect format="html">
      <text><![CDATA[<span style="font-size: 1.5em; color:green;"><i class="fa fa-check"></i></span> Correct answer, well done.]]></text>
    </prtcorrect>
    <prtpartiallycorrect format="html">
      <text><![CDATA[<span style="font-size: 1.5em; color:orange;"><i class="fa fa-adjust"></i></span> Your answer is partially correct.]]></text>
    </prtpartiallycorrect>
    <prtincorrect format="html">
      <text><![CDATA[<span style="font-size: 1.5em; color:red;"><i class="fa fa-times"></i></span> Incorrect answer.]]></text>
    </prtincorrect>
    <decimals>.</decimals>
    <scientificnotation>*10</scientificnotation>
    <multiplicationsign>dot</multiplicationsign>
    <sqrtsign>1</sqrtsign>
    <complexno>i</complexno>
    <inversetrig>cos-1</inversetrig>
    <logicsymbol>lang</logicsymbol>
    <matrixparens>[</matrixparens>
    <isbroken>0</isbroken>
    <variantsselectionseed/>
    <input>
      <name>ans1</name>
      <type>algebraic</type>
      <tans>correctanswer</tans>
      <boxsize>10</boxsize>
      <strictsyntax>1</strictsyntax>
      <insertstars>0</insertstars>
      <syntaxhint/>
      <syntaxattribute>0</syntaxattribute>
      <forbidwords/>
      <allowwords/>
      <forbidfloat>1</forbidfloat>
      <requirelowestterms>1</requirelowestterms>
      <checkanswertype>1</checkanswertype>
      <mustverify>1</mustverify>
      <showvalidation>1</showvalidation>
      <options/>
    </input>
    <prt>
      <name>Result</name>
      <value>2.0000000</value>
      <autosimplify>1</autosimplify>
      <feedbackstyle>1</feedbackstyle>
      <feedbackvariables>
        <text/>
      </feedbackvariables>
      <node>
        <name>0</name>
        <description/>
        <answertest>AlgEquiv</answertest>
        <sans>ans1</sans>
        <tans>correctanswer</tans>
        <testoptions/>
        <quiet>0</quiet>
        <truescoremode>=</truescoremode>
        <truescore>1.0000000</truescore>
        <truepenalty/>
        <truenextnode>-1</truenextnode>
        <trueanswernote>Result-0-T </trueanswernote>
        <truefeedback format="html">
          <text/>
        </truefeedback>
        <falsescoremode>=</falsescoremode>
        <falsescore>0.0000000</falsescore>
        <falsepenalty/>
        <falsenextnode>-1</falsenextnode>
        <falseanswernote>Result-0-F </falseanswernote>
        <falsefeedback format="html">
          <text/>
        </falsefeedback>
      </node>
    </prt>
    <deployedseed>680910871</deployedseed>
    <deployedseed>1697968866</deployedseed>
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    <deployedseed>16974645</deployedseed>
    <deployedseed>2013621039</deployedseed>
    <deployedseed>1024472676</deployedseed>
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    <deployedseed>1321704505</deployedseed>
    <deployedseed>1050007925</deployedseed>
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    <deployedseed>2125054945</deployedseed>
    <deployedseed>2120380584</deployedseed>
    <deployedseed>221619436</deployedseed>
    <qtest>
      <testcase>1</testcase>
      <description/>
      <testinput>
        <name>ans1</name>
        <value>correctanswer</value>
      </testinput>
      <expected>
        <name>Result</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>Result-0-T</expectedanswernote>
      </expected>
    </qtest>
    <qtest>
      <testcase>2</testcase>
      <description/>
      <testinput>
        <name>ans1</name>
        <value>correctanswer-10</value>
      </testinput>
      <expected>
        <name>Result</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>Result-0-F</expectedanswernote>
      </expected>
    </qtest>
  </question>
</quiz>
