Make sure that you input the real and imaginary parts as fractions or integers and not as decimals. Also do not include brackets in your answer.
", "partialCredit": 0}, "vsetrangepoints": 5}], "variableReplacementStrategy": "originalfirst", "prompt": "\n\\[\\displaystyle z=\\simplify[!collectNumbers]{({z3}*{z2})/{z1}}\\]
\n$z=\\;\\;$[[0]].
\n ", "scripts": {}, "type": "gapfill", "marks": 0}, {"variableReplacements": [], "showCorrectAnswer": true, "gaps": [{"scripts": {}, "showpreview": true, "vsetrange": [0, 1], "checkingtype": "absdiff", "answer": "{re(conj(z3)*z1*z2)}/{abs(z3)^2}+{im(conj(z3)*z1*z2)}/{abs(z3)^2}*i", "checkvariablenames": false, "showCorrectAnswer": true, "variableReplacements": [], "checkingaccuracy": 0.001, "answersimplification": "std", "variableReplacementStrategy": "originalfirst", "marks": 1, "expectedvariablenames": [], "type": "jme", "notallowed": {"strings": [".", "(", ")"], "showStrings": false, "message": "Make sure that you input the real and imaginary parts as fractions or integers and not as decimals. Also do not include brackets in your answer.
", "partialCredit": 0}, "vsetrangepoints": 5}], "variableReplacementStrategy": "originalfirst", "prompt": "\n\\[\\displaystyle z=\\simplify[!collectNumbers]{({z2}*{z1})}(\\var{z3})^{-1}\\]
\n$z=\\;\\;$[[0]].
\n ", "scripts": {}, "type": "gapfill", "marks": 0}], "ungrouped_variables": ["s3", "s2", "s1", "s4", "a3", "rz3", "c2", "c1", "z1", "z2", "z3"], "variables": {"s2": {"group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything", "name": "s2"}, "c2": {"group": "Ungrouped variables", "definition": "random(1..5)", "description": "", "templateType": "anything", "name": "c2"}, "z2": {"group": "Ungrouped variables", "definition": "re(z1)+s2*random(1,2)+s4*random(1..9)*i", "description": "", "templateType": "anything", "name": "z2"}, "s3": {"group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything", "name": "s3"}, "s4": {"group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything", "name": "s4"}, "s1": {"group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything", "name": "s1"}, "a3": {"group": "Ungrouped variables", "definition": "s3*random(1..9)", "description": "", "templateType": "anything", "name": "a3"}, "z3": {"group": "Ungrouped variables", "definition": "rz3+s1*random(1..9)*i", "description": "", "templateType": "anything", "name": "z3"}, "c1": {"group": "Ungrouped variables", "definition": "s3*random(1..9)", "description": "", "templateType": "anything", "name": "c1"}, "rz3": {"group": "Ungrouped variables", "definition": "if(a3=re(z1),a3+random(1,-1),a3)", "description": "", "templateType": "anything", "name": "rz3"}, "z1": {"group": "Ungrouped variables", "definition": "s2*random(1..9)+s1*random(1..9)*i", "description": "", "templateType": "anything", "name": "z1"}}, "metadata": {"notes": "15/07/2015:
\nAdded tags.
\n9/07/2012:
\nAdded tags.
\nQuestion appears to be working correctly.
\n17/07/2012:
\nAdded more tags.
\nThere were errors, giving the wrong results, in the complex arithmetic code due to the treatment of the unary minus. This has been resolved.
\nImproved display in all content areas by using minimal ruleset of !collectNumbers, especially in the Advice section and the staged solution. This has also solved the problem of conjugates not being displayed properly.
", "description": "Composite multiplication and division of complex numbers. Two parts.
", "licence": "Creative Commons Attribution 4.0 International"}, "functions": {}, "statement": "\nExpress the following complex numbers $z$ in the form $a+bi$.
\nInput $a$ and $b$ as fractions and not as decimals. Also do not include brackets in your answer.
\n ", "advice": "\na)
\\[\\begin{eqnarray*}z=\\simplify[!collectNumbers]{({z3}*{z2})/{z1}} &=&\\simplify[!collectNumbers]{({z3}*{z2}*{conj(z1)})/({z1}*{conj(z1)})}\\\\ &=&\\simplify[!collectNumbers]{({z3*z2}*{conj(z1)})/({abs(z1)^2})}\\\\ &=&\\simplify[!collectNumbers]{{z3*z2*conj(z1)}/{abs(z1)^2}}\\\\ &=& \\simplify[std]{{re(z3*z2*conj(z1))}/{abs(z1)^2}+{im(z3*z2*conj(z1))}/{abs(z1)^2}*i} \\end{eqnarray*} \\]
b)
\\[\\begin{eqnarray*}z= \\simplify[!collectNumbers]{({z2}*{z1})}(\\var{z3})^{-1} &=& \\simplify[!collectNumbers]{({z2}*{z1})/{z3}}\\\\ &=&\\simplify[!collectNumbers]{({z2}*{z1}*{conj(z3)})/({z3}*{conj(z3)})}\\\\ &=&\\simplify[!collectNumbers]{({z2*z1}*{conj(z3)})/({abs(z3)^2})}\\\\ &=&\\simplify[!collectNumbers]{{z2*z1*conj(z3)}/{abs(z3)^2}}\\\\ &=& \\simplify[std]{{re(z2*z1*conj(z3))}/{abs(z3)^2}+{im(z2*z1*conj(z3))}/{abs(z3)^2}*i} \\end{eqnarray*} \\]