$(\\simplify[std]{{a}})(\\simplify[std]{{b}})\\;=\\;$[[0]].
\n\n
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Express the following in the form $a+bi\\;$ where $a$ and $b$ are real.
\nDo not include decimals in your answers, only fractions or integers. Also do not include brackets in your answers.
", "tags": ["addition of complex numbers", "checked2015", "complex numbers", "mas1602", "MAS1602", "multiplication of complex numbers", "product of complex numbers"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "15/07/2015:
\nAdded tags.
\n4/07/2012:
\nAdded tags.
\n16/07/2012:
\nAdded forbidden strings and warnings about not including decimal points or brackets in the answers as otherwise can just repeat the question and be marked correct.
", "licence": "Creative Commons Attribution 4.0 International", "description": "Elementary examples of multiplication and addition of complex numbers. Four parts.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "a)
The formula for multiplying complex numbers is
\\[\\begin{eqnarray*}\\simplify[]{Re((a + ib)(c + id))} &=& ac -bd \\\\ \\simplify[]{Im((a + ib)(c + id))} &=& ad +bc \\end{eqnarray*} \\]
So we have:
\\[\\begin{eqnarray*}\\simplify[]{Re({a}*{b})} &=& \\simplify[]{{Re(a)}*{Re(b)} - {Im( a)}*{Im(b)} = {Re(a*b)}}\\\\ \\simplify[]{Im({a}*{b})} &=& \\simplify[]{{Re(a)}*{Im(b)} + {Im( a)}*{Re(b)} = {Im(a*b)}} \\end{eqnarray*} \\]
Hence the solution is :
\\[(\\simplify[std]{{a}})(\\simplify[std]{{b}})=\\var{a*b}\\]
b)
This is calculated in a similar way once the expression is written as:
\n$(\\simplify[std]{{a1}})^2= (\\simplify[std]{{a1}}) (\\simplify[std]{{a1}})$ then we find:
\n\\[\\begin{eqnarray*}(\\simplify[std]{{a1}})^2&=& (\\simplify[std]{{a1}}) (\\simplify[std]{{a1}})\\\\ &=& \\simplify[]{({Re(a1)}*{Re(a1)} - {Im(a1)}*{Im(a1)})+ ({Re(a1)}*{Im(a1)} + {Im(a1)}*{Re(a1)})i}\\\\ &=& \\simplify[std]{{a1^2}} \\end{eqnarray*} \\]
c)
We know that $i^2=-1$ which gives $i^3=i^2i=-i$.
Hence:
\\[ \\begin{eqnarray*} \\simplify[std,!otherNumbers]{{a3} + {b3} * i + {c3} * i ^ 2 + {d3} * i ^ 3}&=&\\simplify[std]{{a3} + {b3} * i -{c3} -({d3} * i)}\\\\ &=&\\simplify[std]{ {a3} -{c3} + ({b3} -{d3}) * i}\\\\ &=&\\simplify[std]{{a3 -c3} + {b3 -d3} * i} \\end{eqnarray*} \\]
d)
This can be calculated by using the formula twice, firstly to multiply out the first two sets of parentheses,
and then to multiply the result of that calculation by the third set of parentheses.
So we obtain:
\\[ \\begin{eqnarray*} (\\var{z1})(\\var{z2})(\\var{z3})&=&((\\var{z1})(\\var{z2}))(\\var{z3})\\\\ &=&(\\var{z1*z2})(\\var{z3})\\\\ &=&\\var{z1*z2*z3} \\end{eqnarray*} \\]
Multiplication and addition of complex numbers. Four parts.
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\nInput all numbers as fractions or integers. Also do not include brackets in your answers.
", "functions": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "a)
\nThe solution is given by:
\n
$\\simplify[std]{{e6*i}}(\\simplify[std]{{a}})=\\simplify[std]{{a*e6*i}}$
b)
$\\simplify[std]{{a}*{z4}={a*z4}}$
\n
c)
\\[ \\begin{eqnarray*} \\simplify[std,!otherNumbers]{{a}*({a3} + {b3} * i + {c3} * i ^ 2 + {d3} * i ^ 3)}&=&\\simplify[std]{{a}*{a3 + b3 * i + c3 * i ^ 2 + d3 * i ^ 3}}\\\\ &=&\\simplify[std]{{a*(a3 + b3 * i + c3 * i ^ 2 + d3 * i ^ 3)}} \\end{eqnarray*} \\]
d)
This can be calculated by using the formula twice, firstly to multiply out the first two sets of parentheses,
\nand then to multiply the result of that calculation by the third set of parentheses.
\nSo we obtain:
\\[ \\begin{eqnarray*} (\\var{a})(\\var{z1})(\\var{z3})&=&((\\var{a})(\\var{z1}))(\\var{z3})\\\\ &=&(\\var{a*(z1)})(\\var{z3})\\\\ &=&\\var{a*(z1)*(z3)} \\end{eqnarray*} \\]
Inverse and division of complex numbers. Four parts.
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\nInput $a$ and $b$ as fractions or integers and not as decimals.
", "tags": ["checked2015", "complex numbers", "conjugate of a complex number", "division of complex numbers", "inverse of complex numbers", "multiplication of complex numbers"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus", "!collectLikeFractions"]}, "preamble": {"css": "", "js": ""}, "type": "question", "advice": "\n \n \nDivision of two complex numbers can be performed by mutiplying both the numerator and denominator by the conjugate of the denominator.
Suppose that \\[ z = \\frac{a+bi}{c+di},\\;\\; c+di \\neq 0\\] then we have:
\\[\\begin{eqnarray*}\n \n z&=&\\frac{a+bi}{c+di}\\\\\n \n &=&\\frac{(a+bi)(c-di)}{(c+di)(c-di)}\\\\\n \n &=&\\frac{(ac+bd)+(bc-ad)i}{c^2+d^2}\\\\\n \n &=&\\frac{ac+bd}{c^2+d^2}+\\frac{bc-ad}{c^2+d^2}i\n \n \\end{eqnarray*}\n \n \\]
Although this is a formula for the inverse, the best way to find these complex numbers is to remember to multiply top and bottom by the conjugate of the denominator.
(a)
\\[\\begin{eqnarray*}\\simplify[std]{{c1}/{z1}} &=&\\simplify[std]{({c1}*{conj(z1)})/({z1}*{conj(z1)})}\\\\\n \n &=&\\simplify[std]{{c1*conj(z1)}/{abs(z1)^2}}\\\\\n \n &=& \\simplify[std]{{c1*re(z1)}/{abs(z1)^2}-{c1*im(z1)}/{abs(z1)^2}*i}\n \n \\end{eqnarray*} \\]
(b)
\\[\\begin{eqnarray*}\\simplify[std]{{c2}/{z2}} &=&\\simplify[std]{({c2}*{conj(z2)})/({z2}*{conj(z2)})}\\\\\n \n &=&\\simplify[std]{{c2*conj(z2)}/{abs(z2)^2}}\\\\\n \n &=& \\simplify[std]{{c2*re(z2)}/{abs(z2)^2}-{c2*im(z2)}/{abs(z2)^2}*i}\n \n \\end{eqnarray*} \\]
(c)
\\[\\begin{eqnarray*}\\simplify[std]{{z1}/{z3}} &=&\\simplify[std]{({z1}*{conj(z3)})/({z3}*{conj(z3)})}\\\\\n \n &=&\\simplify[std]{{z1*conj(z3)}/{abs(z3)^2}}\\\\\n \n &=& \\simplify[std]{{re(z1*conj(z3))}/{abs(z3)^2}+{im(z1*conj(z3))}/{abs(z3)^2}*i}\n \n \\end{eqnarray*} \\]
(d)
\\[\\begin{eqnarray*}\\simplify[std]{{z3}/{z2}} &=&\\simplify[std]{({z3}*{conj(z2)})/({z2}*{conj(z2)})}\\\\\n \n &=&\\simplify[std]{{z3*conj(z2)}/{abs(z2)^2}}\\\\\n \n &=& \\simplify[std]{{re(z3*conj(z2))}/{abs(z2)^2}+{im(z3*conj(z2))}/{abs(z2)^2}*i}\n \n \\end{eqnarray*} \\]
\\[\\displaystyle z=\\simplify[!collectNumbers]{({z3}*{z2})/{z1}}\\]
\n$z=\\;\\;$[[0]].
\n ", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "useCustomName": false, "customName": "", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"answer": "{re(conj(z1)*z3*z2)}/{abs(z1)^2}+{im(conj(z1)*z3*z2)}/{abs(z1)^2}*i", "mustmatchpattern": {"message": "Your answer is not in the form $a+bi$.", "pattern": "`+-((integer:$n/integer:$n`?))`? + ((`+-integer:$n`?/integer:$n`?)*i `| `+-i)`?", "partialCredit": 0, "nameToCompare": ""}, "vsetRangePoints": 5, "useCustomName": false, "checkingType": "absdiff", "valuegenerators": [], "vsetRange": [0, 1], "showFeedbackIcon": true, "type": "jme", "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "variableReplacements": [], "failureRate": 1, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "showPreview": true, "customName": "", "checkVariableNames": false, "unitTests": [], "scripts": {}, "answerSimplification": "std", "showCorrectAnswer": true, "marks": 1}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "sortAnswers": false}, {"prompt": "\n\\[\\displaystyle z=\\simplify[!collectNumbers]{({z2}*{z1})}(\\var{z3})^{-1}\\]
\n$z=\\;\\;$[[0]].
\n ", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "useCustomName": false, "customName": "", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"answer": "{re(conj(z3)*z1*z2)}/{abs(z3)^2}+{im(conj(z3)*z1*z2)}/{abs(z3)^2}*i", "mustmatchpattern": {"message": "Your answer is not in the form $a+bi$.", "pattern": "`+-((integer:$n/integer:$n`?))`? + ((`+-integer:$n`?/integer:$n`?)*i `| `+-i)`?", "partialCredit": 0, "nameToCompare": ""}, "vsetRangePoints": 5, "useCustomName": false, "checkingType": "absdiff", "valuegenerators": [], "vsetRange": [0, 1], "showFeedbackIcon": true, "type": "jme", "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "variableReplacements": [], "failureRate": 1, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "showPreview": true, "customName": "", "checkVariableNames": false, "unitTests": [], "scripts": {}, "answerSimplification": "std", "showCorrectAnswer": true, "marks": 1}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "sortAnswers": false}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "Express the following complex numbers $z$ in the form $a+bi$.
\nInput $a$ and $b$ as fractions and not as decimals.
", "tags": ["algebra of complex numbers", "checked2015", "complex arithmetic", "complex numbers", "division of complex numbers", "inverse of complex numbers", "multiplication of complex numbers", "product of complex numbers"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus", "!collectlikefractions"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "Composite multiplication and division of complex numbers. Two parts.
"}, "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*} \\]
\\[z=(\\var{z1})^{\\var{c1}}(\\var{conj(z1)})^{\\var{c1}}\\]
$z=\\;\\;$[[0]]
Make sure that you input the real and imaginary parts as fractions and not as decimals
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$z=\\;\\;$[[0]]
\\[z=(\\var{z3})^{\\var{-d1}}\\]
$z=\\;\\;$[[0]]
Make sure that you input the real and imaginary parts as fractions and not as decimals
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$z=\\;\\;$[[0]]
Express the following complex numbers $z$ in the form $a+bi$.
\nInput $a$ and $b$ as fractions and not as decimals.
", "tags": ["addition of complex numbers", "algebra of complex numbers", "checked2015", "complex numbers", "conjugate of a complex number", "division of complex numbers", "mas1602", "MAS1602", "multiplication of complex numbers", "powers of complex numbers"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "15/07/2015:
\nAdded tags.
\n4/07/2012:
\nAdded tags.
\nQuestion appears to be working correctly.
", "licence": "Creative Commons Attribution 4.0 International", "description": "Direct calculation of low positive and negative powers of complex numbers. Calculations involving a complex conjugate. Powers of $i$. Four parts.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "a)
\nNote that for a complex number $z=a+bi$ we have:
\n$z\\overline{z}=|z|^2=a^2+b^2$.
\nBut since $\\var{conj(z1)}=\\overline{\\var{z1}}$ we have:
\\[\\begin{eqnarray*}z&=&(\\var{z1})^{\\var{c1}}(\\var{conj(z1)})^{\\var{c1}}\\\\ &=&((\\var{z1})(\\var{conj(z1)}))^{\\var{c1}}\\\\ &=&\\simplify[]{({re(z1)}^2+{im(z1)}^2)^{c1}}\\\\ &=&\\var{(re(z1)^2+im(z1)^2)^c1} \\end{eqnarray*}\\]
b)
\nNote that $(\\var{z2})^4=((\\var{z2})^2)^2$.
\nSince $(\\var{z2})^2=\\simplify[std]{{z2^2}}$ we have:
\\[(\\var{z2})^4=(\\simplify[std]{{z2^2}})^2=\\simplify[std]{{z2^4}}\\]
c)
We have
\\[ \\begin{eqnarray*} z&=&(\\var{z3})^{\\var{-d1}}\\\\ &=&\\frac{1}{(\\var{z3})^{\\var{d1}}}\\\\ &=&\\frac{(\\var{conj(z3)})^{\\var{d1}}}{(\\var{z3})^{\\var{d1}}(\\var{conj(z3)})^{\\var{d1}}}\\\\ &=&\\frac{\\var{conj(z3)^d1}}{\\var{abs(z3)^(2*d1)}}\\\\ &=&\\simplify[std]{{re(conj(z3^d1))}/{(abs(z3)^(2*d1))}+({im(conj(z3^d1))}/{round(abs(z3)^(2*d1))})*i} \\end{eqnarray*}\\]
d)
We have $i^2=-1,\\;\\;i^3=-i,\\;\\;i^4=1$.
So if $n=4m+r,\\;\\;0\\le r\\le 3$ we have \\[i^n=i^{4m+r}=(i^4)^m \\times i^r=i^r\\]
Hence since $\\var{n}=4\\times\\var{m}+\\var{rem}$ we have:
\\[i^{\\var{n}}=i^{\\var{rem}}=\\simplify{{i^rem}}\\]
Find the modulus and argument of $\\var{z1}$ to 3 decimal places.
\n(i) $|\\var{z1}|\\;=\\;$ [[0]], to 3 decimal places.
\n(ii) $\\arg(\\var{z1})\\;=\\;$[[1]] radians, to 3 decimal places.
\nHence find the following $\\var{n}$th roots of $\\var{z1}$ i.e. solve for $z$, $z^\\var{n}=\\var{z1}$.
\nHow many roots are there? [[2]]
\nAll the roots have the same modulus.
\nInput the modulus here: [[3]] (to 3 decimal places).
\nWhat is the argument of the root with the least argument? [[4]] radians (to 3 decimal places)
\nWhat is the argument of the root with the greatest argument? [[5]] radians (to 3 decimal places).
\nIf the roots are ordered in terms of their increasing arguments, what is the angle between successive roots? [[6]] radians (to 3 decimal places).
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\nImportant: When calculating the roots, you must use non-truncated values for the modulus and argument calculated in parts (i) and (ii) and not the approximated values, otherwise the final answer may not be correct.
\nFor the purposes of this question all arguments of complex numbers are between $0$ and $2\\pi$ radians.
", "tags": ["arctan", "argument of a complex number", "argument of complex number", "argument of complex numbers", "checked2015", "complex numbers", "de moivre's theorem", "de Moivre's theorem", "de Moivre's Theorem", "mas1602", "MAS1602", "modulus of complex numbers", "quadrants", "quadrants in the complex plane", "roots of a complex number"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "15/7/2015:
\nAdded tags.
\n27/08/2012:
\nAdded tags.
\nAdded description.
\nBased on question using DM's theorem for positive powers.
\n", "licence": "Creative Commons Attribution 4.0 International", "description": "
Find modulus and argument of the complex number $z_1$ and find the $n$th roots of $z_1$ where $n=5,\\;6$ or $7$.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "To be completed.
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"all,!collectNumbers,!noLeadingMinus", "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\nFind the modulus and argument of $\\var{z1}$ to 3 decimal places.
\n(i) $|\\var{z1}|\\;=\\;$ [[0]], to 3 decimal places.
\n(ii) $\\arg(\\var{z1})\\;=\\;$[[1]] radians, to 3 decimal places.
\nHence find:
\n(iii) $(\\var{z1})^{\\var{n2}}\\;=\\;$[[2]]
\nInput as a complex number, with real and imaginary parts to 3 decimal places.
\n", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "ans2+tol", "minValue": "ans2-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "arg2+tol", "minValue": "arg2-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"answer": "{a4}+{b4}*i", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "answersimplification": "std", "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\nFind the modulus and argument of $\\var{z2}$ to 3 decimal places.
\n(i) $|\\var{z2}|\\;=\\;$ [[0]], to 3 decimal places.
\n(ii) $\\arg(\\var{z2})\\;=\\;$[[1]] radians, to 3 decimal places.
\nHence find:
\n(iii) $(\\var{z2})^{\\var{n4}}\\;=\\;$[[2]]
\nInput as a complex number, with real and imaginary parts to 3 decimal places.
\n", "showCorrectAnswer": true, "marks": 0}], "statement": "Use de Moivre's theorem to write the following complex numbers in the form $a+bi$.
\nNote that for these questions, arguments of complex numbers lie in the range $-\\pi \\lt \\theta \\le \\pi$.
\nImportant: When calculating the final answer in part (iii) of each question, you must use non-truncated values for the modulus and argument calculated in parts (i) and (ii) and not the approximated values, otherwise the final answer will not be correct to three decimal places.
", "tags": ["arctan", "argument of a complex number", "argument of complex number", "argument of complex numbers", "checked2015", "complex numbers", "de Moivre's theorem", "de Moivre's Theorem", "de moivre's theorem", "MAS1602", "modulus of complex numbers", "quadrants", "quadrants in the complex plane"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "15/7/2015:
\nAdded tags.
\n26/11/2013
\nTurn off fractionNumbers in the answer to part a) 2.
\n5/07/2012:
\nAdded tags.
\nThe question doesn't really make sense. In the instruction we are asked to find modulus and argument of (a+i*b)^n but the question that is displayed is to find the modulus and argument of (a+i*b). Does the question need to be rewrittten to avoid this conflicting instruction?
\n9/07/2012:
\nChanged prompt instructions to make this question clearer.
\nCorrected request from 2dps to 3 dps for last question.
\nAlso set new tolerance variable, tol=0.001 for all numeric answers.
\n13/07/2012:
\n
Not a good question as can be done without using de Moivre. Needs to be recast.
Find modulus and argument of two complex numbers. Then use De Moivre's Theorem to find negative powers of the complex numbers.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "Given a complex number $z=r(\\cos(\\theta)+i\\sin(\\theta))$ de Moivre's theorem states that $z^n=r^n(\\cos(n\\theta)+i\\sin(n\\theta))$ for an integer power $n$.
So if we know the modulus $r$ and the argument $\\theta$ for $z$ then the theorem provides a way of calculating $z^n$.
As usual, you must be careful that the argument is calculated correctly, by paying attention to the quadrant of the complex plane in which lies.
\nAlso remember that for this question, arguments of complex numbers lie in the range $-\\pi \\lt \\theta \\le \\pi$.
\nWith the above in mind we can now answer the questions:
\na)
\n\\[ \\begin{eqnarray*} |\\var{z1}|&=&\\sqrt{(\\var{a1})^2+(\\var{b1})^2}\\\\ &=& \\var{abs(z1)}\\\\ &=&\\var{ans1} \\end{eqnarray*} \\] to 3 decimal places.
\nNote that $r^{\\var{n2}}=|(\\var{z1})^{\\var{n2}}| =\\var{abs(z1)}^{\\var{n2}}=\\var{abs(z1)^n2}$ which we will use in the calculation for $(\\var{z1})^{\\var{n2}}$
\n{m1}.
Hence we see that:
\\[\\begin{eqnarray*} \\arg(\\var{z1}) &=& \\var{arg(z1)}\\\\ &=& \\var{arg1}\\; \\mbox{radians} \\end{eqnarray*} \\] to 3 decimal places.
We have $\\arg((\\var{z1})^{\\var{n2}})=\\var{n2}\\times \\var{arg(z1)} = \\var{n2*arg(z1)}$ radians.
\nHence we have \\[\\begin{eqnarray*}(\\var{z1})^{\\var{n2}} &=& \\var{abs(z1)^n2}(\\cos(\\var{n2*arg(z1)})+\\sin(\\var{n2*arg(z1)})i)\\\\ &=& \\var{abs(z1)^n2}\\cos(\\var{n2*arg(z1)})+\\var{abs(z1)^n2}\\times\\sin(\\var{n2*arg(z1)})i\\\\ &=& \\simplify[std]{{a3}+{b3}i} \\end{eqnarray*} \\] to 3 decimal places for real and imaginary parts.
\nb)
\n\\[ \\begin{eqnarray*} |\\var{z2}|&=&\\sqrt{(\\var{a2})^2+(\\var{b2})^2}\\\\ &=& \\var{abs(z2)}\\\\ &=&\\var{ans2} \\end{eqnarray*} \\] to 3 decimal places.
\nNote that $r^{\\var{n4}}=|(\\var{z2})^{\\var{n4}}| =\\var{abs(z2)}^{\\var{n4}}=\\var{precround(abs(z2)^n4,6)}$ which we will use in the calculation for $(\\var{z2})^{\\var{n4}}$
\n{m2}.
Hence we see that:
\\[\\begin{eqnarray*} \\arg(\\var{z2}) &=& \\var{arg(z2)}\\\\ &=& \\var{arg2}\\; \\mbox{radians} \\end{eqnarray*} \\] to 3 decimal places.
We have $\\arg((\\var{z2})^{\\var{n4}})=\\var{n4}\\times \\var{arg(z2)} = \\var{n4*arg(z2)}$ radians.
\nHence we have \\[\\begin{eqnarray*}(\\var{z2})^{\\var{n4}} &=& \\var{precround(abs(z2)^n4,6)}(\\cos(\\var{n4*arg(z2)})+\\sin(\\var{n4*arg(z2)})i)\\\\ &=& \\var{precround(abs(z2)^n4,6)}\\cos(\\var{n4*arg(z2)})+\\var{precround(abs(z2)^n4,6)}\\times\\sin(\\var{n4*arg(z2)})i\\\\ &=& \\simplify[std]{{a4}+{b4}i} \\end{eqnarray*} \\] to 3 decimal places for real and imaginary parts.
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Find the modulus and argument of $\\var{z1}$ to 3 decimal places.
\n(i) $|\\var{z1}|\\;=\\;$ [[0]], to 3 decimal places.
\n(ii) $\\arg(\\var{z1})\\;=\\;$[[1]] radians, to 3 decimal places.
\nHence find:
\n(iii) $(\\var{z1})^{\\var{n2}}\\;=\\;$[[2]]
\nInput as a complex number, with real and imaginary parts integral values.
", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "ans2+tol", "minValue": "ans2-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "arg2+tol", "minValue": "arg2-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"answer": "{a4}+{b4}*i", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "answersimplification": "std", "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "Find the modulus and argument of $\\var{z2}$ to 3 decimal places.
\n(i) $|\\var{z2}|\\;=\\;$ [[0]], to 3 decimal places.
\n(ii) $\\arg(\\var{z2})\\;=\\;$[[1]] radians, to 3 decimal places.
\nHence find:
\n(iii) $(\\var{z2})^{\\var{n4}}\\;=\\;$[[2]]
\nInput as a complex number, with real and imaginary parts integral values.
", "showCorrectAnswer": true, "marks": 0}], "statement": "Use de Moivre's theorem to write the following complex numbers in the form $a+bi$.
\nNote that for these questions, arguments of complex numbers lie in the range $-\\pi \\lt \\theta \\le \\pi$.
\nImportant: When calculating the final answer in part (iii) of each question, you must use non-truncated values for the modulus and argument calculated in parts (i) and (ii) and not the approximated values, otherwise the final answer may not be correct.
", "tags": ["arctan", "argument of a complex number", "argument of complex number", "argument of complex numbers", "checked2015", "complex numbers", "de Moivre's theorem", "de Moivre's Theorem", "de moivre's theorem", "MAS1602", "modulus of complex numbers", "quadrants", "quadrants in the complex plane"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "15/7/2015:
\nAdded tags.
\n5/07/2012:
\nAdded tags.
\nThe question doesn't really make sense. In the instruction we are asked to find modulus and argument of (a+i*b)^n but the question that is displayed is to find the modulus and argument of (a+i*b). Does the question need to be rewrittten to avoid this conflicting instruction?
\n9/07/2012:
\nChanged prompt instructions to make this question clearer.
\nCorrected request from 2dps to 3 dps for last question.
\nAlso set new tolerance variable, tol=0.001 for all numeric answers.
\n13/07/2012:
\n
Not a good question as can be done without using de Moivre. Needs to be recast.
Find modulus and argument of two complex numbers. Then use De Moivre's Theorem to find positive powers of the complex numbers.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "Given a complex number $z=r(\\cos(\\theta)+i\\sin(\\theta))$ de Moivre's theorem states that $z^n=r^n(\\cos(n\\theta)+i\\sin(n\\theta))$ for an integer power $n$.
So if we know the modulus $r$ and the argument $\\theta$ for $z$ then the theorem provides a way of calculating $z^n$.
As usual, you must be careful that the argument is calculated correctly by paying attention to the quadrant of the complex plane in which the complex number lies.
\nAlso remember that for this question, arguments of complex numbers lie in the range $-\\pi \\lt \\theta \\le \\pi$.
\nWith the above in mind we can now answer the questions:
\na)
\n\\[ \\begin{eqnarray*} |\\var{z1}|&=&\\sqrt{(\\var{a1})^2+(\\var{b1})^2}\\\\ &=& \\var{abs(z1)}\\\\ &=&\\var{ans1} \\end{eqnarray*} \\] to 3 decimal places.
\nNote that $r^{\\var{n2}}=|(\\var{z1})^{\\var{n2}}| =\\var{abs(z1)}^{\\var{n2}}=\\var{abs(z1)^n2}$ which we will use in the calculation for $(\\var{z1})^{\\var{n2}}$
\n{m1}.
Hence we see that:
\\[\\begin{eqnarray*} \\arg(\\var{z1}) &=& \\var{arg(z1)}\\\\ &=& \\var{arg1}\\; \\mbox{radians} \\end{eqnarray*} \\] to 3 decimal places.
We have $\\arg((\\var{z1})^{\\var{n2}})=\\var{n2}\\times \\var{arg(z1)} = \\var{n2*arg(z1)}$ radians.
\nHence we have \\[\\begin{eqnarray*}(\\var{z1})^{\\var{n2}} &=& \\var{abs(z1)^n2}(\\cos(\\var{n2*arg(z1)})+\\sin(\\var{n2*arg(z1)})i)\\\\ &=& \\var{abs(z1)^n2}\\cos(\\var{n2*arg(z1)})+\\var{abs(z1)^n2}\\times\\sin(\\var{n2*arg(z1)})i\\\\ &=& \\simplify[std]{{a3}+{b3}i}. \\end{eqnarray*} \\]
\nb)
\n\\[ \\begin{eqnarray*} |\\var{z2}|&=&\\sqrt{(\\var{a2})^2+(\\var{b2})^2}\\\\ &=& \\var{abs(z2)}\\\\ &=&\\var{ans2} \\end{eqnarray*} \\] to 3 decimal places.
\nNote that $r^{\\var{n4}}=|(\\var{z2})^{\\var{n4}}| =\\var{abs(z2)}^{\\var{n4}}=\\var{abs(z2)^n4}$ which we will use in the calculation for $(\\var{z2})^{\\var{n4}}$
\n{m2}.
Hence we see that:
\\[\\begin{eqnarray*} \\arg(\\var{z2}) &=& \\var{arg(z2)}\\\\ &=& \\var{arg2}\\; \\mbox{radians} \\end{eqnarray*} \\] to 3 decimal places.
We have $\\arg((\\var{z2})^{\\var{n4}})=\\var{n4}\\times \\var{arg(z2)} = \\var{n4*arg(z2)}$ radians.
\nHence we have \\[\\begin{eqnarray*}(\\var{z2})^{\\var{n4}} &=& \\var{abs(z2)^n4}(\\cos(\\var{n4*arg(z2)})+\\sin(\\var{n4*arg(z2)})i)\\\\ &=& \\var{abs(z2)^n4}\\cos(\\var{n4*arg(z2)})+\\var{abs(z2)^n4}\\times\\sin(\\var{n4*arg(z2)})i\\\\ &=& \\simplify[std]{{a4}+{b4}i}. \\end{eqnarray*} \\]
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$|\\var{z1}|=\\;\\;$[[0]], $\\arg(\\var{z1})=\\;\\;$[[1]] radians
\nInput both answers to 3 decimal places.
", "scripts": {}, "gaps": [{"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "ans1+tol", "minValue": "ans1-tol", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}, {"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "arg1+tol", "minValue": "arg1-tol", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"prompt": "$|\\var{z2}|=\\;\\;$[[0]], $\\arg(\\var{z2})=\\;\\;$[[1]] radians
\nInput both answers to 3 decimal places.
", "scripts": {}, "gaps": [{"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "ans2+tol", "minValue": "ans2-tol", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}, {"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "arg2+tol", "minValue": "arg2-tol", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"prompt": "$|\\var{z3}|=\\;\\;$[[0]], $\\arg(\\var{z3})=\\;\\;$[[1]] radians
\nInput both answers to 3 decimal places.
", "scripts": {}, "gaps": [{"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "ans3+tol", "minValue": "ans3-tol", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}, {"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "arg3+tol", "minValue": "arg3-tol", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"prompt": "$|\\var{z4}|=\\;\\;$[[0]], $\\arg(\\var{z4})=\\;\\;$[[1]] radians
\nInput both answers to 3 decimal places.
", "scripts": {}, "gaps": [{"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "ans4+tol", "minValue": "ans4-tol", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}, {"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "arg4+tol", "minValue": "arg4-tol", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}], "statement": "Find the modulus and argument (in radians) of the following complex numbers, where the argument lies between $-\\pi$ and $\\pi$.
\nWhen calculating the argument pay particular attention to the quadrant in which the complex number lies.
\nInput all answers to 3 decimal places.
", "tags": ["arctan", "arg", "argument", "argument of complex numbers", "checked2015", "complex number", "complex numbers", "mas1602", "MAS1602", "mod", "modulus", "modulus argument form", "modulus of complex numbers", "quadrants and complex numbers"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "15/7/2015:
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\n5/07/2012:
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\nQuestion appears to be working correctly.
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\n13/07/2009:
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", "licence": "Creative Commons Attribution 4.0 International", "description": "Finding the modulus and argument (in radians) of four complex numbers; the arguments between $-\\pi$ and $\\pi$ and careful with quadrants!
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "Note that the arguments $\\theta$ of the complex numbers are in radians and have to be in the range $-\\pi < \\theta \\le \\pi$.
\nYou have to be careful with using a standard calculator when you are finding the argument of a complex number.
\nIf $z=a+bi=r(\\cos(\\theta)+i\\sin(\\theta))$ then we have:$r\\cos(\\theta)=a,\\;\\;r\\sin(\\theta)=b$ and so $\\tan(\\theta) = b/a$.
\nUsing a calculator to find the argument via $\\arctan(b/a)$ works in the range $-\\pi < \\theta \\le \\pi$ when the complex number is in the first or fourth quadrants – you get the correct value.
\nHowever, The calculator gives the wrong value for complex numbers in the other quadrants.
\nComplex number in the Second Quadrant.
\nSince $\\arctan(b/a)$ does not distinguish between the second and fourth quadrants and the calculator gives the argument for the fourth quadrant you have to add $\\pi$ onto the calculator value.
\nComplex number in the Third Quadrant.
\nSince $\\arctan(b/a)$ does not distinguish between the first and third quadrants and the calculator gives the argument for the first quadrant you have to take away $\\pi$ from the calculator value.
\na)Modulus.
\n\\[ \\begin{eqnarray*} |\\var{z1}|&=&\\sqrt{(\\var{a1})^2+(\\var{b1})^2}\\\\ &=& \\var{abs(z1)}\\\\ &=&\\var{ans1} \\end{eqnarray*} \\] to 3 decimal places.
\nArgument.
\n{m1}
\nHence we see that: \\[\\begin{eqnarray*} \\arg(\\var{z1}) &=& \\var{arg(z1)}\\\\ &=& \\var{arg1}\\; \\mbox{radians} \\end{eqnarray*} \\] to 3 decimal places.
\nb)Modulus.
\n\\[ \\begin{eqnarray*} |\\var{z2}|&=&\\sqrt{(\\var{a2})^2+(\\var{b2})^2}\\\\ &=& \\var{abs(z2)}\\\\ &=&\\var{ans2} \\end{eqnarray*} \\] to 3 decimal places.
\nArgument.
\n{m2}
\nHence we see that: \\[\\begin{eqnarray*} \\arg(\\var{z2}) &=& \\var{arg(z2)}\\\\ &=& \\var{arg2}\\; \\mbox{radians} \\end{eqnarray*} \\] to 3 decimal places.
\nc)Modulus.
\n\\[ \\begin{eqnarray*} |\\var{z3}|&=&\\sqrt{(\\var{c2})^2+(\\var{d2})^2}\\\\ &=& \\var{abs(z3)}\\\\ &=&\\var{ans3} \\end{eqnarray*} \\] to 3 decimal places.
\nArgument.
\n{m3}
\nHence we see that: \\[\\begin{eqnarray*} \\arg(\\var{z3}) &=& \\var{arg(z3)}\\\\ &=& \\var{arg3}\\; \\mbox{radians} \\end{eqnarray*} \\] to 3 decimal places.
\nd)Modulus.
\n\\[ \\begin{eqnarray*} |\\var{z4}|&=&\\sqrt{(\\var{a3})^2+(\\var{b3})^2}\\\\ &=& \\var{abs(z4)}\\\\ &=&\\var{ans4} \\end{eqnarray*} \\] to 3 decimal places.
\nArgument.
\n{m4}
\nHence we see that: \\[\\begin{eqnarray*} \\arg(\\var{z4}) &=& \\var{arg(z4)}\\\\ &=& \\var{arg4}\\; \\mbox{radians} \\end{eqnarray*} \\] to 3 decimal places.
"}, {"name": "Roots of a cubic real polynomial", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "tags": ["checked2015", "complex numbers", "complex roots of real polynomials", "conjugate roots", "roots of polynomial equations", "roots of polynomials", "roots of real polynomials"], "metadata": {"description": "Using a given list of four complex numbers, find by inspection the one that is a root of a given cubic real polynomial and hence find the other roots.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "From the list of complex numbers, choose the one which is a root of the given equation $f(z)=0$ , and hence find all roots.
", "advice": "a) Finding a root.
\nIn order to find which one of the four choices is a root you have to evaluate $f(z)$ for each choice.If you find for a choice of $z$ that $f(z)=0$ then that choice of $z$ is a root of the equation.
\nNote that\\[\\begin{eqnarray*} \\simplify{f({z1})} &=&\\simplify[std]{{z1}^3+{-2*a1 -c1}*{z1} ^ 2 + {2 * a1 * c1 + a1 ^ 2 + b1 ^ 2} * {z1} -{c1 * (a1 ^ 2 + b1 ^ 2)}}\\\\ &=&\\simplify[std]{{z1^3}+{-2*a1 -c1}{z1 ^ 2} + {2 * a1 * c1 + a1 ^ 2 + b1 ^ 2} * {z1} -{c1 * (a1 ^ 2 + b1 ^ 2)}}\\\\ &=&\\simplify[std]{{z1^3}+{(( -2) * a1 -c1)*z1^2}+ {(2 * a1 * c1 + a1 ^ 2 + b1 ^ 2)*z1}-{c1 * (a1 ^ 2 + b1 ^ 2)}}\\\\ &=&0 \\end{eqnarray*}. \\]So of the list of choices $z_1=\\var{z1}$ is a root.
\nb) The other roots
\nNow that you have found a complex root it is very simple to find another complex root.
\nSince $f(z)$ is a polynomial with real coefficients then if $z=z_0$ is a root we have that the conjugate $z=\\overline{z_0}$ is also a root.
\nHence the complex number $z_2=\\overline{\\var{z1}}=\\var{conj(z1)}$ is a root.
\nTo find the real root $z_3=c$ we note that the constant term of\\[f(z) =(z-z_1)(z-z_2)(z-c)\\]is $-z_1z_2c = -(\\var{z1})(\\var{conj(z1)})c=\\var{-z1*conj(z1)}c$.
\nBut we know that the constant term of $f(z)$ is $\\simplify{-{c1 * (a1 ^ 2 + b1 ^ 2)}} $.
\nHence \\[\\begin{eqnarray*} \\var{-z1*conj(z1)}c &=&\\simplify{-{c1 * (a1 ^ 2 + b1 ^ 2)}}\\\\ \\Rightarrow c &=& \\simplify[]{{c1 * (a1 ^ 2 + b1 ^ 2)}/{abs(z1^2)}}\\\\ &=&\\var{c1} \\end{eqnarray*} \\]
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\nChoose the correct value for $z_1$:[[0]]
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", "$\\simplify{{x3a1}+{x3b1}i}$
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\n \n \n \n\\[\\begin{eqnarray*}\n \n \\simplify{f({z1})}\t&=&\\simplify[std]{{z1}^3+{-2*a1 -c1}*{z1} ^ 2 + {2 * a1 * c1 + a1 ^ 2 + b1 ^ 2} * {z1} -{c1 * (a1 ^ 2 + b1 ^ 2)}}\\\\\t\n \n &=&\\simplify[std]{{z1^3}+{-2*a1 -c1}{z1 ^ 2} + {2 * a1 * c1 + a1 ^ 2 + b1 ^ 2} * {z1} -{c1 * (a1 ^ 2 + b1 ^ 2)}}\\\\\t\n \n &=&\\simplify[std]{{z1^3}+{(( -2) * a1 -c1)*z1^2}+ {(2 * a1 * c1 + a1 ^ 2 + b1 ^ 2)*z1}-{c1 * (a1 ^ 2 + b1 ^ 2)}}\\\\\t\t\n \n &=&0\t\t\t\t\t\t\t\t\n \n \\end{eqnarray*}\n \n \\] Hence is a root.
\n \n ", "\n \n \nNot the correct choice as :\\[\\begin{eqnarray*}\n \n \\simplify{f({z2})}\t&=&\\simplify[std]{{z2}^3+{-2*a1 -c1}*{z2} ^ 2 + {2 * a1 * c1 + a1 ^ 2 + b1 ^ 2} * {z2} -{c1 * (a1 ^ 2 + b1 ^ 2)}}\\\\\t\n \n &=&\\simplify[std]{{z2^3}+{-2*a1 -c1}{z2 ^ 2} + {2 * a1 * c1 + a1 ^ 2 + b1 ^ 2} * {z2} -{c1 * (a1 ^ 2 + b1 ^ 2)}}\\\\\t\n \n &=&\\simplify[std]{{z2^3}+{(( -2) * a1 -c1)*z2^2}+ {(2 * a1 * c1 + a1 ^ 2 + b1 ^ 2)*z2}-{c1 * (a1 ^ 2 + b1 ^ 2)}}\\\\\t\t\n \n &=&\\simplify[std]{{z2^3+(( -2) * a1 -c1)*z2^2+ (2 * a1 * c1 + a1 ^ 2 + b1 ^ 2)*z2-c1 * (a1 ^ 2 + b1 ^ 2)}}\\neq 0\t\t\t\t\t\t\t\t\n \n \\end{eqnarray*}\\]
Hence not a root
Not the correct choice as :\\[\\begin{eqnarray*}\n \n \\simplify{f({z3})}\t&=&\\simplify[std]{{z3}^3+{-2*a1 -c1}*{z3} ^ 2 + {2 * a1 * c1 + a1 ^ 2 + b1 ^ 2} * {z3} -{c1 * (a1 ^ 2 + b1 ^ 2)}}\\\\\t\n \n &=&\\simplify[std]{{z3^3}+{-2*a1 -c1}{z3 ^ 2} + {2 * a1 * c1 + a1 ^ 2 + b1 ^ 2} * {z3} -{c1 * (a1 ^ 2 + b1 ^ 2)}}\\\\\t\n \n &=&\\simplify[std]{{z3^3}+{(( -2) * a1 -c1)*z3^2}+ {(2 * a1 * c1 + a1 ^ 2 + b1 ^ 2)*z3}-{c1 * (a1 ^ 2 + b1 ^ 2)}}\\\\\t\t\n \n &=&\\simplify[std]{{z3^3+(( -2) * a1 -c1)*z3^2+ (2 * a1 * c1 + a1 ^ 2 + b1 ^ 2)*z3-c1 * (a1 ^ 2 + b1 ^ 2)}}\\neq 0\t\t\t\t\t\t\t\t\n \n \\end{eqnarray*}\\]
Hence not a root
Not the correct choice as :\\[\\begin{eqnarray*}\n \n \\simplify{f({z2})}\t&=&\\simplify[std]{{z4}^3+{-2*a1 -c1}*{z4} ^ 2 + {2 * a1 * c1 + a1 ^ 2 + b1 ^ 2} * {z4} -{c1 * (a1 ^ 2 + b1 ^ 2)}}\\\\\t\n \n &=&\\simplify[std]{{z4^3}+{-2*a1 -c1}{z4 ^ 2} + {2 * a1 * c1 + a1 ^ 2 + b1 ^ 2} * {z4} -{c1 * (a1 ^ 2 + b1 ^ 2)}}\\\\\t\n \n &=&\\simplify[std]{{z4^3}+{(( -2) * a1 -c1)*z4^2}+ {(2 * a1 * c1 + a1 ^ 2 + b1 ^ 2)*z4}-{c1 * (a1 ^ 2 + b1 ^ 2)}}\\\\\t\t\n \n &=&\\simplify[std]{{z4^3+(( -2) * a1 -c1)*z4^2+ (2 * a1 * c1 + a1 ^ 2 + b1 ^ 2)*z4-c1 * (a1 ^ 2 + b1 ^ 2)}}\\neq 0\t\t\t\t\t\t\t\t\n \n \\end{eqnarray*}\\]
Hence not a root
The remaining roots of $f(z)$ are:
\n$z_2=\\;\\;$[[0]] (enter the complex root here)
\n$z_3=\\;\\;$[[1]] (enter the real root here)
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\nChoose the correct value for $z_1$:[[0]]
", "stepsPenalty": 0, "scripts": {}, "gaps": [{"displayType": "radiogroup", "choices": ["$\\simplify[std]{{a1}+{b1}i}$
", "$\\simplify[std]{{z3}}$
"], "showCorrectAnswer": true, "displayColumns": 4, "distractors": ["", ""], "variableReplacements": [], "shuffleChoices": true, "scripts": {}, "maxMarks": 0, "type": "1_n_2", "minMarks": 0, "variableReplacementStrategy": "originalfirst", "matrix": [1, 0], "marks": 0}], "type": "gapfill", "showCorrectAnswer": true, "steps": [{"showCorrectAnswer": true, "scripts": {}, "type": "information", "variableReplacementStrategy": "originalfirst", "prompt": "Since you are given that $f(z)$ has a complex root $z_1$ and since $f(z)$ is a polynomial with real coefficients then the complex conjugate $\\overline{z_1}$ must also be a root.
\nSince $(z-z_1)(z-\\overline{z_1})=(z^2-2\\operatorname{Re}(z)+|z_1|^2)$ we have that:\\[f(z)=(z^2-2\\operatorname{Re}(z)+|z_1|^2)(z^2+az+b)=\\simplify{z ^ 4+ {( -2) * r12}*z ^ 3+ {mz1+mz2+4*re(z1)*re(z2)} * z^2 -{2*(re(z2)*mz1+re(z1)*mz2)}z+{mz1*mz2}}\\] where $a$ and $b$ are real.
\nLooking at the constant term we see that :
\n\\[|z_1|^2b = \\var{mz1*mz2}\\]
\nHence $|z_1|^2$ divides $ \\var{mz1*mz2}$.
\nAn easy test to see if one of the complex numbers given is not a root is to see if its modulus squared does not divide $ \\var{mz1*mz2}$. If it does not divide then the other must be the root.
", "variableReplacements": [], "marks": 0}], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"prompt": "Write down the quadratic factor with real coefficients, $q_1(z)$, of $f(z)$ which has $z_1$ as a root:
\n$q_1(z)=\\;$[[0]]
\nApart from $z_1$, $q_1(z)$ has another root $z_2$, which is also a root of $f(z)$.
\n$z_2=\\;$[[1]]
", "stepsPenalty": 0, "scripts": {}, "gaps": [{"answer": "z^2-{2*a1}*z+{mz1}", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}, {"answer": "{a1}-{b1}i", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "steps": [{"showCorrectAnswer": true, "scripts": {}, "type": "information", "variableReplacementStrategy": "originalfirst", "prompt": "If $z_1$ is a root then its conjugate $z_2$= $\\overline{z_1}$ is also a root.
", "variableReplacements": [], "marks": 0}], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"prompt": "Since $q_1(z)$ is a factor of $f(z)$ the other roots are given by finding the other quadratic factor $q_2(z)$ of $f(z)=q_1(z)q_2(z)$
\n$q_2(z)\\;=$[[0]]
\nFind the roots of $q_2(z)$ and hence the remaining two roots $z_3,\\;z_4$ of $f(z)$
\n$z_3=\\;$[[1]] (imaginary part negative)
\n$z_4=\\;$[[2]] (imaginary part positive).
", "stepsPenalty": 0, "scripts": {}, "gaps": [{"answer": "z^2-{2*a2}*z+{mz2}", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}, {"answer": "{a2}-{b2}*i", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}, {"answer": "{a2}+{b2}*i", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "steps": [{"showCorrectAnswer": true, "scripts": {}, "type": "information", "variableReplacementStrategy": "originalfirst", "prompt": "$q_1(z)q_2(z)=f(z)$.
\nOnce you have found $q_1(z)$ then the easiest way to find $q_2(z)$ is to compare the terms in $z^3$ and the constant terms.
", "variableReplacements": [], "marks": 0}], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}], "statement": "Given two complex numbers, find by inspection the one that is a root of a given quartic real polynomial $f(z)$ and hence find the other roots.
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\nAdded tags.
\n25/08/2012:
\nCopied question finding roots of a cubic in order to create new question finding roots of a quartic with 4 complex roots.
\nFunction ch finds the imaginary part of the complex number $z_3$ and ensures that $z_3$ is not a solution by insisting that $|z_3|^2$ does not divide the constant term of the polynomial. This is a simple way for the students to test to see which one of $z_1$ and $z_2$ is a solution.
\nAdded tags.
\nAdded description.
\nChecked calculation.OK.
", "licence": "Creative Commons Attribution 4.0 International", "description": "Given two complex numbers, find by inspection the one that is a root of a given quartic real polynomial and hence find the other roots.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "a) We use the method given in Show steps for part a).
\nNote that $|\\var{z1}|^2=\\var{mz1}$ divides the constant term $\\var{mz1*mz2}$,
\nbut that $|\\var{z3}|^2=\\var{mz3}$ does not divides the constant term $\\var{mz1*mz2}$.
\nHence $\\var{z1}$ is the root we are looking for.
\nb) A quadratic factor of $f(z)$.
\nSince $f(z)$ is a polynomial with real coefficients then if $z=z_1$ is a root we have that the conjugate $z=\\overline{z_1}$ is also a root.
\nHence the complex number $z_2=\\overline{\\var{z1}}=\\var{conj(z1)}$ is a root.
\nHence $q_1(z) = (z-(\\var{z1}))(z-(\\var{conj(z1)}))=\\simplify[std]{z^2-{2*a1}*z+{abs(z1)^2}}$ is a factor of $f(z)$.
\nc)The other quadratic factor and the other roots.
\nWe have that $f(z)=q_1(z)q_2(z)$, where $q_1(z)$ is as above and we have to find the quadratic $q_2(z)=z^2+az+b$ with real coefficients $a$ and $b$.
\n\\[\\begin{eqnarray*}f(z) &=& \\simplify[std]{z ^ 4+ {( -2) * r12}*z ^ 3+ {mz1+mz2+4*re(z1)*re(z2)} * z^2 -{2*(re(z2)*mz1+re(z1)*mz2)}z+{mz1*mz2}}\\\\&=&q_1(z)q_2(z)\\\\&=&(\\simplify[std]{z^2-{2*a1}*z+{mz1}})(z^2+az+b)\\\\&=&\\simplify[std]{z^4+(a-{2*a1})z^3+(b-{2*a1}*a+{mz1})*z^2+({mz1}a-{2*a1}b)*z+{mz1}*b}\\end{eqnarray*}\\]
\nIdentifying the constant terms and the coefficients of $z^3$ on both sides of this equation gives:
\n$a=\\var{-2*a2},\\;\\;b=\\var{mz2}$
\nHence $q_2(z)=\\simplify[std]{z^2-{2*a2}*z+{mz2}}$
\nYou can then find the roots of this quadratic, giving the other roots of $f(z)$:
\n$z_3=\\simplify[std]{{a2}-{b2}*i}$ (negative imaginary part)
\n$z_4=\\simplify[std]{{a2}+{b2}*i}$ (positive imaginary part)
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"all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "showQuestionGroupNames": false, "functions": {}, "parts": [{"showCorrectAnswer": true, "scripts": {}, "gaps": [{"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "ans1+tol", "minValue": "ans1-tol", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "prompt": "\n \n \n
Find the distance between $\\var{z1}$ and $\\var{z2}$.
\n \n \n \nDistance = [[0]]
\n \n \n ", "variableReplacements": [], "marks": 0}, {"showCorrectAnswer": true, "scripts": {}, "gaps": [{"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "ans2+tol", "minValue": "ans2-tol", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "prompt": "\n \n \nFind the distance between $\\var{z3}$ and $\\var{z4}$.
\n \n \n \nDistance = [[0]]
\n \n \n ", "variableReplacements": [], "marks": 0}, {"showCorrectAnswer": true, "scripts": {}, "gaps": [{"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "ans3+tol", "minValue": "ans3-tol", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "prompt": "\n \n \nFind the distance between $\\var{z5}$ and $\\var{z6}$.
\n \n \n \nDistance = [[0]]
\n \n \n ", "variableReplacements": [], "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "Find the distance between the following complex numbers, leaving your answer in decimal form to 3 decimal places:
", "tags": ["checked2015", "complex numbers", "distance between complex numbers", "mas1602", "MAS1602", "modulus", "modulus of a complex number", "modulus of complex numbers"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "15/07/2015:
\nAdded tags.
\n5/07/2012:
\nAdded tags.
\nPerhaps more steps are needed in the solutions? It isn't explained how to find the modulus of the complex number. Explanation included in Advice.
\nQuestion appears to be working correctly.
\n13/07/2012:
\nSet new variable tol=0 for all numeric input.
", "licence": "Creative Commons Attribution 4.0 International", "description": "Finding the distance between two complex numbers using the modulus of their difference. Three parts.
"}, "advice": "\n \n \nThe distance D between two complex numbers $z_1=a+bi$ and $z_2=c+di$ is given by the modulus of the difference i.e.
\\[D=|z_1-z_2| = |(a-c)+(b-d)i|=\\sqrt{(a-c)^2+(b-d)^2}\\]
Applying to the questions we have:\t\t\t\t\t
a) \\[ \\begin{eqnarray*} D&=&|(\\var{z1})-(\\var{z2})|\\\\\n \n &=&|\\var{z1-z2}|\\\\\n \n &=& \\var{abs(z1-z2)}\\\\\n \n &=&\\var{ans1}\n \n \\end{eqnarray*} \\] to 3 decimal places.
b) \\[ \\begin{eqnarray*} D&=&|(\\var{z3})-(\\var{z4})|\\\\\n \n &=&|\\var{z3-z4}|\\\\\n \n &=& \\var{abs(z3-z4)}\\\\\n \n &=&\\var{ans2}\n \n \\end{eqnarray*} \\] to 3 decimal places.
\n \n \n \nc) \\[ \\begin{eqnarray*} D&=&|(\\var{z5})-(\\var{z6})|\\\\\n \n &=&|\\var{z5-z6}|\\\\\n \n &=& \\var{abs(z5-z6)}\\\\\n \n &=&\\var{ans3}\n \n \\end{eqnarray*} \\] to 3 decimal places.
\n \n \n "}, {"name": "The modulus of complex numbers", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s1", "description": ""}, "tol": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0.001", "name": "tol", "description": ""}, "b3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(1..4)", "name": "b3", "description": ""}, "s5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s5", "description": ""}, "c4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(a4=f,f+1,f)", "name": "c4", "description": ""}, "b4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(1..9)", "name": "b4", "description": ""}, "s4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s4", "description": ""}, "s2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s2", "description": ""}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(2..9)", "name": "a1", "description": ""}, "z5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a4+b4*i", "name": "z5", "description": ""}, "c2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s6*random(1..9)", "name": "c2", "description": ""}, "d2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s7*random(1..9)", "name": "d2", "description": ""}, "n": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(3..5)", "name": "n", "description": ""}, "z3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "c2+d2*i", "name": "z3", "description": ""}, "z1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a1+b1*i", "name": "z1", "description": ""}, "z4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a3+b3*i", "name": "z4", "description": ""}, "ans2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(abs(z2*z3),3)", "name": "ans2", "description": ""}, "ans3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(abs(z4)^n,3)", "name": "ans3", "description": ""}, "d4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s5*random(1..9)", "name": "d4", "description": ""}, "f": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "name": "f", "description": ""}, "z2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a2+b2*i", "name": "z2", "description": ""}, "b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s2*random(3..9)", "name": "b1", "description": ""}, "a2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s4*random(1..9)", "name": "a2", "description": ""}, "ans1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(abs(z1),3)", "name": "ans1", "description": ""}, "b2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s5*random(1..9)", "name": "b2", "description": ""}, "s8": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s8", "description": ""}, "ans4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(abs(z5/z6),3)", "name": "ans4", "description": ""}, "z6": {"templateType": "anything", "group": "Ungrouped variables", "definition": "c4+d4*i", "name": "z6", "description": ""}, "a3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s3*random(1..4)", "name": "a3", "description": ""}, "a4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s8*random(1..9)", "name": "a4", "description": ""}, "s3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s3", "description": ""}, "s6": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s6", "description": ""}, "s7": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s7", "description": ""}}, "ungrouped_variables": ["ans1", "ans2", "ans3", "ans4", "b4", "b1", "b2", "b3", "d4", "d2", "s8", "s3", "s2", "s1", "s7", "s6", "s5", "s4", "z2", "tol", "c4", "a1", "a3", "a2", "a4", "z4", "z5", "z6", "z1", "c2", "z3", "f", "n"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "functions": {}, "showQuestionGroupNames": false, "parts": [{"showCorrectAnswer": true, "scripts": {}, "gaps": [{"showCorrectAnswer": true, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "correctAnswerFraction": false, "minValue": "ans1-tol", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "maxValue": "ans1+tol"}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "prompt": "$|\\var{z1}|=\\;\\;$[[0]]
", "variableReplacements": [], "marks": 0}, {"showCorrectAnswer": true, "scripts": {}, "gaps": [{"showCorrectAnswer": true, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "correctAnswerFraction": false, "minValue": "ans2-tol", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "maxValue": "ans2+tol"}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "prompt": "$|(\\var{z2})(\\var{z3})|=\\;\\;$[[0]]
", "variableReplacements": [], "marks": 0}, {"showCorrectAnswer": true, "scripts": {}, "gaps": [{"showCorrectAnswer": true, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "correctAnswerFraction": false, "minValue": "ans3-tol", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "maxValue": "ans3+tol"}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "prompt": "$|(\\var{z4})^{\\var{n}}|=\\;\\;$[[0]]
", "variableReplacements": [], "marks": 0}, {"showCorrectAnswer": true, "scripts": {}, "gaps": [{"showCorrectAnswer": true, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "correctAnswerFraction": false, "minValue": "ans4-tol", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "maxValue": "ans4+tol"}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "prompt": "Let \\[z=\\frac{\\var{z5}}{\\var{z6}}\\]
$|z|=\\;\\;$[[0]]
Find the modulus of each of the following complex numbers, leaving your answer in decimal form to 3 decimal places:
", "tags": ["checked2015", "complex number", "complex numbers", "division of complex numbers", "mas1602", "MAS1602", "modulus of a complex number", "modulus of complex numbers", "modulus of the division of complex numbers", "modulus of the power of complex numbers", "modulus of the product of complex numbers", "multiplication of complex numbers", "multiply complex numbers", "product of complex numbers", "properties of the modulus of complex numbers", "rationalise the denominator", "rationalising the denominator"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "15/07/2015:
\nAdded tags.
\n5/07/2012:
\nAdded tags.
\nQuestion appears to be working correctly.
", "licence": "Creative Commons Attribution 4.0 International", "description": "Finding the modulus of four complex numbers; includes finding the modulus of a product, a power and a quotient.
"}, "advice": "Recall that $|a+bi|=\\sqrt{a^2+b^2}$ and that:
\n1. $ |z^n| = |z|^n$
\n2. $ |z_1z_2|=|z_1|\\;|z_2|$
\n3. $ |z_1/z_2|=|z_1|/|z_2|$
\na) \\[ \\begin{eqnarray*} |\\var{z1}|&=&\\sqrt{(\\var{a1})^2+(\\var{b1})^2}\\\\ &=& \\var{abs(z1)}\\\\ &=&\\var{ans1} \\end{eqnarray*} \\] to 3 decimal places.
\nb) \\[ \\begin{eqnarray*} |(\\var{z2})(\\var{z3})|&=&|\\var{z2}|\\;|\\var{z3}|\\\\ &=& \\var{abs(z2)}\\times \\var{abs(z3)}\\\\ &=&\\var{abs(z2*z3)}\\\\ &=&\\var{ans2} \\end{eqnarray*} \\] to 3 decimal places.
\nc) \\[ \\begin{eqnarray*} |(\\var{z4})^{\\var{n}}|&=&|\\var{z4}|^{\\var{n}}\\\\ &=& \\var{abs(z4)}^{\\var{n}}\\\\ &=& \\var{abs(z4)^n}\\\\ &=&\\var{ans3} \\end{eqnarray*} \\] to 3 decimal places.
\nd) \\[ \\begin{eqnarray*} \\left|\\frac{\\var{z5}}{\\var{z6}}\\right|&=&\\frac{|\\var{z5}|}{|\\var{z6}|}\\\\ &=& \\frac{\\var{abs(z5)}}{\\var{abs(z6)}}\\\\ &=& \\var{abs(z5/z6)}\\\\ &=&\\var{ans4} \\end{eqnarray*} \\] to 3 decimal places.
"}]}], "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "Questions about complex arithmetic; argument and modulus of complex numbers; complex roots of polynomials; de Moivre's theorem.
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