// Numbas version: exam_results_page_options {"name": "WM104 in-class assessment : part (i) complex numbers - 50% extra timing", "metadata": {"description": "

You are allowed 38 minutes to complete this test.

You may use rough paper for your calculations.

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Multiplication and addition of complex numbers. Four parts.

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

Find the following complex numbers in the form $a+bi\\;$ where $a$ and $b$ are real.

\n

Input all numbers as fractions or integers (whole numbers). Also do not include brackets in your answers.

a)

\n

The solution is given by:

\n

$\\simplify[std]{{e6*i}}(\\simplify[std]{{a}})=\\simplify[std]{{a*e6*i}}$

\n

b)

\n

$\\simplify[std]{{a}*{z4}={a*z4}}$

\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)

\n

This can be calculated by using the formula twice, firstly to multiply out the first two sets of parentheses,

\n

and then to multiply the result of that calculation by the third set of parentheses.

\n

So 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*} \$

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$\\var{e6*i}(\\simplify[std]{{a}})\\;=\\;$[[0]].

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Input all numbers as fractions or integers. Also do not include brackets in your answers.

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$(\\simplify[std]{{a}})(\\simplify[std]{{z4}})\\;=\\;$[[0]].

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Input all numbers as fractions or integers. Also do not include brackets in your answers.

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This question provides practice at adding, subtracting, dividing and multiplying complex numbers in rectangular form.

", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "

Given the complex numbers

\n

\$a=\\var{a},\\quad b=\\var{b},\\quad c=\\var{c},\\quad d=\\var{d},\$

\n

\n

\$x+yi\$

\n

Note, ensure that you write i (not j) at the end of the imaginary part!

\$z_1+z_2=(\\var{z1})+(\\var{z2})=\\simplify{{z1}+{z2}}\$

\n

\$z_1-z_2=(\\var{z1})-(\\var{z2})=\\simplify{{z1}-{z2}}\$

\n

\$z_1z_3=(\\var{z1})(\\var{z3})=\\var{z1z3}\$

\n

\$\\frac{z_3}{z_4}=\\frac{\\var{z3}}{\\var{z4}}=\\frac{(\\var{z3})(\\var{z4bar})}{(\\var{z4})(\\var{z4bar})}=\\simplify{{z3z4bar}/{modz4sq}}\\approx \\var{re_z3onz4}+\\var{im_z3onz4}\$

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$ab=$

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Expand the brackets $(\\var{a})(\\var{b})$.

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Simplify using the fact $i^2=-1$.

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No brackets are required or allowed.

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For the question below, please give your answers for both the real and imaginary parts either as whole numbers or decimals correct to 4 decimal places, as appropriate:

\n

$\\frac{b}{d}=$

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No brackets are required or allowed.

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Firstly, $|z| = \\sqrt{(\\var{a1})^2+(\\var{b1})^2} = \\sqrt{\\var{a1*a1+b1*b1}} = \\var{dpformat(modz,2)}$

\n

Secondly, since the real part of $z$ is {rez} and the imaginary part of $z$ is {imz}, {quad} So,

\n

$\\arg(z) = \\var{dpformat(argz,2)}^\\circ$.

\n

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im

Find the modulus and argument (in degrees) of the complex number $z=\\var{z1}$.

\n

$|z| =$ [[0]]

\n

$\\arg(z) =$ [[1]]$^\\circ$.

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Practice to decide which quadrant a complex number lies in.

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Practice to decide which quadrant a complex number lies in.

Find the modulus and argument (in radians) of the complex number $z=\\var{z1}$.

\n

$|z| =$ [[0]]

\n

$\\arg(z) =$ [[1]] radians.

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Firstly, $|z| = \\sqrt{(\\var{a1})^2+(\\var{b1})^2} = \\sqrt{\\var{a1*a1+b1*b1}} = \\var{dpformat(modz,2)}$

\n

Secondly, since the real part of $z$ is {rez} and the imaginary part of $z$ is {imz}, {quad} So,

\n

$\\arg(z) = \\var{dpformat(argz,2)}$ radians.

\n

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im

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Find modulus and argument of two complex numbers.

Then use De Moivre's Theorem to find powers of the complex numbers.

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

Use de Moivre's theorem to write the following complex number in the form $a+bi$.

\n

Remember that the argument of a complex numbers lies in the range $-\\pi \\lt \\theta \\le \\pi$.

\n

Important

\n

When calculating the final answer in part (iii), you must use whole number values (integers), otherwise the final answer will not be marked as correct.

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 $z^n$.

\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.

\n

Also remember that for this question, arguments of complex numbers lie in the range $-\\pi \\lt \\theta \\le \\pi$.

\n

With the above in mind we can now answer the questions:

\n

a)

\n

#### Modulus

\n

\$\\begin{eqnarray*} |\\var{z1}|&=&\\sqrt{(\\var{a1})^2+(\\var{b1})^2}\\\\ &=& \\var{abs(z1)}\\\\ &=&\\var{ans1} \\end{eqnarray*} \$ to 3 decimal places.

\n

Note 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

#### Argument

\n

{m1}.
Hence we see that:
\$\\begin{eqnarray*} \\arg(\\var{z1}) &=& \\var{arg(z1)}\\\\ &=& \\var{arg1}\\; \\mbox{radians} \\end{eqnarray*} \$ to 3 decimal places.

\n

We have $\\arg((\\var{z1})^{\\var{n2}})=\\var{n2}\\times \\var{arg(z1)} = \\var{n2*arg(z1)}$ radians.

\n

Hence 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*} \$

\n

b)

\n

#### Modulus

\n

\$\\begin{eqnarray*} |\\var{z2}|&=&\\sqrt{(\\var{a2})^2+(\\var{b2})^2}\\\\ &=& \\var{abs(z2)}\\\\ &=&\\var{ans2} \\end{eqnarray*} \$ to 3 decimal places.

\n

Note 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

#### Argument

\n

{m2}.
Hence we see that:
\$\\begin{eqnarray*} \\arg(\\var{z2}) &=& \\var{arg(z2)}\\\\ &=& \\var{arg2}\\; \\mbox{radians} \\end{eqnarray*} \$ to 3 decimal places.

\n

We have $\\arg((\\var{z2})^{\\var{n4}})=\\var{n4}\\times \\var{arg(z2)} = \\var{n4*arg(z2)}$ radians.

\n

Hence 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*} \$

\n

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Find the modulus and argument of $\\var{z1}$  correct to 3 decimal places.

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(i) $|\\var{z1}|\\;=\\;$ [[0]], correct to 3 decimal places.

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(ii) $\\arg(\\var{z1})\\;=\\;$[[1]] radians, correct to 3 decimal places.

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Hence find:

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(iii) $(\\var{z1})^{\\var{n2}}\\;=\\;$[[2]]

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Input as a complex number, with integer values for the real and imaginary parts.

"}, "preventleave": true, "startpassword": "Y1Ai"}, "timing": {"allowPause": false, "timeout": {"action": "warn", "message": "

The complex numbers assessment has now ended.

"}, "timedwarning": {"action": "warn", "message": "

You have 5 minutes left to complete this complex numbers assessment.

WM104 in-class assessment: part (i) - complex numbers

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Content assessed :

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• complex arithmetic
• \n
• argument and modulus of complex numbers
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• De Moivre's theorem
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This complex numbers in-class assesment counts 20% towards your final maths grade for the WM104 module.

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Note, although questions are randomised for each student, all questions test the same learning outcomes at the same level for each student.

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If you have any questions during the test, please put up your hand to alert the invigilator that you need attention.

", "feedbackmessages": []}, "contributors": [{"name": "Shaheen Charlwood", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1819/"}], "extensions": [], "custom_part_types": [], "resources": []}