1 Marks Question

Question 11 Mark

Let $z_1 = 2 - i, z_2 = -2 + i.$ Find $\operatorname{Im} \left( {\frac{1}{{{z_1}{z_1}}}} \right)$

Answer

Here $z_1 = 2 - i$ and $z_2 = -2 + i$
$\therefore \overline {{z_1}} = 2 + i$$\frac{1}{{{z_1}\overline {{z_1}} }} = \frac{1}{{(2 - i)(2 + i)}} = \frac{1}{{4 - {i^2}}} = \frac{1}{5}$ $= \frac 1 5 + 0i$
$\therefore \operatorname{Im} \left( {\frac{1}{{{z_1}\overline {{z_1}} }}} \right) = 0$

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Question 21 Mark

Express the complex numbers$ (1 - i)^4$ in the form of $a + ib.$

Answer

$(1 - i)^4 = [(1 - i)^2]^2$
$= (1 + i^2 - 2i)^2$
$= (1 - 1 - 2i)^2 = (-2i)^2$
$ = 4{i^2} = 4 \times - 1 = - 4$

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Question 31 Mark

Express the complex number$\left[ {\left( {\frac{1}{3} + \frac{7}{3}i} \right) + \left( {4 + \frac{1}{3}i} \right)} \right] - \left[ {\frac{{ - 4}}{3} + i} \right]$ in the form of a + ib.

Answer

$\left[ {\left( {\frac{1}{3} + \frac{7}{3}i} \right) + \left( {4 + \frac{1}{3}i} \right)} \right] - \left[ {\frac{{ - 4}}{3} + i} \right]$
$ =\left(\frac13+4+\frac43\right)+\left(\frac73i+\frac13i\;-i\right)$
$\;=\left(\frac13+\frac{4\times3}3+\frac43\right)+\left(\frac73i\;+\frac i3-\frac{3i}3\right)$
$=\left(\frac{1+12+4}3\right)+\left(\frac{7i+i\;-3i}3\right)$
$ = \frac{{17}}{3} + \frac{5}{3}i$

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Question 41 Mark

Express the complex number $\left( {\frac{1}{5} + \frac{2}{5}i} \right) - \left( {4 + \frac{5}{2}i} \right)$ in the form of a + ib.

Answer

$\left( {\frac{1}{5} + \frac{2}{5}i} \right) - \left( {4 + \frac{5}{2}i} \right)$
$ = \frac{1}{5} + \frac{2}{5}i - 4 - \frac{5}{2}i$
$ = \left( {\frac{1}{5} - 4} \right) + \left( {\frac{2}{5} - \frac{5}{2}} \right)i$
$ = \frac{{ - 19}}{5} - \frac{{21}}{{10}}i$

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Question 51 Mark

Express the complex number (1 + i) - (- 1 + i6) in form of a + ib.

Answer

(1 + i) - (-1 + i6)
1 + i + 1 - 6i = 2 - 5i

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Question 61 Mark

Express the complex numbers $3(7 + i7) + i(7 + i7)$ in the form of $a + ib.$

Answer

$3(7 + i7) + i (7 + i7)$
$= 21 + 21i + 7i + 7i^2= 21 + 28i - 7$
$= 14 + 28 i$

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Question 71 Mark

Express the complex numbers $i^{-39}$ in the form of $a + ib.$

Answer

$i^{-39}= i^{4 \times -9-3}$
$= (i^4)^{-9} \times i^{-3}$
$=(1)^{-9} \times i^{-3}$
$=\frac {1}{i^3}$
$=\frac {1}{-i} \times \frac {i}{i} $
$= \frac {-i}{i^2} = \frac {-i}{-1} = i$

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Question 81 Mark

Express the complex numbers $i^9 + i^{19}$ in the form $a + ib.$

Answer

$i^9 + i^{19} = {({i^2})^4} \cdot i + {({i^2})^9} \cdot i$
$ = {( - 1)^4} \cdot i + {( - 1)^9} \cdot i$
$= i - i = 0$

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Question 91 Mark

Find the multiplicative inverse of the complex numbers -i

Answer

M.I. of $ - i = \frac{1}{{ - i}} = \frac{i}{{ - {i^2}}} = \frac{i}{{ - (i)}} = i$

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Question 101 Mark

Express the complex numbers $(5i)\left( { - \frac{3}{5}i} \right)$ in the form a + ib.

Answer

$(5i)\left( { - \frac{3}{5}i} \right) = - 3{i^2} = - 3 \times - 1$$(\because {i^2} = - 1)$

= 3 = 3+0i

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Question 111 Mark

Express the (-i)(2i)$\left (- \frac 18 i\right)^3$ in the form of a + bi:

Answer

(-i)(2i)$\left (- \frac 18 i\right)^3$ = $2i^2 \times \frac 1{8\times 8\times 8} \times i^3$ = $2 \times \frac 1{8\times 8\times 8} \times i^5$
$= \frac 1{256} (i^2)^2$i
$i = \frac {1}{256} i$

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Question 121 Mark

Express the $(-5i)\left(\frac 18 i\right)$ in the form of $a+bi.$

Answer

$(-5i)\left(\frac 18 i\right) = \frac {-5}{8} i^2$
$i^2 = -1$
$= \frac {-5}{8} (-1) = \frac 58 = \frac 58 + i0$

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