Question 11 Mark
Simplify the following and express in the form $a+i b$ : $\frac{4+3 i}{1-i}$
Answer$
\begin{aligned}
\frac{4+3 i}{1-i} & =\frac{(4+3 i)(1+i)}{(1-i)(1+i)} \\
& =\frac{4+4 i+3 i+3 i^2}{1-i^2} \\
& =\frac{4+7 i+3(-1)}{1-(-1)} \quad \ldots\left[\because i^2=-1\right] \\
& =\frac{1+7 i}{2}=\frac{1}{2}+\frac{7}{2} i
\end{aligned}
$
View full question & answer→Question 21 Mark
Simplify the following and express in the form $a+i b$ : $(1+3 i) 2(3+i)$
Answer$(1+3 i)^2(3+i)$
$=\left(1+6 i+9 i^2\right)(3+i)$
$\left.=(1+6 i-9)(3+i) \ldots . . \cdot \because i^2=-1\right]$
$=(-8+6 i)(3+i)$
$=-24-8 i+18 i+6 i^2$
$=-24+10 i+6(-1)$
$=-24+10 i-6$
$=-30+10 i$
View full question & answer→Question 31 Mark
Simplify the following and express in the form $a+i b$ : $\frac{5}{2} i (-4-3 i )$
Answer$
\begin{aligned}
& \frac{5}{2} i(-4-3 i) \\
= & \frac{5}{2}\left(-4 i-3 i^2\right) \\
= & \frac{5}{2}[-4 i-3(-1)] \ldots \ldots . \cdot\left[ i ^2=-1\right] \\
= & \frac{5}{2}(3-4 i) \\
= & \frac{15}{2}-10 i
\end{aligned}
$
View full question & answer→Question 41 Mark
Simplify the following and express in the form $a+i b$ : $(2+3 i )(1-4 i )$
Answer$(2+3 i)(1-4 i)=2-8 i+3 i-12 i^2$
$=2-5 i-12(-1) \ldots . .\left[\because i^2=-1\right]$
$=14-5 i$
View full question & answer→Question 51 Mark
Simplify the following and express in the form $a+i b$ : $\left(2 i^3\right)^2$
Answer$
\begin{aligned}
& \left.2 i^3\right)^2 \\
= & 4 i^6 \\
= & 4\left(i^2\right)^3 \\
= & 4(-1)^3 \ldots \ldots\left[\because i^2=-1\right] \\
= & -4 \\
= & -4+0 i
\end{aligned}
$
View full question & answer→Question 61 Mark
Simplify the following and express in the form $a+i b$ : $3+\sqrt{ -64}$
Answer$3+\sqrt{ -64}$
$=3+\sqrt{ 64} \cdot \sqrt{ -1}$
$=3+8 i $
View full question & answer→Question 71 Mark
Find the value of $\sqrt{ }-3 \times \sqrt{ }-6$.
Answer$\sqrt{ -3} \times \sqrt{ }-6=\sqrt{ 3} \times \sqrt{ -1}+\sqrt{ 6} \times \sqrt{ -1}$
$=\sqrt{ 3 i} \times \sqrt{ 6 i}$
$=\sqrt{ 18} i^2$
$=-3 \sqrt{ 2} \ldots \ldots\left[\because i^2=-1\right]$
View full question & answer→Question 81 Mark
Find the value of $\frac{i^{592}+i^{590}+i^{588}+i^{586}+i^{584}}{i^{582}+i^{580}+i^{578}+i^{576}+i^{574}}$
Answer$ \frac{i^{592}+i^{590}+i^{588}+i^{586}+i^{584}}{i^{582}+i^{580}+i^{578}+i^{576}+i^{574}}$
$=\frac{i^{10}\left(i^{582}+i^{580}+i^{578}+i^{576}+i^{574}\right)}{i^{582}+i^{580}+i^{578}+i^{576}+i^{574}}$
$=i^{10}$
$=\left(i^4\right)^2 \cdot i^2$
$=(1)^2(-1)$
$=-1 $
View full question & answer→Question 91 Mark
If $\omega$ is a complex cube root of unity, find the value of : $(1+\omega)\left(1+\omega^2\right)\left(1+\omega^4\right)\left(1+\omega^8\right)$
Answer$\begin{aligned} & \text { (v) }(1+\omega)\left(1+\omega^2\right)\left(1+\omega^4\right)\left(1+\omega^8\right) \\ & =(1+\omega)\left(1+\omega^2\right)(1+\omega)\left(1+\omega^2\right) \ldots . . .\left[\omega^3=1, \omega^4=\omega\right] \\ & =\left(-\omega^2\right)(-\omega)\left(-\omega^2\right)(-\omega) \\ & =\omega^6 \\ & =\left(\omega^3\right)^2 \\ & =(1)^2 \\ & =1\end{aligned}$
View full question & answer→Question 101 Mark
If $\omega$ is a complex cube root of unity, find the value of : $\left(1-w-\omega^2\right)^3+\left(1-w+\omega^2\right)^3$
Answer$\begin{aligned} & \text { (iv) }\left(1-\omega-\omega^2\right)^3+\left(1-\omega+\omega^2\right)^3 \\ & =\left[1-\left(\omega+\omega^2\right)\right]^3+\left[\left(1+\omega^2\right)-\omega\right]^3 \\ & =[1-(-1)]^3+(-\omega-\omega)^3 \\ & =2^3+(-2 \omega)^3 \\ & =8-8 \omega^3 \\ & =8-8(1) \\ & =0\end{aligned}$
View full question & answer→Question 111 Mark
If $\omega$ is a complex cube root of unity, find the value of : $\left(1+\omega^2\right)^3$
Answer$
\begin{aligned}
& \left(1+\omega^2\right)^3 \\
= & (-\omega)^3 \\
= & -\omega^3 \\
= & -1
\end{aligned}
$
View full question & answer→Question 121 Mark
If $\omega$ is a complex cube root of unity, find the value of : $\omega^2+\omega^3+\omega^4$
Answer$
\begin{aligned}
& \omega^2+\omega^3+\omega^4 \\
= & \omega^2\left(1+\omega+\omega^2\right) \\
= & \omega^2(0) \\
= & 0
\end{aligned}
$
View full question & answer→Question 131 Mark
If $\omega$ is a complex cube root of unity, find the value of : $\omega+\frac{1}{\omega}$
Answer$\omega$ is the complex cube root of unity.
$
\therefore \omega^3=1 \text { and } 1+\omega+\omega^2=0
$
Also, $1+\omega^2=-\omega, 1+\omega=-\omega^2$ and $\omega+\omega^2=-1$
$
\begin{aligned}
& \omega+\frac{1}{\omega} \\
= & \frac{\omega^2+1}{\omega} \\
= & \frac{-\omega}{\omega} \\
= & -1
\end{aligned}
$
View full question & answer→Question 141 Mark
If $\omega$ is a complex cube root of unity, show that : $\frac{\left(\mathbf{a}+\mathbf{b} \omega+\mathbf{c} \omega^2\right)}{\mathbf{c}+\mathbf{a} \omega+\mathbf{b} \omega^2}=\omega^2$
Answer\begin{aligned}
& \text { L.H.S. }=\frac{\left(\mathbf{a}+\mathbf{b} \omega+\mathbf{c} \omega^2\right)}{\mathbf{c}+\mathbf{a} \omega+\mathbf{b} \omega^2} \\
& =\frac{a \omega^3+b \omega^4+c \omega^2}{c+a \omega+b \omega^2} \ldots \ldots . .\left[\omega^3=1, \omega^4=\omega\right] \\
& =\frac{\omega^2\left(c+a \omega+b \omega^2\right)}{c+a \omega+b \omega^2} \\
& =\omega^2 \\
& =\text { R.H.S. }
\end{aligned}
View full question & answer→Question 151 Mark
If $\omega$ is a complex cube root of unity, show that : $\left(2+\omega+\omega^2\right)^3-\left(1-3 \omega+\omega^2\right)^3=65$
Answer\begin{aligned}
& \text { L.H.S. }=\left(2+\omega+\omega^2\right)^3-\left(1-3 \omega+\omega^2\right)^3 \\
& =\left[2+\left(\omega+\omega^2\right)\right]^3-\left[-3 \omega+\left(1+\omega^2\right)\right]^3 \\
& =(2-1)^3-(-3 \omega-\omega)^3 \\
& =13-(-4 \omega)^3 \\
& =1+64 \omega^3 \\
& =1+64(1) \\
& =65 \\
& =\text { R.H.S. }
\end{aligned}
View full question & answer→Question 161 Mark
If $\omega$ is a complex cube root of unity, show that : $(2-\omega)\left(2-\omega^2\right)=7$
Answer$\omega$ is the complex cube root of unity.
$
\therefore \omega^3=1 \text { and } 1+\omega+\omega^2=0
$
Also, $1+\omega^2=-\omega, 1+\omega=-\omega^2$ and $\omega+\omega^2=-1$
$
\begin{aligned}
& \text { L.H.S. }=(2-\omega)\left(2-\omega^2\right) \\
& =4-2 \omega^2-2 \omega+\omega^3 \\
& =4-2\left(\omega^2+\omega\right)+1 \\
& =4-2(-1)+1 \\
& =4+2+1 \\
& =7 \\
& =\text { R.H.S. }
\end{aligned}
$
View full question & answer→Question 171 Mark
Find the value of : $i+i^2+i^3+i^4$
Answer$\begin{aligned} & i+i^2+i^3+i^4 \\ & =i+i^2+i^2 \cdot i+i^4 \\ & =i-1-i+1\left[\because i^2=-1, i^4=1\right] \\ & =0\end{aligned}$
View full question & answer→Question 181 Mark
Find the value of : $i^{49}+i^{68}+i^{89}+i^{110}$
Answer$\begin{aligned} & i^{49}+i^{68}+i^{89}+i^{110} \\ & =\left(i^4\right)^{12} \cdot i+\left(i^4\right)^{17}+\left(i^4\right)^{22} \cdot i+\left(i^4\right)^{27} \cdot i^2 \\ & \left.=(1)^{12} \cdot i+(1)^{17}+(1)^{22} \cdot i+(1)^{27}(-1) \ldots \ldots . . \because i^4=1, i^2=-1\right] \\ & =\mathrm{i}+1+\mathrm{i}-1 \\ & =2 \mathrm{i} \\ & \end{aligned}$
View full question & answer→Question 191 Mark
Show that $1+\mathrm{i}^{10}+\mathrm{i}^{20}+\mathrm{i}^{30}$ is a real number.
Answer$\begin{aligned} & 1+i^{10}+i^{20}+i^{30} \\ & =1+\left(i^4\right)^2 \cdot i^2+\left(i^4\right)^5+\left(i^4\right)^7 \cdot i^2 \\ & =1+(1)^2(-1)+(1)^5+(1)^7(-1)\left[\because i^4=1, i^2=-1\right] \\ & =1-1+1-1 \\ & =0, \text { which is a real number. }\end{aligned}$
View full question & answer→Question 201 Mark
Evaluate the following : $i^{30}+i^{40}+i^{50}+i^{60}$
Answer$i^{30}+i^{40}+i^{50}+i^{60}$
$
\begin{aligned}
& =\left(i^4\right)^7 i^2+\left(i^4\right)^{10}+\left(i^4\right)^{12} i^2+\left(i^4\right)^{15} \\
& =(1)^7(-1)+(1)^{10}+(1)^{12}(-1)+(1)^{15} \\
& =-1+1-1+1 \\
& =0
\end{aligned}
$
View full question & answer→Question 211 Mark
Evaluate the following : $\frac{1}{i^{58}}$
Answer$\frac{1}{i^{88}}=\frac{1}{\left(i^4\right)^{14} \cdot i^2}=\frac{1}{(1)^{14}(-1)}=-1$
View full question & answer→Question 221 Mark
Evaluate the following : $\mathrm{i}^{403}$
Answer$i^{403}=\left(i^4\right)^{100}\left(i^2\right) i=(1)^{100}(-1) i=-i$
View full question & answer→Question 231 Mark
Evaluate the following : $\mathrm{i}^{116}$
Answer$i^{116}=\left(i^4\right)^{29}=(1)^{29}=1$
View full question & answer→Question 241 Mark
Evaluate the following : $\mathrm{i}^{93}$
Answer$i^{93}=\left(i^4\right)^{23} \cdot i=(1)^{23} \cdot i=i$
View full question & answer→Question 251 Mark
Evaluate the following : $\mathrm{i}^{888}$
Answer$i^{888}=\left(i^4\right)^{222}=(1)^{222}=1$
View full question & answer→Question 261 Mark
Evaluate the following : $\mathrm{i}^{35}$
AnswerWe know that, $i^2=-1, i^3=-i, i^4=1$
$\mathrm{i}^{35}=\left(\mathrm{i}^4\right)^8\left(\mathrm{i}^2\right) \mathrm{i}=(1)^8(-1) \mathrm{i}=-\mathrm{i}$
View full question & answer→Question 271 Mark
Express the following in the form of $a+i b, a, b \in R, i=\sqrt{ }-1$. State the values of $a$ and $b$ : $(2+3 \mathrm{i})(2-3 \mathrm{i})$
View full question & answer→Question 281 Mark
Express the following in the form of $a+i b, a, b \in R, i=\sqrt{ }-1$. State the values of $a$ and $b$ : $(1+2 i)(-2+i)$
View full question & answer→Question 291 Mark
Write the conjugates of the following complex numbers : $\sqrt{ } 2+\sqrt{ } 3 \mathrm{i}$
AnswerConjugate of $\sqrt{ } 2+\sqrt{ } 3 i$ is $\sqrt{ } 2-\sqrt{ } 3 i$
View full question & answer→Question 301 Mark
Write the conjugates of the following complex numbers : \$\sqrt{5}-\mathrm{i}$
AnswerConjugate of $\sqrt{5}-\mathrm{i}$ is $\sqrt{5}+\mathrm{i}$
View full question & answer→Question 311 Mark
Write the conjugates of the following complex numbers : $5 \mathrm{i}$
AnswerConjugate of $5 \mathrm{i}$ is $-5 \mathrm{i}$
View full question & answer→Question 321 Mark
Write the conjugates of the following complex numbers : $-\sqrt{ }-5$
Answer$-\sqrt{ }-5=-\sqrt{ } 5 \times \sqrt{ }-1=-\sqrt{5} \mathrm{i}$
Conjugate of $-\sqrt{ }-5$ is $\sqrt{ } 5 \mathrm{i}$
View full question & answer→Question 331 Mark
Write the conjugates of the following complex numbers : $-\sqrt{ } 5-\sqrt{ } 7 i$
AnswerConjugate of $(-\sqrt{ } 5-\sqrt{7} \mathrm{i})$ is $(-\sqrt{ } 5+\sqrt{7} \mathrm{i})$
View full question & answer→Question 341 Mark
Write the conjugates of the following complex numbers : $3-\mathrm{i}$
AnswerConjugate of $(3-\mathrm{i})$ is $(3+\mathrm{i})$
View full question & answer→Question 351 Mark
Write the conjugates of the following complex numbers : $3+\mathrm{i}$
AnswerConjugate of $(3+i)$ is $(3-i)$
View full question & answer→