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Maths Solve the following Question.(1 Marks)

Complex Numbers · Maharashtra Board STD 11 (MSBSHSE - Semi English Medium). 63 questions.

Questions · Page 1 of 2

Solve the following Question.(1 Marks)

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Question 11 Mark
Simplify
$\frac{ i ^{238}+ i ^{236}+ i ^{234}+ i ^{232}+ i ^{2290}}{ i ^{2228}+ i ^{226}+ i ^{224}+ i ^{222}+ i ^{2200}}$
Answer
$\frac{i^{238}+i^{236}+i^{234}+i^{232}+i^{230}}{i^{228}+i^{226}+i^{224}+i^{222}+i^{220}}$
$=\frac{i^{10}\left(i^{228}+i^{226}+i^{224}+i^{222}+i^{230}\right)}{i^{228}+i^{226}+i^{224}+i^{222}+i^{220}}$
$=i^{10}=\left(i^4\right)^2 \cdot i^2=(1)^2(-1)$
$=-1$
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Question 21 Mark
Simplify
$\left(i^{65}+\frac{1}{i^{145}}\right)$
Answer
$\left(i^{65}+\frac{1}{i^{145}}\right)$
$=\left[\left(i^4\right)^{16} \cdot i+\frac{1}{\left(i^4\right)^{36} \cdot i}\right]=i+\frac{1}{i}=\frac{i^2+1}{i}$
$=\frac{-1+1}{i}$
$=0$
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Question 31 Mark
Simplify
$\frac{ i ^{29}+ i ^{39}+ i ^{49}}{ i ^{30}+ i ^{40}+ i ^{50}}$
Answer
$\frac{i^{29}+i^{39}+i^{49}}{i^{30}+i^{49}+i^{30}}$
$=\frac{i^{29}+i^{39}+i^{49}}{i\left(i^{29}+i^{39}+i^{49}\right)}=\frac{1}{i}=\frac{i}{i^2}=\frac{i}{-1}$
$=- i$
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Question 41 Mark
Evaluate:
$i^{131}+i^{49}$
Answer
$i^{131}+i^{49}$
$=\left(i^4\right)^{32} \cdot i^3+\left(i^4\right)^{12} \cdot i$
$=(1)^{32}(-i)+(1)^{12} \cdot i$
$=-i+i$
$=0$
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Question 51 Mark
Evaluate:
$\left(1-i+i^2\right)^{-15}$
Answer
$\left(1-i+i^2\right)^{-15}=(1-i-1)^{-15}$
$=(-i)^{-15}=\frac{1}{(-i)^{15}}$
$=\frac{-1}{\left( i ^4\right)^3 \cdot i ^3}=\frac{-1}{(1)^3(- i )}$
$=\frac{1}{ i }=\frac{ i }{ i ^2}=\frac{ i }{-1}=- i $
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Question 71 Mark
Simplify the following and express in the form a + ib:

$\frac{4+3 i}{1-i}$

Answer
$\begin{aligned} \frac{4+3 i}{1-i} & =\frac{(4+3 i)(1+i)}{(1-i)(1+i)} \\ & =\frac{4+4 i+3 i+3 i^2}{1-i^2} \\ & =\frac{4+7 i+3(-1)}{1-(-1)} \quad \ldots\left[\because i^2=-1\right]\end{aligned}$

$=\frac{1+7 i}{2}=\frac{1}{2}+\frac{7}{2} i$

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Question 81 Mark
Simplify the following and express in the form $a + ib:$
$\frac{5}{2} i(-4-3 i)$
Answer
$\frac{5}{2} i(-4-3 i)=\frac{5}{2}\left(-4 i-3 i^2\right)$
$=\frac{5}{2}[-4 i-3(-1)] \ldots\left[\because i^2=-1\right]$
$=\frac{5}{2}(3-4 i)$
$=\frac{15}{2}-10 i $
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Question 91 Mark
Simplify the following and express in the form $a + ib:$
$(2 + 3i) (1 – 4i)$
Answer
$(2+3 i)(1-4 i)$
$=2-8 i+3 i-12 i^2$
$=2-5 i-12(-1) \ldots . .\left[\because i^2=-1\right]$
$=14-5 i$
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Question 101 Mark
Simplify the following and express in the form a + ib:
$\left(2 i^3\right)^2$
Answer
$\left(2 i^3\right)^2$
$=4 i^6$
$=4\left(i^2\right)^3$
$=4(-1)^3$
$=-4 \ldots . .\left[\because i^2=-1\right]$
$=-4+0 i$
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Question 111 Mark
Simplify the following and express in the form a + ib: 3 + √-64
Answer
3 + √-64 = 3 + √64 √-1 = 3 + 8i
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Question 121 Mark
Find the equation in cartesian coordinates of the locus of $z$ if $: |z – 5 + 6i| = 5$
Answer
Let $z = x + iy$
$|z-5+6 i|=5$
$|x+i y-5+6 i|=5$
$|(x-5)+i(y+6)|=5$
$\sqrt{(x-5)^2+(y+6)^2}=5$
$\therefore(x-5)^2+(y+6)^2=25$
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Question 131 Mark
Find the equation in cartesian coordinates of the locus of $z$ if $: |z – 3| = 2$
Answer
Let $z = x + iy$
$|z-3|=2$
$|x+i y-3|=2$
$|(x-3)+i y|=2$
$\sqrt{(x-3)^2+y^2}=2$
$\therefore(x-3)^2+y^2=4$
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Question 141 Mark
Find the equation in cartesian coordinates of the locus of $z$ if $: |z| = 10$
Answer
Let $z=x+i y$
$|z|=10$
$|x+i y|=10$
$\sqrt{x^2+y^2}=10$
$\therefore x^2+y^2=100$
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Question 151 Mark
If ω is the complex cube root of unity, find the value of

$(1+\omega)\left(1+\omega^2\right)\left(1+\omega^4\right)\left(1+\omega^8\right)$

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Question 161 Mark
If ω is the complex cube root of unity, find the value of

$\left(1-\omega-\omega^2\right)^3+\left(1-\omega+\omega^2\right)^3$

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Question 201 Mark
Find the value of : $\omega^{-105}$
Answer
$amp; \omega^3=1$ $\omega^{-105}=\left(\omega^3\right)^{-35}=(1)^{-35}=\frac{1}{(1)^{35}}=1$
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Question 211 Mark
Find the value of : $\omega^{-30}$
Answer
$amp; \omega^3=1$ $amp; \omega^{-30}=\left(\omega^3\right)^{-10}=(1)^{-10}=\frac{1}{(1)^{10}}=1$
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Question 221 Mark
Find the value of : $\omega^{21}$
Answer
$amp; \omega^3=1$ $amp; \omega^{21}=\left(\omega^3\right)^7=(1)^7=1$
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Question 231 Mark
Find the value of : $\omega^{18}$
Answer
$amp; \omega^3=1$ $amp; \omega^{18}=\left(\omega^3\right)^6=(1)^6=1$
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Question 241 Mark
$z_1=1+i_1 z_2=2-3 i$, verify the following:

$\overline{\left(\frac{ z _1}{ z _2}\right)}=\frac{\overline{ z }_1}{\overline{ z }_2}$

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Question 251 Mark
$z_1=1+i_1 z_2=2-3 i$, verify the following:

$\overline{Z_1 \cdot Z_2}=\overline{Z_1} \cdot \overline{Z_2}$

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Question 281 Mark
For $z = 2 + 3i,$ verify the following:
$z-\bar{z}=6 i$
Answer
$z-\bar{z}=(2+3 i)-(2-3 i)$
$=2+3 i-2+3 i$
$=6 i$
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Question 291 Mark
For $z = 2 + 3i,$ verify the following:
$(z+\bar{z})$ is real
Answer
$(z+\bar{z})=(2+3 i)+(2-3 i)$
$=2+3 i+2-3 i$
$=4, \text { which is a real number. }$
$\therefore z+z \text { is real. }$
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Question 301 Mark
For $z = 2 + 3i,$ verify the following:
$\overline{z \bar{z}}=|z|^2$
Answer
$z \bar{z}=(2+3 i)(2-3 i)$
$=4-9 i^2$
$=4-9(-1) \ldots \ldots\left[\because i^2=-1\right]$
$=13$
$|z|^2=\left(\sqrt{2^2+3^2}\right)^2$
$=2^2+3^2$
$=4+9$
$=13$
$\therefore \overline{z \bar{z}}=|z|^2$
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Question 311 Mark
For $z = 2 + 3i,$ verify the following:
$\overline{(\bar{z})}=z$
Answer
$z=2+3 i$
$\therefore \bar{z}=2-3 i$
$\therefore \overline{\bar{z}}=2+3 i = z$
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Question 351 Mark
Express the following numbers in the form x + iy:

$7\left(\cos \left(-\frac{5 \pi}{6}\right)+i \sin \left(-\frac{5 \pi}{6}\right)\right)$

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Question 361 Mark
Express the following numbers in the form x + iy:

$\sqrt{2} \cdot\left(\cos \frac{7 \pi}{4}+i \sin \frac{7 \pi}{4}\right)$

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Question 371 Mark
Express the following numbers in the form x + iy:

$\sqrt{3}\left(\cos \frac{\pi}{6}+i \sin \frac{\pi}{6}\right)$

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Question 411 Mark
Simplify: $\frac{ i ^{592}+ i ^{580}+ i ^{588}+ i ^{586}+ i ^{584}}{ i ^{582}+ i ^{580}+ i ^{578}+ i ^{576}+ i ^{574}}$
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Question 421 Mark
Find the value of

$i+i^2+i^3+i^4$

Answer
$\begin{aligned}
i+i^2+i^3+i^4 & =i+i^2+i^2 \cdot i+i^4 \\
& =i-1-i+1
\end{aligned}$
$\ldots\left[\because i^2=-1, i^4=1\right]$
$=0$
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Question 431 Mark
Find the value of$i^{49}+i^{68}+i^{89}+i^{110}$
Answer
$ i^{49}+i^{68}+i^{89}+i^{110}$
$=\left(i^4\right)^{12} \cdot i+\left(i^4\right)^{17}+\left(i^4\right)^{22} \cdot i+\left(i^4\right)^{27} \cdot i^2$
$=(1)^{12} \cdot i+(1)^{17}+(1)^{22} \cdot i+(1)^{27}(-1)$
$\quad \ldots\left[\because i^4=1, i^2=-1\right]$
$=i+1+i-1=2 i$
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Question 451 Mark
Evaluate the following:$i^{30}+i^{40}+i^{50}+i^{60}$
Answer
We know that, $i^2=-1, i^3=-i, i^4=1$
$ i^{30}+i^{40}+i^{50}+i^{60}$
$=\left(i^4\right)^7 i^2+\left(i^4\right)^{10}+\left(i^4\right)^{12} i^2+\left(i^4\right)^{15}$
$=(1)^7(-1)+(1)^{10}+(1)^{12}(-1)+(1)^{15}$
$=-1+1-1+1=0$
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Question 461 Mark
Evaluate the following:

$i ^{-888}$

Answer
We know that, $i^2=-1, i^3=-i, i^4=1$

$\mathrm{i}^{-888}=\left(\mathrm{i}^4\right)^{-222}=(1)^{-222}=\frac{1}{(1)^{222}}=1$

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Question 471 Mark
Evaluate the following:

$\frac{1}{i^{58}}$

Answer
We know that, $i^2=-1, i^3=-i, i^4=1$

$\frac{1}{\mathrm{i}^{58}}=\frac{1}{\left(\mathrm{i}^4\right)^{14} \mathrm{i}^2}=\frac{1}{(1)^{14}(-1)}=-1$

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Question 481 Mark
Evaluate the following:

$i^{403}$

Answer
We know that, $i^2=-1, i^3=-i, i^4=1$

$i^{403}=\left(i^4\right)^{100}\left(i^2\right) i=(1)^{100}(-1) i=-i$

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Question 491 Mark
Evaluate the following:

$i ^{116}$

Answer
We know that, $i^2=-1, i^3=-i, i^4=1$

$i^{116}=\left(i^4\right)^{29}=(1)^{29}=1$

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Question 501 Mark
Evaluate the following:

$i^{93}$

Answer
We know that, $i^2=-1, i^3=-i, i^4=1$

$\mathrm{i}^{93}=\left(\mathrm{i}^4\right)^{23} \cdot \mathrm{i}=(1)^{23} \cdot \mathrm{i}=\mathrm{i}$

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