1 Marks Question

Question 11 Mark

If $z_1, z_2$ and $z_3 \in C$ then write the value of $\overline{z_1+z_2}$.

Answer

$\overline{z_1+z_2}=\overline{z_1}+\overline{z_2}$

Question 21 Mark

Write the value of $1+i^2+i^4+\ldots\ldots +i^{2 n}.$

Answer

$\quad1+i^2+i^4+i^6+\ldots \ldots+i^{2 n}$
$=1-1+1-1+\ldots \ldots +(-1)^{2 n}$
Clearly the required value depends on the value of $n$.

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Question 31 Mark

Write the product of the complex number $3-2 i$ and its conjugate.

Answer

$9+4=13$

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Question 41 Mark

Find the value of $x$ and $y$ in equation $(3 x-7)+2 i y$ $=-5 y+(5+x) i$.

Answer

$x=-1$ and $y=2$

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Question 51 Mark

Write the expression $\frac{3-\sqrt{-16}}{1-\sqrt{-9}}$ in the form of $a+i b$.

Answer

$\begin{array}{l}\frac{3-\sqrt{-16}}{1-\sqrt{-9}}=\frac{3-4 i}{1-3 i}=\left(\frac{3-4 i}{1-3 i}\right)\left(\frac{1+3 i}{1+3 i}\right) \\ =\frac{3+9 i-4 i-12 i^2}{(1)^2-(3 i)^2} \\ =\frac{15+5 i}{1+9}=\frac{15+5 i}{10} \\ =\frac{3}{2}+\frac{1}{2} i\end{array}$

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Question 61 Mark

If $z_1=2+3 i$ and $z_2=1+2 i$ then, write the value of $\frac{z_1}{z_2}$.

Answer

$\begin{array}{l}\frac{z_1}{z_2}=\frac{2+3 i}{1+2 i}=\frac{(2+3 i)(1-2 i)}{(1+2 i)(1-2 i)} \\ =\frac{2-4 i+3 i-6 i^2}{1-4 i^2}=\frac{2-i+6}{1+4} \\ \because i^2=-1 \\ \frac{8-i}{5}=\frac{8}{5}-\frac{1}{5} i\end{array}$

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Question 71 Mark

Write the condition for which the sum of two complex numbers $x_1+i y_1$ and $x_2+i y_2$ is a pure real number?

Answer

Sum is real if $y_1+y_2=0$

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Question 81 Mark

Write the condition for which the sum of two complex numbers $x_1+i y_1$ and $x_2+i y_2$ is a pure imaginary number?

Answer

The required condition for imaginary number is $x_1+x_2=0$.

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Question 91 Mark