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Complex Numbers and Quadratic Equations

Short Answer Type

11.

Find the multiplicative inverse of each of the complex number.

4 - 3i

175 Views

12.

Find the multiplicative inverse of each of the complex number.

$space space space space space space space space space space space space square root of 5 plus space 3 straight i$

126 Views

13.

Express the following in a + ib form.

$space space space space space space space space space space space space space space space space space space space space fraction numerator left parenthesis 3 space plus space straight i square root of 5 right parenthesis space left parenthesis 3 minus straight i square root of 5 right parenthesis end root over denominator left parenthesis square root of 3 plus square root of 2 straight i end root right parenthesis space minus space left parenthesis square root of 3 minus straight i square root of 2 right parenthesis end root end fraction$

120 Views

14.

Find the modulus and the arguments of each of the complex numbers.

$space space space space space space space space space space minus 1 minus straight i square root of 3$

132 Views

15.
Convert each of the complex numbers given in the Polar form.

$space space space space space space space space space space space space space square root of 3 plus space straight i$

Here, z = $space space space space space space space square root of 3 space end root plus space straight i$
Let z = $space space space space space space space space square root of 3 plus space straight i space equals space straight r space left parenthesis cos space straight theta space plus space straight i space sin space straight theta right parenthesis$ ...(1)

$space space space space space space space space space space space space space space rightwards double arrow$ $space space space space space space space space space space space space space space space space straight r space cos space straight theta space equals space square root of 3$ ...(ii)
and $space space space space space space space space space space space space space space space straight r space sin space straight theta space equals space 1$ ...(iii)

Squaring and adding (ii) and (iii), we get

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$rightwards double arrow$ r²= 4 $rightwards double arrow$ r = 2

From (ii), $space space space space space space space space space space space space space space space space space space 2 space cos space straight theta space.. square root of 3 space space space space space space space space space space space space space rightwards double arrow space space space space space space space cos space straight theta space space equals space fraction numerator square root of 3 over denominator 2 end fraction$

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since both $space space space space space sin space straight theta space and space cos space straight theta$ are positive

∴ $space space space theta$ lies in 1st quadrant

∴ $space space space space space space space space space space space space space space space space space space sin space theta space equals space 1 half space equals space sin space 30 degree space equals space sin space straight pi over 6 space space space space space space space space space space space space space space space space space space rightwards double arrow space space space space space space theta space space space equals space straight pi over 6$