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Complex Numbers and Quadratic Equations

Short Answer Type

31.

Find the value of the following :

$space space space space space space space space space space space space space space space open square brackets straight i to the power of 19 plus open parentheses 1 over straight i close parentheses to the power of 25 close square brackets squared$

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32.

Find the value of the following :

$space space space space space space space space space space space space space space space space space space space open parentheses 1 space plus straight i close parentheses to the power of 4 open parentheses 1 plus 1 over straight i close parentheses to the power of 4$

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33.

Simplify

$space space space space space space space space space space space space space space space space fraction numerator 1 over denominator left parenthesis 1 minus straight i right parenthesis squared end fraction minus fraction numerator 1 over denominator left parenthesis 1 plus straight i right parenthesis squared end fraction$

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34.

Simplify

$space space space space space space space space sum from straight n space equals space 1 to 200 of straight i to the power of straight n$

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35.

Simplify

$space space space space space space space space space space space space fraction numerator left parenthesis 1 space minus straight i right parenthesis cubed over denominator 1 minus straight i cubed end fraction equals fraction numerator left parenthesis 1 minus straight i right parenthesis cubed over denominator straight i cubed minus straight i cubed end fraction$

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36.

Simplify

$space space space space space space space space space space space space space space space space fraction numerator straight i to the power of 4 straight n space plus space 1 end exponent minus straight i to the power of 4 straight n minus 1 end exponent over denominator 2 end fraction$

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37.

find the smallest positive integer n for which $space space space space space open parentheses fraction numerator 1 plus straight i over denominator 1 minus straight i end fraction close parentheses to the power of straight n equals space 1$

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38.

If $space space space space open parentheses fraction numerator 1 plus straight i over denominator 1 minus straight i end fraction close parentheses to the power of straight m equals 1$ , then find the least positive integral value of m.

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39.
Find the modulus of $space space space space space space space space fraction numerator 1 plus straight i over denominator 1 minus straight i end fraction minus fraction numerator 1 minus straight i over denominator 1 plus straight i end fraction$ .

∴ z = 2i = 0 + 2i

∴

Hence,

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40.

If (x + iy)³= u + iv, then show that $space space space space space straight u over straight x plus straight v over straight y equals 4 left parenthesis straight x squared minus straight y squared right parenthesis$

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Short Answer Type

39. Find the modulus of .

39.
Find the modulus of $space space space space space space space space fraction numerator 1 plus straight i over denominator 1 minus straight i end fraction minus fraction numerator 1 minus straight i over denominator 1 plus straight i end fraction$ .