Find the value of 4 + 5 - 1 2 + i 3 2 334 + 3 - 1 2 + i 3 2 365 .

As, we know that ω = - 1 + 3 i 2 and 1 + ω + ω 2 = 0 & ω 3 = 1 Now, 4 + 5 - 1 2 + i 3 2 334 + 3 - 1 2 + i 3 2 365 = 4 + 5 ω 334 + 3 ω 365 = 4 + 5 . ω 3 × 111 + 1 + 3 ω 3 × 121 + 2 = 4 + 5 ω 3 111 ω + 3 ω 3 121 ω 2 = 4 + 5 ω + 3 ω 2 = 3 1 + ω + ω 2 + 1 + 2 ω = 1 + 2 ω = 1 + 2 - 1 + 3 i 2 = 3 i

Let s , t , r be non-zero complex numbers and L be the set of solutions z = x + i y x , y ∈ ℝ , i = - 1 of the equation s z + t z ¯ + r = 0 , where z ¯ = x - i y . Then, which of the following statement(s) is (are) TRUE?

If L has more than one element, then L has infinitely many elements

If s = t , then L has infinitely many elements

E M B I B E

Gamma Question Bank for Engineering Mathematics>Chapter 5 - Complex Numbers>EXERCISE>Q 7

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If $ω$ is an imaginary cube root of unity, then find the value of $(1 + ω) (1 + ω^{2}) (1 + ω^{3}) (1 + ω^{4}) (1 + ω^{5}) . . . . . . . (1 + ω^{3 n})$ .

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Important Questions on Complex Numbers

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Gamma Question Bank for Engineering Mathematics>Chapter 5 - Complex Numbers>EXERCISE>Q 8

If

x^{2} + x + 1 = 0

, then find the value of

\sum_{r = 1}^{27} {(x^{r} + \frac{1}{x^{r}})}^{2}

.

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Gamma Question Bank for Engineering Mathematics>Chapter 5 - Complex Numbers>EXERCISE>Q 9

Find the value of

4 + 5 {(- \frac{1}{2} + \frac{i \sqrt{3}}{2})}^{334} + 3 {(- \frac{1}{2} + \frac{i \sqrt{3}}{2})}^{365}

.

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Gamma Question Bank for Engineering Mathematics>Chapter 5 - Complex Numbers>EXERCISE>Q 10

If

α = \cos \frac{2 π}{5} + i \sin \frac{2 π}{5}

, then find the value of

α + α^{2} + α^{3} + α^{4}

.

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Gamma Question Bank for Engineering Mathematics>Chapter 5 - Complex Numbers>EXERCISE>Q 11

If

|z| = 2,

then the points representing the complex numbers

- 1 + 5 z

will lie on a

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Gamma Question Bank for Engineering Mathematics>Chapter 5 - Complex Numbers>EXERCISE>Q 12

If

z_{1}, z_{2}, z_{3}, z_{4}

are the affixes of four points in the Argand plane and

z

is the affix of a point such that

|z - z_{1}| = |z - z_{2}| = |z - z_{3}| = |z - z_{4}|,

then

z_{1}, z_{2}, z_{3}, z_{4}

are

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Gamma Question Bank for Engineering Mathematics>Chapter 5 - Complex Numbers>EXERCISE>Q 13

If

A (z_{1})

and

B (z_{2})

are two fixed points in the argand plane the locus of point

P (z)

satisfying

|z - z_{1}| + |z - z_{2}| = |z_{1} - z_{2}|

, is

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Gamma Question Bank for Engineering Mathematics>Chapter 5 - Complex Numbers>EXERCISE>Q 14

The vertices of a square are

z_{1}, z_{2}, z_{3}

and

z_{4}

taken in the anticlockwise order, then

z_{3} =

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Gamma Question Bank for Engineering Mathematics>Chapter 5 - Complex Numbers>EXERCISE>Q 16

A rectangle is constructed in the complex plane with its sides parallel to the axes and its centre is situated at the origin. If one of the vertices of the rectangle is