if sqrt 3 i n sqrt 3 i nn in n then the least value of n is

If $${\left( {\sqrt 3 + i} \right)^n} = {\left( {\sqrt 3 - i} \right)^n},n \in N$$ then the least value of $$n$$ is

A. 3

B. 4

C. 6

D. None of these

Answer : 6

Solution :
$$\eqalign{ & {\left( {\frac{{\sqrt 3 + i}}{{\sqrt 3 - i}}} \right)^n} = {\left\{ {\frac{1}{i} \cdot \frac{{ - 1 + \sqrt {3}i }}{{ - 1 - \sqrt {3}i }} \cdot \left( { - i} \right)} \right\}^n} \cr & {\left( {\frac{{\sqrt 3 + i}}{{\sqrt 3 - i}}} \right)^n} = {\left\{ {\frac{{\frac{{ - 1 + \sqrt {3}i }}{2}}}{{\frac{{ - 1 - \sqrt {3}i }}{2}}}} \right\}^n} = {\left( { - \frac{1}{\omega }} \right)^n} = 1,\,\,{\text{when }}n = 0,6.....\,\,. \cr} $$

Releted MCQ Question on
Algebra >> Complex Number

Releted Question 1

If the cube roots of unity are $$1,\omega ,{\omega ^2},$$ then the roots of the equation $${\left( {x - 1} \right)^3} + 8 = 0\,\,{\text{are}}$$

A. $$ - 1,1 + 2\omega ,1 + 2{\omega ^2}$$

B. $$ - 1,1 - 2\omega ,1 - 2{\omega ^2}$$

C. $$- 1, - 1, - 1$$

D. none of these

View Answer

Releted Question 2

The smallest positive integer $$n$$ for which $${\left( {\frac{{1 + i}}{{1 - i}}} \right)^n} = 1\,{\text{is}}$$

A. $$n = 8$$

B. $$n = 16$$

C. $$n = 12$$

D. none of these

View Answer

Releted Question 3

The complex numbers $$z = x+ iy$$ which satisfy the equation $$\left| {\frac{{z - 5i}}{{z + 5i}}} \right| = 1$$ lie on

A. the $$x$$ - axis

B. the straight line $$y = 5$$

C. a circle passing through the origin

D. none of these

View Answer

Releted Question 4

If $$z = {\left( {\frac{{\sqrt 3 }}{2} + \frac{i}{2}} \right)^5} + {\left( {\frac{{\sqrt 3 }}{2} - \frac{i}{2}} \right)^5},\,{\text{then}}$$

A. $${\text{Re}}\left( z \right) = 0$$

B. $${\text{Im}}\left( z \right) = 0$$

C. $${\text{Re}}\left( z \right) > 0,{\text{Im}}\left( z \right) > 0$$

D. $${\text{Re}}\left( z \right) > 0,{\text{Im}}\left( z \right) < 0$$

View Answer

If $${\left( {\sqrt 3 + i} \right)^n} = {\left( {\sqrt 3 - i} \right)^n},n \in N$$ then the least value of $$n$$ is

Releted MCQ Question on
Algebra >> Complex Number

If the cube roots of unity are $$1,\omega ,{\omega ^2},$$ then the roots of the equation $${\left( {x - 1} \right)^3} + 8 = 0\,\,{\text{are}}$$

The smallest positive integer $$n$$ for which $${\left( {\frac{{1 + i}}{{1 - i}}} \right)^n} = 1\,{\text{is}}$$

The complex numbers $$z = x+ iy$$ which satisfy the equation $$\left| {\frac{{z - 5i}}{{z + 5i}}} \right| = 1$$ lie on

If $$z = {\left( {\frac{{\sqrt 3 }}{2} + \frac{i}{2}} \right)^5} + {\left( {\frac{{\sqrt 3 }}{2} - \frac{i}{2}} \right)^5},\,{\text{then}}$$

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Complex Number

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If $${\left( {\sqrt 3 + i} \right)^n} = {\left( {\sqrt 3 - i} \right)^n},n \in N$$ then the least value of $$n$$ is

Releted MCQ Question on Algebra >> Complex Number

If the cube roots of unity are $$1,\omega ,{\omega ^2},$$ then the roots of the equation $${\left( {x - 1} \right)^3} + 8 = 0\,\,{\text{are}}$$

The smallest positive integer $$n$$ for which $${\left( {\frac{{1 + i}}{{1 - i}}} \right)^n} = 1\,{\text{is}}$$

The complex numbers $$z = x+ iy$$ which satisfy the equation $$\left| {\frac{{z - 5i}}{{z + 5i}}} \right| = 1$$ lie on

If $$z = {\left( {\frac{{\sqrt 3 }}{2} + \frac{i}{2}} \right)^5} + {\left( {\frac{{\sqrt 3 }}{2} - \frac{i}{2}} \right)^5},\,{\text{then}}$$

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Releted MCQ Question on
Algebra >> Complex Number

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Complex Number