the minimum value of z z i is 982030713

The minimum value of $$\left| z \right| + \left| {z - i} \right|$$ is

A. 0

B. 1

C. 2

D. None of these

Answer : 1

Solution :
Complex Number mcq solution image

Using the result $$\left| {{z_1} + {z_2}} \right| \leqslant \left| {{z_1}} \right| + \left| {{z_2}} \right|,$$
we get, $$\left| z \right| + \left| {z - i} \right| = \left| z \right| + \left| {i - z} \right|\,\,\,\,\left[ {{\text{since}}\,\left| z \right| = \left| { - z} \right|} \right]$$
$$ \leqslant \left| {z + i - z} \right| = \left| i \right| = 1$$
∴ minimum value of $$\left| z \right| + \left| {z - i} \right|\,\,{\text{is}}\,\,1$$

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

The minimum value of $$\left| z \right| + \left| {z - i} \right|$$ 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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The minimum value of $$\left| z \right| + \left| {z - i} \right|$$ 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 $$z = {\left( {\frac{{\sqrt 3 }}{2} + \frac{i}{2}} \right)^5} + {\left( {\frac{{\sqrt 3 }}{2} - \frac{i}{2}} \right)^5},\,{\text{then}}$$

Practice More Releted MCQ Question on Complex Number

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Releted MCQ Question on
Algebra >> Complex Number

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Complex Number