if 2z1 3z2 z3 0 then z1z2z3 are represented by 597021529

If $$2{z_1} - 3{z_2} + {z_3} = 0$$ then $${z_1},{z_2},{z_3}$$ are represented by

A. three vertices of a triangle

B. three collinear points

C. three vertices of a rhombus

D. None of these

Answer : three collinear points

Solution :
$${z_2} = \frac{{2{z_1} + {z_3}}}{{2 + 1}}$$
$$ \Rightarrow {z_2}$$ divides the line segment joining $${z_1},{z_3}$$ in the ratio $$2 : 1$$ internally.
Note : If $${z_1},{z_2},{z_3}$$ are related by $$a{z_1} + b{z_2} + c{z_3} = 0,$$ where $$a + b \pm c = 0,$$ then $${z_1},{z_2},{z_3}$$ will be collinear points.
ALTERNATE SOLUTION
$$\eqalign{ & {\text{Given, }}2{z_1} + {z_3} = 3{z_2} \cr & {z_2} = \frac{{2{z_1} + {z_3}}}{{2 + 1}} \cr} $$
So, $${z_2}$$ divided the line joining the point $${z_1},\,{z_3}$$ are in the ratio $$1 : 2$$
So, $${z_1},{z_2},{z_3}$$ are collinear.
Complex Number mcq solution image

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

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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

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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

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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 $$2{z_1} - 3{z_2} + {z_3} = 0$$ then $${z_1},{z_2},{z_3}$$ are represented by

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 $$2{z_1} - 3{z_2} + {z_3} = 0$$ then $${z_1},{z_2},{z_3}$$ are represented by

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
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