Tag: complex numbers and quadratic equations

Questions Related to complex numbers and quadratic equations

The number of solution of $z^2 + \bar{z} = 0$ is

maths complex numbers and linear inequations argand plane and polar representation geometric representation of a complex number complex numbers and quadratic equations

  1. $5$

  2. $4$

  3. $2$

  4. $3$

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Correct Option: B
Explanation:
Let $z=x+iy$.
Now,
$z^2+\overline{z}=0$
or, $x^2-y^2+2ixy+(x-iy)=0$
or, $(x^2-y^2+x)+i(2xy-y)=0$
Now comparing the real and imaginary part both sides we get,
$x^2-y^2+x=0$.....(1) and $2xy-y=0$.....(2).
From (2) we get, $x=\dfrac{1}{2}$ or $y=0$.
Now $x=\dfrac{1}{2}$ gives from (1) we get, $y=\pm \dfrac{\sqrt{3}}{2}$.
And $y=0$ gives from (1) we get, $x=0, 1$.
So the solution s are $(0,0), (1,0), \left(\dfrac{1}{2},\pm \dfrac{\sqrt{3}}{2}\right)$.
So we have $4$ solutions.

If $z \neq 0$, then $ \overset{100}{\underset{0}{\int}}arg(-|z|)dx =$

maths complex numbers and linear inequations argand plane and polar representation geometric representation of a complex number complex numbers and quadratic equations

  1. $0$

  2. Not defined

  3. $100$

  4. $100\pi$

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Correct Option: A

The complex no. $\dfrac{1+2i}{1-i}$ lies in which quadrant of the complex plane

maths complex numbers and linear inequations argand plane and polar representation geometric representation of a complex number complex numbers and quadratic equations

  1. first

  2. second

  3. third

  4. fourth

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Correct Option: B

If $|z^2-1|=|z^2|+1$, then z lies on?

maths complex numbers and linear inequations argand plane and polar representation geometric representation of a complex number complex numbers and quadratic equations

  1. The real axis

  2. The imaginary axis

  3. A circle

  4. An ellipse

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Correct Option: A