Tag: complex numbers and quadratic equations

Quizzes (3) Questions (24)

Questions Related to complex numbers and quadratic equations

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

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

  1. $5$
  2. $4$
  3. $2$
  4. $3$
Reveal answer Fill a bubble to check yourself
B Correct answer
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.
Multiple choice maths complex numbers and linear inequations argand plane and polar representation geometric representation of a complex number complex numbers and quadratic equations

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

  1. $0$

  2. Not defined
  3. $100$
  4. $100\pi$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

For any non-zero complex number z, -|z| is a negative real number. The argument of any negative real number is pi. Therefore, arg(-|z|) = pi. Integrating this constant function with respect to x from 0 to 100 gives the integral of pi dx from 0 to 100, which evaluates to pi * (100 - 0) = 100pi.

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

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

  1. first

  2. second

  3. third

  4. fourth
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

To find the quadrant, multiply the numerator and denominator by the conjugate (1+i). (1+2i)(1+i) / (1-i)(1+i) = (1+i+2i-2) / 2 = (-1+3i) / 2 = -0.5 + 1.5i. This point lies in the second quadrant.

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

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

  1. The real axis

  2. The imaginary axis

  3. A circle

  4. An ellipse
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let z = x + iy. Then z^2 = (x^2 - y^2) + i(2xy), so |z^2| = x^2 + y^2 and |z^2 - 1| = sqrt((x^2 - y^2 - 1)^2 + (2xy)^2). The given equation becomes |z^2 - 1| = |z^2| + 1. Squaring both sides yields (x^2 - y^2 - 1)^2 + 4x^2y^2 = (x^2 + y^2 + 1)^2, which simplifies to x^2 = 0, meaning x = 0. Thus, z lies on the imaginary axis.