Tag: complex numbers and linear inequations

Quizzes (4) Questions (127)

Questions Related to complex numbers and linear inequations

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

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 identities of complex numbers powers of imaginary unit i algebra of complex numbers

Which  of the following is correct ?

  1. $2 + 3i > 1 + 4i$
  2. $2 + 2i > 3 + 3i$
  3. $5 + 8i > 5 + 7i$

  4. None of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Option $A$

$2+3i>1+4i$
$2^2+3^2>1^2+4^2$
$4+9>1+16$
$13>17$

It is not correct.

Option $B$
$2+2i>3+3i$

$2^2+2^2>3^2+3^2$
$4+4>9+9$
$8>18$

It is not correct.


Option $C$
$5+8i>5+7i$

$5^2+^82>5^2+7^2$
$25+64>25+49$
$89>74$

It is correct.

Hence, this is the answer.

Multiple choice maths complex numbers and linear inequations identities of complex numbers powers of imaginary unit i algebra of complex numbers

Find the value of $\begin{vmatrix} 2+i & 2-i \ 1+i & 1-i \end{vmatrix}$ if $i^2=-1$.

  1. A complex quantity

  2. real quantity
  3. $0$

  4. cannot be determined
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$\left| \begin{matrix} 2+i & 2-i \ 1+i & 1-i \end{matrix} \right| \ =(2+i)(1-i)-(2+i)(1+i)\ =2-2i+i-{ i }^{ 2 }-2-2i-i-{ i }^{ 2 }\ =-4i+2$

So it is complex quantity

Multiple choice maths complex numbers and linear inequations identities of complex numbers powers of imaginary unit i algebra of complex numbers

$\displaystyle i+\frac{1}{i}=$

  1. $1$
  2. $-1$
  3. $0$
  4. $2i$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let $Z= i+\dfrac{1}{i}$


Mutiplying numerator and denominator by i. We get,

          $=i+\dfrac{i}{i^{2}}$

          $=i+\dfrac{i}{-1} \quad \dots (i^2=-1)$

          $=i-i$

      $Z=0$

Hence, 

$i+\dfrac{1}{i}=0$.