Advertisements
Advertisements
प्रश्न
Find the conjugate of the following complex number:
\[\frac{(3 - 2i)(2 + 3i)}{(1 + 2i)(2 - i)}\]
उत्तर
\[\text { Let } z = \frac{\left( 3 - 2i \right)\left( 2 + 3i \right)}{\left( 1 + 2i \right)\left( 2 - i \right)}\]
\[ = \frac{6 + 9i - 4i - 6 i^2}{2 - i + 4i - 2 i^2}\]
\[ = \frac{6 + 6 + 5i}{2 + 2 + 3i}\]
\[ = \frac{12 + 5i}{4 + 3i} \times \frac{4 - 3i}{4 - 3i}\]
\[ = \frac{48 - 36i + 20i - 15 i^2}{16 - 9 i^2}\]
\[ = \frac{63 - 16i}{25}\]
`therefore overlineZ =(63 +16i)/25`
APPEARS IN
संबंधित प्रश्न
Find the modulus and argument of the complex number `(1 + 2i)/(1-3i)`
Find the modulus of `(1+i)/(1-i) - (1-i)/(1+i)`
If (x + iy)3 = u + iv, then show that `u/x + v/y =4(x^2 - y^2)`
Find the conjugate of the following complex number:
4 − 5 i
Find the conjugate of the following complex number:
\[\frac{1}{1 + i}\]
Find the conjugate of the following complex number:
\[\frac{(3 - i )^2}{2 + i}\]
Find the conjugate of the following complex number:
\[\frac{(1 + i)(2 + i)}{3 + i}\]
Find the modulus and argument of the following complex number and hence express in the polar form:
1 + i
Find the modulus and argument of the following complex number and hence express in the polar form:
1 − i
Find the modulus and argument of the following complex number and hence express in the polar form:
\[\frac{1 - i}{1 + i}\]
Find the modulus and argument of the following complex number and hence express in the polar form:
\[\frac{1}{1 + i}\]
Find the modulus and argument of the following complex number and hence express in the polar form:
\[\frac{1 + 2i}{1 - 3i}\]
Find the modulus and argument of the following complex number and hence express in the polar form:
sin 120° - i cos 120°
If z1, z2 and z3, z4 are two pairs of conjugate complex numbers, prove that \[\arg\left( \frac{z_1}{z_4} \right) + \arg\left( \frac{z_2}{z_3} \right) = 0\].
Write the conjugate of \[\frac{2 - i}{\left( 1 - 2i \right)^2}\] .
If (1+i)(1 + 2i)(1+3i)..... (1+ ni) = a+ib,then 2 ×5 ×10 ×...... ×(1+n2) is equal to.
If (1 + i) (1 + 2i) (1 + 3i) .... (1 + ni) = a + ib, then 2.5.10.17.......(1+n2)=
If \[\frac{( a^2 + 1 )^2}{2a - i} = x + iy, \text { then } x^2 + y^2\] is equal to
If \[\frac{1 - ix}{1 + ix} = a + ib\] then \[a^2 + b^2\]
Solve the equation `z^2 = barz`, where z = x + iy.
If |z2 – 1| = |z|2 + 1, then show that z lies on imaginary axis.
If a complex number z lies in the interior or on the boundary of a circle of radius 3 units and centre (–4, 0), find the greatest and least values of |z + 1|.
The conjugate of the complex number `(1 - i)/(1 + i)` is ______.
If a complex number lies in the third quadrant, then its conjugate lies in the ______.
What is the conjugate of `(sqrt(5 + 12i) + sqrt(5 - 12i))/(sqrt(5 + 12i) - sqrt(5 - 12i))`?
Solve the system of equations Re(z2) = 0, z = 2.
State True or False for the following:
If z is a complex number such that z ≠ 0 and Re(z) = 0, then Im(z2) = 0.
What is the conjugate of `(2 - i)/(1 - 2i)^2`?
sinx + icos2x and cosx – isin2x are conjugate to each other for ______.
