Advertisements
Advertisements
प्रश्न
Express the following complex number in the standard form a + i b:
\[\frac{3 - 4i}{(4 - 2i)(1 + i)}\]
उत्तर
\[\frac{3 - 4i}{\left( 4 - 2i \right)\left( 1 + i \right)}\]
\[ = \frac{3 - 4i}{4 + 2i - 2 i^2} \left( \because i^2 = - 1 \right)\]
\[ = \frac{3 - 4i}{6 + 2i}\]
\[ = \frac{3 - 4i}{6 + 2i} \times \frac{6 - 2i}{6 - 2i}\]
\[ = \frac{18 - 6i - 24i + 8 i^2}{36 - 4 i^2}\]
\[ = \frac{18 - 30i - 8}{36 + 4} \]
\[ = \frac{10 - 30i}{40}\]
\[ = \frac{1}{4} - \frac{3}{4}i\]
APPEARS IN
संबंधित प्रश्न
Express the given complex number in the form a + ib: (1 – i)4
Express the given complex number in the form a + ib: `(1/3 + 3i)^3`
Let z1 = 2 – i, z2 = –2 + i. Find `"Im"(1/(z_1barz_1))`
Evaluate the following:
(ii) i528
Evaluate the following:
\[i^{37} + \frac{1}{i^{67}}\].
Show that 1 + i10 + i20 + i30 is a real number.
Find the value of the following expression:
i + i2 + i3 + i4
Find the value of the following expression:
i5 + i10 + i15
Express the following complex number in the standard form a + i b:
\[\frac{(1 - i )^3}{1 - i^3}\]
Find the real value of x and y, if
\[\frac{(1 + i)x - 2i}{3 + i} + \frac{(2 - 3i)y + i}{3 - i}\]
Find the multiplicative inverse of the following complex number:
\[(1 + i\sqrt{3} )^2\]
Find the least positive integral value of n for which \[\left( \frac{1 + i}{1 - i} \right)^n\] is real.
If \[\left( \frac{1 + i}{1 - i} \right)^3 - \left( \frac{1 - i}{1 + i} \right)^3 = x + iy\] find (x, y).
If \[a = \cos\theta + i\sin\theta\], find the value of \[\frac{1 + a}{1 - a}\].
Evaluate the following:
\[x^4 - 4 x^3 + 4 x^2 + 8x + 44,\text { when } x = 3 + 2i\]
Solve the system of equations \[\text { Re }\left( z^2 \right) = 0, \left| z \right| = 2\].
If \[\frac{z - 1}{z + 1}\] is purely imaginary number (\[z \neq - 1\]), find the value of \[\left| z \right|\].
Express the following complex in the form r(cos θ + i sin θ):
\[\frac{1 - i}{\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}}\]
Express \[\sin\frac{\pi}{5} + i\left( 1 - \cos\frac{\pi}{5} \right)\] in polar form.
Find the principal argument of \[\left( 1 + i\sqrt{3} \right)^2\] .
If n ∈ \[\mathbb{N}\] then find the value of \[i^n + i^{n + 1} + i^{n + 2} + i^{n + 3}\] .
If i2 = −1, then the sum i + i2 + i3 +... upto 1000 terms is equal to
The argument of \[\frac{1 - i\sqrt{3}}{1 + i\sqrt{3}}\] is
\[\text { If } z = \frac{1}{(1 - i)(2 + 3i)}, \text { than } \left| z \right| =\]
\[\text { If }z = 1 - \text { cos }\theta + i \text { sin }\theta, \text { then } \left| z \right| =\]
If \[z = \frac{1 + 7i}{(2 - i )^2}\] , then
Which of the following is correct for any two complex numbers z1 and z2?
Express the following in the form of a + ib, a, b ∈ R, i = `sqrt(−1)`. State the values of a and b:
(1 + 2i)(– 2 + i)
Express the following in the form of a + ib, a, b∈R i = `sqrt(−1)`. State the values of a and b:
(1 + i)−3
Evaluate the following : `1/"i"^58`
Evaluate the following : i–888
Evaluate the following : i30 + i40 + i50 + i60
Show that 1 + i10 + i20 + i30 is a real number
If z1 and z2 both satisfy `z + barz = 2|z - 1|` arg`(z_1 - z_2) = pi/4`, then find `"Im" (z_1 + z_2)`.
State true or false for the following:
If a complex number coincides with its conjugate, then the number must lie on imaginary axis.
If a = cosθ + isinθ, find the value of `(1 + "a")/(1 - "a")`.
If w is a complex cube-root of unity, then prove the following
(w2 + w − 1)3 = −8
Show that `(-1+sqrt3i)^3` is a real number.
