Advertisements
Advertisements
प्रश्न
If \[z = \frac{- 2}{1 + i\sqrt{3}}\],then the value of arg (z) is
पर्याय
π
\[\frac{\pi}{3}\]
\[\frac{2\pi}{3}\]
\[\frac{\pi}{4}\]
Advertisements
उत्तर
\[\frac{2\pi}{3}\]
z =\[\frac{- 2}{1 + i\sqrt{3}}\]
Rationalising z, we get,
\[z = \frac{- 2}{1 + i\sqrt{3}} \times \frac{1 - i\sqrt{3}}{1 - i\sqrt{3}}\]
\[ \Rightarrow z = \frac{- 2 + i2\sqrt{3}}{1 + 3}\]
\[ \Rightarrow z = \frac{- 1 + i\sqrt{3}}{2} \]
\[ \Rightarrow z = \frac{- 1}{2} + \frac{i\sqrt{3}}{2}\]
\[\tan \alpha = \left| \frac{Im(z)}{Re(z)} \right|\]
\[ = \sqrt{3}\]
\[ \Rightarrow \alpha = \frac{\pi}{3}\]
\[\text { Since, z lies in the second quadrant } . \]
\[\text { Therefore,}\arg (z) = \pi - \frac{\pi}{3}\]
\[ = \frac{2\pi}{3}\]
APPEARS IN
संबंधित प्रश्न
Express the given complex number in the form a + ib: `(5i) (- 3/5 i)`
Express the given complex number in the form a + ib: i–39
Express the given complex number in the form a + ib:
`[(1/3 + i 7/3) + (4 + i 1/3)] -(-4/3 + i)`
Let z1 = 2 – i, z2 = –2 + i. Find Re`((z_1z_2)/barz_1)`
Evaluate the following:
i457
Evaluate the following:
(ii) i528
Evaluate the following:
\[\frac{1}{i^{58}}\]
Evaluate the following:
\[i^{49} + i^{68} + i^{89} + i^{110}\]
Show that 1 + i10 + i20 + i30 is a real number.
Find the value of the following expression:
\[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}}\]
Find the value of the following expression:
(1 + i)6 + (1 − i)3
Express the following complex number in the standard form a + i b:
\[\frac{1}{(2 + i )^2}\]
Express the following complex number in the standard form a + i b:
\[\frac{1 - i}{1 + i}\]
Express the following complex number in the standard form a + i b:
\[\frac{(1 - i )^3}{1 - i^3}\]
Express the following complex number in the standard form a + i b:
\[\frac{3 - 4i}{(4 - 2i)(1 + i)}\]
Express the following complex number in the standard form a + i b:
\[\left( \frac{1}{1 - 4i} - \frac{2}{1 + i} \right)\left( \frac{3 - 4i}{5 + i} \right)\]
If \[\frac{\left( 1 + i \right)^2}{2 - i} = x + iy\] find x + y.
If \[\left( \frac{1 - i}{1 + i} \right)^{100} = a + ib\] find (a, b).
Evaluate the following:
\[2 x^3 + 2 x^2 - 7x + 72, \text { when } x = \frac{3 - 5i}{2}\]
Evaluate the following:
\[x^4 - 4 x^3 + 4 x^2 + 8x + 44,\text { when } x = 3 + 2i\]
Evaluate the following:
\[x^4 + 4 x^3 + 6 x^2 + 4x + 9, \text { when } x = - 1 + i\sqrt{2}\]
Solve the system of equations \[\text { Re }\left( z^2 \right) = 0, \left| z \right| = 2\].
Write the argument of −i.
Write the least positive integral value of n for which \[\left( \frac{1 + i}{1 - i} \right)^n\] is real.
Write the sum of the series \[i + i^2 + i^3 + . . . .\] upto 1000 terms.
For any two complex numbers z1 and z2 and any two real numbers a, b, find the value of \[\left| a z_1 - b z_2 \right|^2 + \left| a z_2 + b z_1 \right|^2\].
Write the argument of \[\left( 1 + i\sqrt{3} \right)\left( 1 + i \right)\left( \cos\theta + i\sin\theta \right)\].
Disclaimer: There is a misprinting in the question. It should be \[\left( 1 + i\sqrt{3} \right)\] instead of \[\left( 1 + \sqrt{3} \right)\].
The polar form of (i25)3 is
The principal value of the amplitude of (1 + i) is
\[\text { If }z = 1 - \text { cos }\theta + i \text { sin }\theta, \text { then } \left| z \right| =\]
The value of \[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}} - 1\] is
Find a and b if a + 2b + 2ai = 4 + 6i
Evaluate the following : i116
Evaluate the following : i403
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:
2 is not a complex number.
The real value of θ for which the expression `(1 + i cos theta)/(1 - 2i cos theta)` is a real number is ______.
