Advertisements
Advertisements
Question
The principal value of the amplitude of (1 + i) is
Options
\[\frac{\pi}{4}\]
\[\frac{\pi}{12}\]
\[\frac{3\pi}{4}\]
π
Advertisements
Solution
\[\frac{\pi}{4}\]
Let z = (1+i)
\[\tan \alpha = \left| \frac{\text { Im }(z)}{Re(z)} \right|\]
\[ = 1\]
\[ \Rightarrow \alpha = \frac{\pi}{4}\]
\[\text { Since, z lies in the first quadrant }. \]
Therefore, arg (z) =\[\frac{\pi}{4}\]
APPEARS IN
RELATED QUESTIONS
Express the given complex number in the form a + ib: 3(7 + i7) + i(7 + i7)
Express the given complex number in the form a + ib: `(1/5 + i 2/5) - (4 + i 5/2)`
Evaluate the following:
\[\frac{1}{i^{58}}\]
Evaluate the following:
\[( i^{77} + i^{70} + i^{87} + i^{414} )^3\]
Find the value of the following expression:
i5 + i10 + i15
Express the following complex number in the standard form a + i b:
\[(1 + 2i )^{- 3}\]
Express the following complex number in the standard form a + i b:
\[\left( \frac{1}{1 - 4i} - \frac{2}{1 + i} \right)\left( \frac{3 - 4i}{5 + i} \right)\]
Find the real value of x and y, if
\[(x + iy)(2 - 3i) = 4 + i\]
If \[z_1 = 2 - i, z_2 = 1 + i,\text { find } \left| \frac{z_1 + z_2 + 1}{z_1 - z_2 + i} \right|\]
Evaluate the following:
\[x^6 + x^4 + x^2 + 1, \text { when }x = \frac{1 + i}{\sqrt{2}}\]
If z1 is a complex number other than −1 such that \[\left| z_1 \right| = 1\] and \[z_2 = \frac{z_1 - 1}{z_1 + 1}\] then show that the real parts of z2 is zero.
If \[\frac{z - 1}{z + 1}\] is purely imaginary number (\[z \neq - 1\]), find the value of \[\left| z \right|\].
Solve the equation \[\left| z \right| = z + 1 + 2i\].
Express the following complex in the form r(cos θ + i sin θ):
tan α − i
Write −1 + i \[\sqrt{3}\] in polar form .
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 value of \[\arg\left( z \right) + \arg\left( \bar{z} \right)\].
If \[z = \frac{- 2}{1 + i\sqrt{3}}\],then the value of arg (z) is
If a = cos θ + i sin θ, then \[\frac{1 + a}{1 - a} =\]
The least positive integer n such that \[\left( \frac{2i}{1 + i} \right)^n\] is a positive integer, is.
If \[z = \frac{1 + 2i}{1 - (1 - i )^2}\], then arg (z) equal
The amplitude of \[\frac{1}{i}\] is equal to
The amplitude of \[\frac{1 + i\sqrt{3}}{\sqrt{3} + i}\] is
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
Express the following in the form of a + ib, a, b ∈ R, i = `sqrt(−1)`. State the values of a and b:
`("i"(4 + 3"i"))/((1 - "i"))`
Express the following in the form of a + ib, a, b∈R i = `sqrt(−1)`. State the values of a and b:
(2 + 3i)(2 – 3i)
Evaluate the following : i116
Evaluate the following : i403
Evaluate the following : i–888
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)`.
Match the statements of column A and B.
| Column A | Column B |
| (a) The value of 1 + i2 + i4 + i6 + ... i20 is | (i) purely imaginary complex number |
| (b) The value of `i^(-1097)` is | (ii) purely real complex number |
| (c) Conjugate of 1 + i lies in | (iii) second quadrant |
| (d) `(1 + 2i)/(1 - i)` lies in | (iv) Fourth quadrant |
| (e) If a, b, c ∈ R and b2 – 4ac < 0, then the roots of the equation ax2 + bx + c = 0 are non real (complex) and |
(v) may not occur in conjugate pairs |
| (f) If a, b, c ∈ R and b2 – 4ac > 0, and b2 – 4ac is a perfect square, then the roots of the equation ax2 + bx + c = 0 |
(vi) may occur in conjugate pairs |
If `((1 - i)/(1 + i))^100` = a + ib, then find (a, b).
If a = cosθ + isinθ, find the value of `(1 + "a")/(1 - "a")`.
State True or False for the following:
2 is not a complex number.
