Advertisements
Advertisements
प्रश्न
Find z, if \[\left| z \right| = 4 \text { and }\arg(z) = \frac{5\pi}{6} .\]
Advertisements
उत्तर
We know that,
\[z = \left| z \right|\left\{ \cos\left[ \arg(z) \right] + i\sin\left[ \arg(z) \right] \right\}\]
\[ \Rightarrow z = 4\left( \cos\frac{5\pi}{6} + i\sin\frac{5\pi}{6} \right)\]
\[ = 4\left( - \cos\frac{\pi}{6} + i\sin\frac{\pi}{6} \right)\]
\[ = 4\left( - \frac{\sqrt{3}}{2} + \frac{1}{2}i \right)\]
\[ = - 2\sqrt{3} + 2i\]
Thus,
\[z = - 2\sqrt{3} + 2i\]
APPEARS IN
संबंधित प्रश्न
Express the given complex number in the form a + ib: (1 – i) – (–1 + i6)
Evaluate the following:
i457
Evaluate the following:
(ii) i528
Find the value of the following expression:
i5 + i10 + i15
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:
\[\left( \frac{1}{1 - 4i} - \frac{2}{1 + i} \right)\left( \frac{3 - 4i}{5 + i} \right)\]
Find the multiplicative inverse of the following complex number:
1 − i
Find the multiplicative inverse of the following complex number:
\[(1 + i\sqrt{3} )^2\]
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:
\[x^4 + 4 x^3 + 6 x^2 + 4x + 9, \text { when } x = - 1 + i\sqrt{2}\]
Find the number of solutions of \[z^2 + \left| z \right|^2 = 0\].
Write (i25)3 in polar form.
Express the following complex in the form r(cos θ + i sin θ):
tan α − i
If π < θ < 2π and z = 1 + cos θ + i sin θ, then write the value of \[\left| z \right|\] .
Write 1 − i in polar form.
Write the least positive integral value of n for which \[\left( \frac{1 + i}{1 - i} \right)^n\] is real.
Find the principal argument of \[\left( 1 + i\sqrt{3} \right)^2\] .
Find the real value of a for which \[3 i^3 - 2a i^2 + (1 - a)i + 5\] is real.
If \[\left| z \right| = 2 \text { and } \arg\left( z \right) = \frac{\pi}{4}\],find z.
The principal value of the amplitude of (1 + i) is
\[(\sqrt{- 2})(\sqrt{- 3})\] is equal to
If \[z = \frac{1}{1 - cos\theta - i sin\theta}\] then Re (z) =
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
A real value of x satisfies the equation \[\frac{3 - 4ix}{3 + 4ix} = a - ib (a, b \in \mathbb{R}), if a^2 + b^2 =\]
Which of the following is correct for any two complex numbers z1 and z2?
Simplify : `4sqrt(-4) + 5sqrt(-9) - 3sqrt(-16)`
Find a and b if a + 2b + 2ai = 4 + 6i
Find a and b if (a – b) + (a + b)i = a + 5i
Find a and b if (a+b) (2 + i) = b + 1 + (10 + 2a)i
Find a and b if abi = 3a − b + 12i
Find a and b if `1/("a" + "ib")` = 3 – 2i
Express the following in the form of a + ib, a, b∈R i = `sqrt(−1)`. State the values of a and b:
`((1 + "i")/(1 - "i"))^2`
Express the following in the form of a + ib, a, b ∈ R i = `sqrt(−1)`. State the values of a and b:
`(2 + sqrt(-3))/(4 + sqrt(-3))`
Find the value of `(3 + 2/i) (i^6 - i^7) (1 + i^11)`.
Evaluate the following : i116
Answer the following:
Show that z = `5/((1 - "i")(2 - "i")(3 - "i"))` is purely imaginary number.
State true or false for the following:
If a complex number coincides with its conjugate, then the number must lie on imaginary axis.
If `((1 - i)/(1 + i))^100` = a + ib, then find (a, b).
State True or False for the following:
The order relation is defined on the set of complex numbers.
