Advertisements
Advertisements
प्रश्न
If \[z = \frac{1 + 7i}{(2 - i )^2}\] , then
पर्याय
\[\left| z \right| = 2\]
\[\left| z \right| = \frac{1}{2}\]
amp (z) = \[\frac{\pi}{4}\]
amp (z) = \[\frac{3\pi}{4}\]
Advertisements
उत्तर
amp (z) = \[\frac{3\pi}{4}\]
\[z = \frac{1 + 7i}{\left( 2 - i \right)^2}\]
\[ \Rightarrow z = \frac{1 + 7i}{4 + i^2 - 4i}\]
\[ \Rightarrow z = \frac{1 + 7i}{4 - 1 - 4i} \left[ \because i^2 = - 1 \right]\]
\[ \Rightarrow z = \frac{1 + 7i}{3 - 4i}\]
\[ \Rightarrow z = \frac{1 + 7i}{3 - 4i} \times \frac{3 + 4i}{3 + 4i}\]
\[ \Rightarrow z = \frac{3 + 4i + 21i + 28 i^2}{9 - 16 i^2}\]
\[ \Rightarrow z = \frac{3 - 28 + 25i}{9 + 16}\]
\[ \Rightarrow z = \frac{- 25 + 25i}{25}\]
\[ \Rightarrow z = - 1 + i\]
\[\tan \alpha = \left| \frac{Im\left( z \right)}{Re\left( z \right)} \right|\]
\[ = 1\]
\[ \Rightarrow \alpha = \frac{\pi}{4}\]
\[\text { Since, z lies in the second quadrant }. \]
\[\text { Therefore, amp } (z) = \pi - \alpha\]
\[ = \pi - \frac{\pi}{4} \]
\[ = \frac{3\pi}{4} \]
APPEARS IN
संबंधित प्रश्न
Express the given complex number in the form a + ib: (1 – i) – (–1 + i6)
Express the given complex number in the form a + ib: `(1/5 + i 2/5) - (4 + i 5/2)`
Let z1 = 2 – i, z2 = –2 + i. Find `"Im"(1/(z_1barz_1))`
Evaluate the following:
(ii) i528
Find the value of the following expression:
i49 + i68 + i89 + i110
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:
\[\frac{3 - 4i}{(4 - 2i)(1 + i)}\]
Express the following complex number in the standard form a + i b:
\[\left( \frac{1}{1 - 4i} - \frac{2}{1 + i} \right)\left( \frac{3 - 4i}{5 + i} \right)\]
Find the real value of x and y, if
\[(1 + i)(x + iy) = 2 - 5i\]
If \[z_1 = 2 - i, z_2 = - 2 + i,\] find
Re \[\left( \frac{z_1 z_2}{z_1} \right)\]
If \[z_1 = 2 - i, z_2 = - 2 + i,\] find
Im `(1/(z_1overlinez_1))`
Solve the system of equations \[\text { Re }\left( z^2 \right) = 0, \left| z \right| = 2\].
If z1, z2, z3 are complex numbers such that \[\left| z_1 \right| = \left| z_2 \right| = \left| z_3 \right| = \left| \frac{1}{z_1} + \frac{1}{z_2} + \frac{1}{z_3} \right| = 1\] then find the value of \[\left| z_1 + z_2 + z_3 \right|\] .
Express the following complex in the form r(cos θ + i sin θ):
1 + i tan α
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.
If π < θ < 2π and z = 1 + cos θ + i sin θ, then write the value of \[\left| z \right|\] .
Write the value of \[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}}\] .
Write −1 + i \[\sqrt{3}\] in polar form .
Find the principal argument of \[\left( 1 + i\sqrt{3} \right)^2\] .
If \[\frac{\left( a^2 + 1 \right)^2}{2a - i} = x + iy\] find the value of \[x^2 + y^2\].
Write the value of \[\sqrt{- 25} \times \sqrt{- 9}\].
If `(3+2i sintheta)/(1-2 i sin theta)`is a real number and 0 < θ < 2π, then θ =
The principal value of the amplitude of (1 + i) is
The argument of \[\frac{1 - i\sqrt{3}}{1 + i\sqrt{3}}\] is
If \[z = \frac{1 + 2i}{1 - (1 - i )^2}\], then arg (z) equal
If \[z = \frac{1}{1 - cos\theta - i sin\theta}\] then Re (z) =
The argument of \[\frac{1 - i}{1 + i}\] 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
If the complex number \[z = x + iy\] satisfies the condition \[\left| z + 1 \right| = 1\], then z lies on
Find a and b if abi = 3a − b + 12i
Find a and b if `1/("a" + "ib")` = 3 – 2i
Find a and b if (a + ib) (1 + i) = 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")/(1 - "i"))^2`
If `((1 - i)/(1 + i))^100` = a + ib, then find (a, b).
Show that `(-1+ sqrt(3)i)^3` is a real number.
