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: i9 + i19
Express the given complex number in the form a + ib: `(1/5 + i 2/5) - (4 + i 5/2)`
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`
Find the value of the following expression:
i30 + i80 + i120
Express the following complex number in the standard form a + i b:
\[(1 + i)(1 + 2i)\]
Express the following complex number in the standard form a + i b:
\[\frac{(1 + i)(1 + \sqrt{3}i)}{1 - i}\] .
Find the smallest positive integer value of m for which \[\frac{(1 + i )^n}{(1 - i )^{n - 2}}\] is a real number.
Evaluate the following:
\[x^4 - 4 x^3 + 4 x^2 + 8x + 44,\text { when } x = 3 + 2i\]
Evaluate the following:
\[x^6 + x^4 + x^2 + 1, \text { when }x = \frac{1 + i}{\sqrt{2}}\]
If \[\left( 1 + i \right)z = \left( 1 - i \right) \bar{z}\],then show that \[z = - i \bar{z}\].
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 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 \[\left| z_1 + z_2 + z_3 \right|\] .
Write (i25)3 in polar form.
Express the following complex in the form r(cos θ + i sin θ):
1 + i tan α
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 z, if \[\left| z \right| = 4 \text { and }\arg(z) = \frac{5\pi}{6} .\]
If \[\left| z - 5i \right| = \left| z + 5i \right|\] , then find the locus of z.
Write the sum of the series \[i + i^2 + i^3 + . . . .\] upto 1000 terms.
The value of \[(1 + i)(1 + i^2 )(1 + i^3 )(1 + i^4 )\] is.
If `(3+2i sintheta)/(1-2 i sin theta)`is a real number and 0 < θ < 2π, then θ =
The polar form of (i25)3 is
If \[z = \frac{- 2}{1 + i\sqrt{3}}\],then the value of arg (z) is
\[(\sqrt{- 2})(\sqrt{- 3})\] is equal to
The argument of \[\frac{1 - i\sqrt{3}}{1 + i\sqrt{3}}\] is
If \[f\left( z \right) = \frac{7 - z}{1 - z^2}\] , where \[z = 1 + 2i\] then \[\left| f\left( z \right) \right|\] is
If z is a complex number, then
Find a and b if (a+b) (2 + i) = b + 1 + (10 + 2a)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`
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))`
Evaluate the following : `1/"i"^58`
Show that 1 + i10 + i20 + i30 is a real number
Answer the following:
Show that z = `5/((1 - "i")(2 - "i")(3 - "i"))` is purely imaginary number.
If `((1 - i)/(1 + i))^100` = a + ib, then find (a, b).
The real value of θ for which the expression `(1 + i cos theta)/(1 - 2i cos theta)` is a real number is ______.
