Advertisements
Advertisements
प्रश्न
If \[z = \frac{1 + 7i}{(2 - i )^2}\] , then
विकल्प
\[\left| z \right| = 2\]
\[\left| z \right| = \frac{1}{2}\]
amp (z) = \[\frac{\pi}{4}\]
amp (z) = \[\frac{3\pi}{4}\]
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: 3(7 + i7) + i(7 + i7)
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 – i)4
Express the given complex number in the form a + ib: `(-2 - 1/3 i)^3`
Evaluate the following:
\[\frac{1}{i^{58}}\]
Evaluate the following:
\[i^{37} + \frac{1}{i^{67}}\].
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:
\[\frac{3 - 4i}{(4 - 2i)(1 + i)}\]
Express the following complex number in the standard form a + i b:
\[\frac{5 + \sqrt{2}i}{1 - 2\sqrt{i}}\]
If \[\left( \frac{1 + i}{1 - i} \right)^3 - \left( \frac{1 - i}{1 + i} \right)^3 = x + iy\] find (x, y).
Evaluate the following:
\[x^4 + 4 x^3 + 6 x^2 + 4x + 9, \text { when } x = - 1 + i\sqrt{2}\]
For a positive integer n, find the value of \[(1 - i )^n \left( 1 - \frac{1}{i} \right)^n\].
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 \[\left| z_1 + z_2 + z_3 \right|\] .
Find the number of solutions of \[z^2 + \left| z \right|^2 = 0\].
Express the following complex in the form r(cos θ + i sin θ):
1 − sin α + i cos α
If z1 and z2 are two complex numbers such that \[\left| z_1 \right| = \left| z_2 \right|\] and arg(z1) + arg(z2) = \[\pi\] then show that \[z_1 = - \bar{{z_2}}\].
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 the least positive integral value of n for which \[\left( \frac{1 + i}{1 - i} \right)^n\] is real.
Find z, if \[\left| z \right| = 4 \text { and }\arg(z) = \frac{5\pi}{6} .\]
The value of \[(1 + i)(1 + i^2 )(1 + i^3 )(1 + i^4 )\] is.
If \[z = \frac{- 2}{1 + i\sqrt{3}}\],then the value of arg (z) is
\[(\sqrt{- 2})(\sqrt{- 3})\] is equal to
\[\text { If }z = 1 - \text { cos }\theta + i \text { sin }\theta, \text { then } \left| z \right| =\]
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
The complex number z which satisfies the condition \[\left| \frac{i + z}{i - z} \right| = 1\] lies on
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 – b) + (a + b)i = a + 5i
Express the following in the form of a + ib, a, b ∈ R, i = `sqrt(−1)`. State the values of a and b:
(1 + 2i)(– 2 + i)
Express the following in the form of a + ib, a, b∈R i = `sqrt(−1)`. State the values of a and b:
`((2 + "i"))/((3 - "i")(1 + 2"i"))`
Find the value of `(3 + 2/"i")("i"^6 - "i"^7)(1 + "i"^11)`
Answer the following:
Show that z = `5/((1 - "i")(2 - "i")(3 - "i"))` is purely imaginary number.
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:
The order relation is defined on the set of complex numbers.
Match the statements of Column A and Column B.
| Column A | Column B |
| (a) The polar form of `i + sqrt(3)` is | (i) Perpendicular bisector of segment joining (–2, 0) and (2, 0). |
| (b) The amplitude of `-1 + sqrt(-3)` is | (ii) On or outside the circle having centre at (0, –4) and radius 3. |
| (c) If |z + 2| = |z − 2|, then locus of z is | (iii) `(2pi)/3` |
| (d) If |z + 2i| = |z − 2i|, then locus of z is | (iv) Perpendicular bisector of segment joining (0, –2) and (0, 2). |
| (e) Region represented by |z + 4i| ≥ 3 is | (v) `2(cos pi/6 + i sin pi/6)` |
| (f) Region represented by |z + 4| ≤ 3 is | (vi) On or inside the circle having centre (–4, 0) and radius 3 units. |
| (g) Conjugate of `(1 + 2i)/(1 - i)` lies in | (vii) First quadrant |
| (h) Reciprocal of 1 – i lies in | (viii) Third quadrant |
Show that `(-1+sqrt3i)^3` is a real number.
