Advertisements
Advertisements
प्रश्न
Write 1 − i in polar form.
Advertisements
उत्तर
\[z = 1 - i \]
\[r = \left| z \right|\]
\[ = \sqrt{1 + 1}\]
\[ = \sqrt{2}\]
\[\text { Let } \tan \alpha = \left| \frac{Im\left( z \right)}{Re\left( z \right)} \right|\]
\[ \therefore \tan\alpha = \left| \frac{- 1}{1} \right|\]
\[ = \frac{\pi}{4}\]
\[ \Rightarrow \alpha = \frac{\pi}{4}\]
\[\text { Since point (1, - 1) lies in the fourth quadrant, the argument of z is given by } \]
\[\theta = - \alpha = - \frac{\pi}{4}\]
\[\text { Polar form = } r\left( \cos \theta + i\sin \theta \right) \]
\[ = \sqrt{2}\left\{ \cos\left( - \frac{\pi}{4} \right) + i\sin\left( - \frac{\pi}{4} \right) \right\}\]
\[ = \sqrt{2}\left( \cos\frac{\pi}{4} - i\sin\frac{\pi}{4} \right)\]
APPEARS IN
संबंधित प्रश्न
Express the given complex number in the form a + ib: `(1/3 + 3i)^3`
Evaluate: `[i^18 + (1/i)^25]^3`
Evaluate the following:
\[( i^{77} + i^{70} + i^{87} + i^{414} )^3\]
Evaluate the following:
\[i^{30} + i^{40} + i^{60}\]
Show that 1 + i10 + i20 + i30 is a real number.
Find the value of the following expression:
i5 + i10 + i15
Find the value of the following expression:
\[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}}\]
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:
\[\frac{(1 + i)(1 + \sqrt{3}i)}{1 - i}\] .
Express the following complex number in the standard form a + i b:
\[\frac{3 - 4i}{(4 - 2i)(1 + i)}\]
If \[z_1 = 2 - i, z_2 = - 2 + i,\] find
Re \[\left( \frac{z_1 z_2}{z_1} \right)\]
If \[a = \cos\theta + i\sin\theta\], find the value of \[\frac{1 + a}{1 - a}\].
Solve the system of equations \[\text { Re }\left( z^2 \right) = 0, \left| z \right| = 2\].
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.
What is the smallest positive integer n for which \[\left( 1 + i \right)^{2n} = \left( 1 - i \right)^{2n}\] ?
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 \[z_1 = - \bar{{z_2}}\].
If π < θ < 2π and z = 1 + cos θ + i sin θ, then write the value of \[\left| z \right|\] .
Find the principal argument of \[\left( 1 + i\sqrt{3} \right)^2\] .
If \[\left| z - 5i \right| = \left| z + 5i \right|\] , then find the locus of z.
If n ∈ \[\mathbb{N}\] then find the value of \[i^n + i^{n + 1} + i^{n + 2} + i^{n + 3}\] .
If\[z = \cos\frac{\pi}{4} + i \sin\frac{\pi}{6}\], then
The polar form of (i25)3 is
If \[z = \frac{1 + 2i}{1 - (1 - i )^2}\], then arg (z) equal
\[\text { If }z = 1 - \text { cos }\theta + i \text { sin }\theta, \text { then } \left| z \right| =\]
Simplify : `sqrt(-16) + 3sqrt(-25) + sqrt(-36) - sqrt(-625)`
Simplify : `4sqrt(-4) + 5sqrt(-9) - 3sqrt(-16)`
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:
`((1 + "i")/(1 - "i"))^2`
Find the value of `(3 + 2/i) (i^6 - i^7) (1 + i^11)`.
Evaluate the following : `1/"i"^58`
Evaluate the following : i–888
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).
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 |
