Advertisements
Advertisements
Question
If z is a non-zero complex number, then \[\left| \frac{\left| z \right|^2}{zz} \right|\] is equal to
Options
`|overlinez/z|`
\[\left| z \right|\]
`|overlinez|`
none of these
Advertisements
Solution
`|overlinez/z|`
`||overlinez|^2/(zoverlinez)| = ||overlinez|^2/|z|^2| (becausez overlinez= |z|^2)`
Let `z = a - ib`
⇒ `|z| = sqrt(a^2 +b^2)`
Let `overlinez = a - ib`
⇒ `|overlinez| = sqrt(a^2 +b^2)`
`therefore ||overlinez|^2/(zoverlinez)| = ||overlinez|^2/|z|^2|`
= `|overlinez/z|`
APPEARS IN
RELATED QUESTIONS
Evaluate the following:
\[( i^{77} + i^{70} + i^{87} + i^{414} )^3\]
Find the value of the following expression:
i + i2 + i3 + i4
Find the value of the following expression:
1+ i2 + i4 + i6 + i8 + ... + i20
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:
\[(1 + 2i )^{- 3}\]
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 values of θ for which the complex number \[\frac{1 + i cos\theta}{1 - 2i cos\theta}\] is purely real.
If \[\left( \frac{1 + i}{1 - i} \right)^3 - \left( \frac{1 - i}{1 + i} \right)^3 = x + iy\] find (x, y).
If \[\frac{\left( 1 + i \right)^2}{2 - i} = x + iy\] find x + y.
If \[a = \cos\theta + i\sin\theta\], find the value of \[\frac{1 + a}{1 - a}\].
Evaluate the following:
\[x^4 - 4 x^3 + 4 x^2 + 8x + 44,\text { when } x = 3 + 2i\]
Evaluate the following:
\[x^6 + x^4 + x^2 + 1, \text { when }x = \frac{1 + i}{\sqrt{2}}\]
For a positive integer n, find the value of \[(1 - i )^n \left( 1 - \frac{1}{i} \right)^n\].
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.
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 + i tan α
Find the principal argument of \[\left( 1 + i\sqrt{3} \right)^2\] .
If n ∈ \[\mathbb{N}\] then find the value of \[i^n + i^{n + 1} + i^{n + 2} + i^{n + 3}\] .
Write the argument of \[\left( 1 + i\sqrt{3} \right)\left( 1 + i \right)\left( \cos\theta + i\sin\theta \right)\].
Disclaimer: There is a misprinting in the question. It should be \[\left( 1 + i\sqrt{3} \right)\] instead of \[\left( 1 + \sqrt{3} \right)\].
The value of \[(1 + i)(1 + i^2 )(1 + i^3 )(1 + i^4 )\] is.
If\[z = \cos\frac{\pi}{4} + i \sin\frac{\pi}{6}\], then
The principal value of the amplitude of (1 + i) is
The argument of \[\frac{1 - i\sqrt{3}}{1 + i\sqrt{3}}\] is
If \[z = \left( \frac{1 + i}{1 - i} \right)\] then z4 equals
If \[x + iy = \frac{3 + 5i}{7 - 6i},\] then y =
The argument of \[\frac{1 - i}{1 + i}\] is
The value of \[(1 + i )^4 + (1 - i )^4\] is
If z is a complex number, then
If the complex number \[z = x + iy\] satisfies the condition \[\left| z + 1 \right| = 1\], then z lies on
Express the following in the form of a + ib, a, b ∈ R, i = `sqrt(−1)`. State the values of a and b:
`("i"(4 + 3"i"))/((1 - "i"))`
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))`
Express the following in the form of a + ib, a, b ∈ R, i = `sqrt(−1)`. State the values of a and b:
`(4"i"^8 - 3"i"^9 + 3)/(3"i"^11 - 4"i"^10 - 2)`
Show that `(-1 + sqrt(3)"i")^3` is a real number
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).
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+ sqrt(3)i)^3` is a real number.
