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
Express the given complex number in the form a + ib: i9 + i19
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`
If a + ib = `(x + i)^2/(2x^2 + 1)` prove that a2 + b2 = `(x^2 + 1)^2/(2x + 1)^2`
Evaluate the following:
\[\left( i^{41} + \frac{1}{i^{257}} \right)^9\]
Show that 1 + i10 + i20 + i30 is a real number.
Find the value of the following expression:
(1 + i)6 + (1 − i)3
Express the following complex number in the standard form a + i b:
\[\frac{1 - i}{1 + i}\]
Find the multiplicative inverse of the following complex number:
\[(1 + i\sqrt{3} )^2\]
If \[z_1 = 2 - i, z_2 = - 2 + i,\] find
Re \[\left( \frac{z_1 z_2}{z_1} \right)\]
Find the least positive integral value of n for which \[\left( \frac{1 + i}{1 - i} \right)^n\] is real.
Evaluate the following:
\[x^6 + x^4 + x^2 + 1, \text { when }x = \frac{1 + i}{\sqrt{2}}\]
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 \[z^2 + \left| z \right|^2 = 0\].
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.
If n ∈ \[\mathbb{N}\] then find the value of \[i^n + i^{n + 1} + i^{n + 2} + i^{n + 3}\] .
Find the real value of a for which \[3 i^3 - 2a i^2 + (1 - a)i + 5\] is real.
If\[z = \cos\frac{\pi}{4} + i \sin\frac{\pi}{6}\], then
The polar form of (i25)3 is
If (x + iy)1/3 = a + ib, then \[\frac{x}{a} + \frac{y}{b} =\]
If \[x + iy = \frac{3 + 5i}{7 - 6i},\] then y =
The amplitude of \[\frac{1}{i}\] is equal to
The value of \[(1 + i )^4 + (1 - i )^4\] is
If the complex number \[z = x + iy\] satisfies the condition \[\left| z + 1 \right| = 1\], then z lies on
Simplify : `sqrt(-16) + 3sqrt(-25) + sqrt(-36) - sqrt(-625)`
Find a and b if `1/("a" + "ib")` = 3 – 2i
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 : i35
Evaluate the following : i403
Evaluate the following : `1/"i"^58`
Evaluate the following : i–888
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)`.
Match the statements of column A and B.
| Column A | Column B |
| (a) The value of 1 + i2 + i4 + i6 + ... i20 is | (i) purely imaginary complex number |
| (b) The value of `i^(-1097)` is | (ii) purely real complex number |
| (c) Conjugate of 1 + i lies in | (iii) second quadrant |
| (d) `(1 + 2i)/(1 - i)` lies in | (iv) Fourth quadrant |
| (e) If a, b, c ∈ R and b2 – 4ac < 0, then the roots of the equation ax2 + bx + c = 0 are non real (complex) and |
(v) may not occur in conjugate pairs |
| (f) If a, b, c ∈ R and b2 – 4ac > 0, and b2 – 4ac is a perfect square, then the roots of the equation ax2 + bx + c = 0 |
(vi) may occur in conjugate pairs |
If `((1 - i)/(1 + i))^100` = a + ib, then find (a, b).
