Advertisements
Advertisements
प्रश्न
If `((1 - i)/(1 + i))^100` = a + ib, then find (a, b).
उत्तर
We have `((1 - i)/(1 + i))^100` = a + bi
⇒ `((1 - i)/(1 + i) xx (1 - i)/(1 - i))^100` = a + bi
⇒ `((1 + i^2 - 2i)/(1 - i^2))^100` = a + bi
⇒ `((1 - 1 - 2i)/(1 + 1))^100` = a + bi
⇒ `((-2i)/2)^100` = a + bi
⇒ (–i)100 = a + bi
⇒ i100 = a + bi
⇒ (i4)25 = a + bi
⇒ (1)25 = a + bi
⇒ 1 = a + bi
⇒ 1 + 0i = a + bi
Comparing the real and imaginary parts,
We have a = 1, b = 0
Hence (a, b) = (1, 0)
APPEARS IN
संबंधित प्रश्न
Express the given complex number in the form a + ib: (1 – i) – (–1 + i6)
Let z1 = 2 – i, z2 = –2 + i. Find `"Im"(1/(z_1barz_1))`
Evaluate the following:
\[\left( i^{41} + \frac{1}{i^{257}} \right)^9\]
Evaluate the following:
\[i^{49} + i^{68} + i^{89} + i^{110}\]
Find the value of the following expression:
i5 + i10 + i15
Find the multiplicative inverse of the following complex number:
1 − i
If \[z_1 = 2 - i, z_2 = - 2 + i,\] find
Im `(1/(z_1overlinez_1))`
Find the least positive integral value of n for which \[\left( \frac{1 + i}{1 - i} \right)^n\] is real.
If \[\frac{\left( 1 + i \right)^2}{2 - i} = x + iy\] 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 \[(1 - i )^n \left( 1 - \frac{1}{i} \right)^n\].
If \[\frac{z - 1}{z + 1}\] is purely imaginary number (\[z \neq - 1\]), find the value of \[\left| z \right|\].
Express the following complex in the form r(cos θ + i sin θ):
tan α − i
If n is any positive integer, write the value of \[\frac{i^{4n + 1} - i^{4n - 1}}{2}\].
If \[\left| z - 5i \right| = \left| z + 5i \right|\] , then find the locus of z.
For any two complex numbers z1 and z2 and any two real numbers a, b, find the value of \[\left| a z_1 - b z_2 \right|^2 + \left| a z_2 + b z_1 \right|^2\].
The least positive integer n such that \[\left( \frac{2i}{1 + i} \right)^n\] is a positive integer, is.
The argument of \[\frac{1 - i\sqrt{3}}{1 + i\sqrt{3}}\] is
The amplitude of \[\frac{1}{i}\] is equal to
If \[z = a + ib\] lies in third quadrant, then \[\frac{\bar{z}}{z}\] also lies in third quadrant if
If z is a complex number, then
Evaluate the following : i35
Evaluate the following : i116
State True or False for the following:
2 is not a complex number.
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 + sqrt3 "i")^3` is a real number.
If w is a complex cube-root of unity, then prove the following
(w2 + w − 1)3 = −8
