Advertisements
Advertisements
प्रश्न
For a positive integer n, find the value of \[(1 - i )^n \left( 1 - \frac{1}{i} \right)^n\].
Advertisements
उत्तर
\[(1 - i )^n \left( 1 - \frac{1}{i} \right)^n = \left( 1 - i \right)^n \left( 1 - \frac{i^4}{i} \right)^n [ \because i^4 = 1]\]
\[ = \left( 1 - i \right)^n \left( 1 - i^3 \right)^n \]
\[ = \left( 1 - i \right)^n \left( 1 + i \right)^n [ \because i^3 = - i]\]
\[ = \left[ (1 - i)(1 + i) \right]^n \]
\[ = (1 - i^2 )^n \]
\[ = 2^n [ \because i^2 = - 1]\]
Thus, the value of
\[(1 - i )^n \left( 1 - \frac{1}{i} \right)^n\] is 2n.
APPEARS IN
संबंधित प्रश्न
Express the given complex number in the form a + ib: i9 + i19
Express the given complex number in the form a + ib: (1 – i) – (–1 + i6)
Evaluate the following:
(ii) i528
Evaluate the following:
\[i^{37} + \frac{1}{i^{67}}\].
Find the value of the following expression:
i + i2 + i3 + i4
Express the following complex number in the standard form a + i b:
\[(1 + i)(1 + 2i)\]
Express the following complex number in the standard form a + ib:
\[\frac{(2 + i )^3}{2 + 3i}\]
Express the following complex number in the standard form a + i b:
\[\frac{2 + 3i}{4 + 5i}\]
Express the following complex number in the standard form a + i b:
\[\frac{3 - 4i}{(4 - 2i)(1 + i)}\]
Find the real value of x and y, if
\[(1 + i)(x + iy) = 2 - 5i\]
If \[x + iy = \frac{a + ib}{a - ib}\] prove that x2 + y2 = 1.
Find the real values of θ for which the complex number \[\frac{1 + i cos\theta}{1 - 2i cos\theta}\] is purely real.
Find the smallest positive integer value of m for which \[\frac{(1 + i )^n}{(1 - i )^{n - 2}}\] is a real number.
If \[\frac{\left( 1 + i \right)^2}{2 - i} = x + iy\] find x + y.
Evaluate the following:
\[x^4 + 4 x^3 + 6 x^2 + 4x + 9, \text { when } x = - 1 + i\sqrt{2}\]
If \[\left( 1 + i \right)z = \left( 1 - i \right) \bar{z}\],then show that \[z = - i \bar{z}\].
If \[\frac{z - 1}{z + 1}\] is purely imaginary number (\[z \neq - 1\]), find the value of \[\left| z \right|\].
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
Express the following complex in the form r(cos θ + i sin θ):
\[\frac{1 - i}{\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}}\]
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 argument of −i.
If \[\frac{\left( a^2 + 1 \right)^2}{2a - i} = x + iy\] find the value of \[x^2 + y^2\].
If \[\left| z \right| = 2 \text { and } \arg\left( z \right) = \frac{\pi}{4}\],find z.
If (x + iy)1/3 = a + ib, then \[\frac{x}{a} + \frac{y}{b} =\]
\[(\sqrt{- 2})(\sqrt{- 3})\] is equal to
\[\text { If } z = \frac{1}{(2 + 3i )^2}, \text { than } \left| z \right| =\]
If \[x + iy = \frac{3 + 5i}{7 - 6i},\] then y =
If \[f\left( z \right) = \frac{7 - z}{1 - z^2}\] , where \[z = 1 + 2i\] then \[\left| f\left( z \right) \right|\] is
The complex number z which satisfies the condition \[\left| \frac{i + z}{i - z} \right| = 1\] lies on
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:
(1 + i)(1 − i)−1
Express the following in the form of a + ib, a, b ∈ R i = `sqrt(−1)`. State the values of a and b:
`(- sqrt(5) + 2sqrt(-4)) + (1 -sqrt(-9)) + (2 + 3"i")(2 - 3"i")`
Evaluate the following : i93
Evaluate the following : `1/"i"^58`
If a = cosθ + isinθ, find the value of `(1 + "a")/(1 - "a")`.
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 |
If w is a complex cube-root of unity, then prove the following
(w2 + w − 1)3 = −8
