Advertisements
Advertisements
प्रश्न
Find the least positive integral value of n for which \[\left( \frac{1 + i}{1 - i} \right)^n\] is real.
Advertisements
उत्तर
`((1+i)/(1-i))^n`
Simplify the base
`(1+i)/(1-i)`
`= ((1+i)(1+i))/((1-i)(1+i)) = (1+i)^2/(1^2-i^2)`
(1 + i)2 = 1 + 2i + i2 = 1 + 2i − 1 = 2i
12 − i2 = 1 − (−1) = 2
`(1+i)/(1-i) = (2i)/2 = i`
`((1+i)/(1-i))^n = i^n`
Find the least positive n
n = 2
n = 4 ...
n = 2
APPEARS IN
संबंधित प्रश्न
Express the given complex number in the form a + ib: `(5i) (- 3/5 i)`
Express the given complex number in the form a + ib: i9 + i19
Express the given complex number in the form a + ib: 3(7 + i7) + i(7 + i7)
Find the value of the following expression:
i30 + i80 + i120
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:
\[(1 + 2i )^{- 3}\]
Find the real value of x and y, if `((1+i)x-2i)/(3+i) + ((2-3i)y+i)/(3-i) = i, xy ∈ R, i = sqrt-1`
If \[a = \cos\theta + i\sin\theta\], find the value of \[\frac{1 + a}{1 - a}\].
Solve the system of equations \[\text { Re }\left( z^2 \right) = 0, \left| z \right| = 2\].
What is the smallest positive integer n for which \[\left( 1 + i \right)^{2n} = \left( 1 - i \right)^{2n}\] ?
Write (i25)3 in polar form.
Express the following complex in the form r(cos θ + i sin θ):
tan α − i
Express the following complex in the form r(cos θ + i sin θ):
\[\frac{1 - i}{\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}}\]
Write the value of \[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}}\] .
Write 1 − i in polar form.
Find z, if \[\left| z \right| = 4 \text { and }\arg(z) = \frac{5\pi}{6} .\]
Write the value of \[\arg\left( z \right) + \arg\left( \bar{z} \right)\].
If \[\left| z \right| = 2 \text { and } \arg\left( z \right) = \frac{\pi}{4}\],find z.
The value of \[(1 + i)(1 + i^2 )(1 + i^3 )(1 + i^4 )\] is.
If (x + iy)1/3 = a + ib, then \[\frac{x}{a} + \frac{y}{b} =\]
\[\text { If } z = \frac{1}{(1 - i)(2 + 3i)}, \text { than } \left| z \right| =\]
\[\text { If }z = 1 - \text { cos }\theta + i \text { sin }\theta, \text { then } \left| z \right| =\]
\[\frac{1 + 2i + 3 i^2}{1 - 2i + 3 i^2}\] equals
If \[f\left( z \right) = \frac{7 - z}{1 - z^2}\] , where \[z = 1 + 2i\] then \[\left| f\left( z \right) \right|\] is
A real value of x satisfies the equation \[\frac{3 - 4ix}{3 + 4ix} = a - ib (a, b \in \mathbb{R}), if a^2 + b^2 =\]
Which of the following is correct for any two complex numbers z1 and z2?
Find a and b if abi = 3a − b + 12i
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)
Evaluate the following : i35
Evaluate the following : i888
Evaluate the following : `1/"i"^58`
Evaluate the following : i30 + i40 + i50 + i60
If `((1 + "i"sqrt3)/(1 - "i"sqrt3))^"n"` is an integer, then n is ______.
If z1 = 3 – 2i and z2 = –1 + 3i, then Im(z1z2) = ______.
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).
State True or False for the following:
The order relation is defined on the set of complex numbers.
Show that `(-1+ sqrt(3)i)^3` is a real number.
