Advertisements
Advertisements
प्रश्न
The least positive integer n such that \[\left( \frac{2i}{1 + i} \right)^n\] is a positive integer, is.
पर्याय
16
8
4
2
Advertisements
उत्तर
\[8\]
\[\text { Let } z = \left( \frac{2i}{1 + i} \right)\]
\[ \Rightarrow z = \frac{2i}{1 + i} \times \frac{1 - i}{1 - i}\]
\[ \Rightarrow z = \frac{2i\left( 1 - i \right)}{1 - i^2}\]
\[ \Rightarrow z = \frac{2i\left( 1 - i \right)}{1 + 1} \left[ \because i^2 = - 1 \right]\]
\[ \Rightarrow z = \frac{2i\left( 1 - i \right)}{2}\]
\[ \Rightarrow z = i - i^2 \]
\[ \Rightarrow z = i + 1\]
\[\text { Now }, z^n = \left( 1 + i \right)^n \]
\[\text { For } n = 2, \]
\[ z^2 = \left( 1 + i \right)^2 \]
\[ = 1 + i^2 + 2i\]
\[ = 1 - 1 + 2i\]
\[ = 2i . . . (1) \]
\[\text { Since this is not a positive integer }, \]
\[\text { For } n = 4, \]
\[ z^4 = \left( 1 + i \right)^4 \]
\[ = \left[ \left( 1 + i \right)^2 \right]^2 \]
\[ = \left( 2i \right)^2 \left[ \text { Using } (1) \right] \]
\[ = 4 i^2 \]
\[ = - 4 . . . (2)\]
\[\text { This is a negative integer }. \]
\[\text { For } n = 8, \]
\[ z^8 = \left( 1 + i \right)^8 \]
\[ = \left[ \left( 1 + i \right)^4 \right]^2 \]
\[ = \left( - 4 \right)^2 \left[ \text { Using } (2) \right]\]
\[ = 16\]
\[\text { This is a positive integer } . \]
\[\text { Thus }, z = \left( \frac{2i}{1 + i} \right)^n\text { is positive for } n = 8 . \]
\[\text { Therefore, 8 is the least positive integer such that } \left( \frac{2i}{1 + i} \right)^n\text { is a positive integer } .\]
APPEARS IN
संबंधित प्रश्न
Express the given complex number in the form a + ib: i–39
Express the given complex number in the form a + ib: `(1/5 + i 2/5) - (4 + i 5/2)`
Express the given complex number in the form a + ib: `(-2 - 1/3 i)^3`
Evaluate: `[i^18 + (1/i)^25]^3`
If a + ib = `(x + i)^2/(2x^2 + 1)` prove that a2 + b2 = `(x^2 + 1)^2/(2x + 1)^2`
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:
\[\left( \frac{1}{1 - 4i} - \frac{2}{1 + i} \right)\left( \frac{3 - 4i}{5 + i} \right)\]
If \[z_1 = 2 - i, z_2 = - 2 + i,\] find
Im `(1/(z_1overlinez_1))`
If \[a = \cos\theta + i\sin\theta\], find the value of \[\frac{1 + a}{1 - a}\].
Evaluate the following:
\[x^6 + x^4 + x^2 + 1, \text { when }x = \frac{1 + i}{\sqrt{2}}\]
Evaluate the following:
\[2 x^4 + 5 x^3 + 7 x^2 - x + 41, \text { when } x = - 2 - \sqrt{3}i\]
If \[\left| z + 1 \right| = z + 2\left( 1 + i \right)\],find z.
What is the smallest positive integer n for which \[\left( 1 + i \right)^{2n} = \left( 1 - i \right)^{2n}\] ?
Express the following complex in the form r(cos θ + i sin θ):
1 − sin α + i cos α
If π < θ < 2π and z = 1 + cos θ + i sin θ, then write the value of \[\left| z \right|\] .
Find z, if \[\left| z \right| = 4 \text { and }\arg(z) = \frac{5\pi}{6} .\]
If n ∈ \[\mathbb{N}\] then find the value of \[i^n + i^{n + 1} + i^{n + 2} + i^{n + 3}\] .
If `(3+2i sintheta)/(1-2 i sin theta)`is a real number and 0 < θ < 2π, then θ =
The argument of \[\frac{1 - i\sqrt{3}}{1 + i\sqrt{3}}\] is
If θ is the amplitude of \[\frac{a + ib}{a - ib}\] , than tan θ =
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
Which of the following is correct for any two complex numbers z1 and z2?
If the complex number \[z = x + iy\] satisfies the condition \[\left| z + 1 \right| = 1\], then z lies on
Simplify : `4sqrt(-4) + 5sqrt(-9) - 3sqrt(-16)`
Find a and b if abi = 3a − b + 12i
Find a and b if (a + ib) (1 + i) = 2 + i
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)
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)`
Evaluate the following : i403
Evaluate the following : i–888
If z1 = 3 – 2i and z2 = –1 + 3i, then Im(z1z2) = ______.
The real value of θ for which the expression `(1 + i cos theta)/(1 - 2i cos theta)` is a real number is ______.
If w is a complex cube-root of unity, then prove the following
(w2 + w − 1)3 = −8
