Advertisements
Advertisements
प्रश्न
Find the value of the following expression:
\[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}}\]
उत्तर
\[ \frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}}\]
\[ = \frac{i^{4 \times 148} + i^{4 \times 147 + 2} + i^{4 \times 147} + i^{4 \times 146 + 2} + i^{4 \times 146}}{i^{4 \times 145 + 2} + i^{4 \times 145} + i^{4 \times 144 + 2} + i^{4 \times 144} + i^{4 \times 143 + 2}}\]
\[ = \frac{\left( i^4 \right)^{148} + \left\{ \left( i^4 \right)^{147} \times i^2 \right\} + \left\{ \left( i^4 \right)^{146} \right\} + \left\{ \left( i^4 \right)^{146} \times i^2 \right\} + \left\{ \left( i^4 \right)^{146} \right\}}{\left\{ \left( i^4 \right)^{145} \times i^2 \right\} + \left\{ \left( i^4 \right)^{145} \right\} + \left\{ \left( i^4 \right)^{144} \times i^2 \right\} + \left\{ \left( i^4 \right)^{144} \right\} + \left\{ \left( i^4 \right)^{143} \times i^2 \right\}}\]
\[ = \frac{1 + i^2 + 1 + i^2 + 1}{i^2 + 1 + i^2 + 1 + i^2} \left[ \because i^4 = 1 \right]\]
\[ = \frac{1 - 1 + 1 - 1 + 1}{- 1 + 1 - 1 + 1 - 1} \left[ \because i^2 = - 1 \right]\]
\[ = - 1\]
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: (1 – i) – (–1 + i6)
Express the given complex number in the form a + ib: `(-2 - 1/3 i)^3`
Evaluate: `[i^18 + (1/i)^25]^3`
Let z1 = 2 – i, z2 = –2 + i. Find `"Im"(1/(z_1barz_1))`
Evaluate the following:
(ii) i528
Evaluate the following:
\[i^{30} + i^{40} + i^{60}\]
Show that 1 + i10 + i20 + i30 is a real number.
Find the value of the following expression:
i49 + i68 + i89 + i110
Find the value of the following expression:
i30 + i80 + i120
Express the following complex number in the standard form a + i b:
\[\frac{3 + 2i}{- 2 + i}\]
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 + i b:
\[\frac{1 - i}{1 + i}\]
Express the following complex number in the standard form a + i b:
\[\left( \frac{1}{1 - 4i} - \frac{2}{1 + i} \right)\left( \frac{3 - 4i}{5 + i} \right)\]
Evaluate the following:
\[x^4 + 4 x^3 + 6 x^2 + 4x + 9, \text { when } x = - 1 + i\sqrt{2}\]
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):
\[\frac{1 - i}{\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}}\]
Express \[\sin\frac{\pi}{5} + i\left( 1 - \cos\frac{\pi}{5} \right)\] in polar form.
Write −1 + i \[\sqrt{3}\] in polar form .
Write the value of \[\sqrt{- 25} \times \sqrt{- 9}\].
The polar form of (i25)3 is
The principal value of the amplitude of (1 + i) is
The least positive integer n such that \[\left( \frac{2i}{1 + i} \right)^n\] is a positive integer, is.
\[\text { If } z = \frac{1}{(2 + 3i )^2}, \text { than } \left| z \right| =\]
If θ is the amplitude of \[\frac{a + ib}{a - ib}\] , than tan θ =
If \[z = \frac{1 + 7i}{(2 - i )^2}\] , then
If z is a complex number, then
Simplify : `4sqrt(-4) + 5sqrt(-9) - 3sqrt(-16)`
Express the following in the form of a + ib, a, b∈R i = `sqrt(−1)`. State the values of a and b:
`((2 + "i"))/((3 - "i")(1 + 2"i"))`
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")`
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 : i93
Evaluate the following : `1/"i"^58`
If `((1 - i)/(1 + i))^100` = a + ib, then find (a, b).
If a = cosθ + isinθ, find the value of `(1 + "a")/(1 - "a")`.
