Advertisements
Advertisements
प्रश्न
If `((1 + i)/(1 - i))^x` = 1, then ______.
विकल्प
x = 2n + 1
x = 4n
x = 2n
x = 4n + 1, where n ∈ N
Advertisements
उत्तर
If `((1 + i)/(1 - i))^x` = 1, then x = 4n.
Explanation:
Given that: `((1 + i)/(1 - i))^x` = 1
⇒ `(((1 + i)(1 + i))/((1 - i)(1 - i)))^x` = 1
⇒ `((1 + i^2 + 2i)/(1 - i^2))^x` = 1
⇒ `((1 - 1 + 2i)/(1 + 1))^x` = 1
⇒ `((2i)/2)^x` = 1
⇒ (i)x = (i)4n
⇒ x = 4n, n ∈ N
APPEARS IN
संबंधित प्रश्न
Reduce `(1/(1-4i) - 2/(1+i))((3-4i)/(5+i))` to the standard form.
If `x – iy = sqrt((a-ib)/(c - id))` prove that `(x^2 + y^2) = (a^2 + b^2)/(c^2 + d^2)`
If α and β are different complex numbers with |β| = 1, then find `|(beta - alpha)/(1-baralphabeta)|`
Find the number of non-zero integral solutions of the equation `|1-i|^x = 2^x`.
If (a + ib) (c + id) (e + if) (g + ih) = A + iB, then show that (a2 + b2) (c2 + d2) (e2 + f2) (g2 + h2) = A2 + B2.
If `((1+i)/(1-i))^m` = 1, then find the least positive integral value of m.
Find the value of: 2x3 – 11x2 + 44x + 27, if x = `25/(3 - 4"i")`
Simplify the following and express in the form a + ib:
(2i3)2
Simplify the following and express in the form a + ib:
(1 + 3i)2 (3 + i)
Simplify the following and express in the form a + ib:
`(5 + 7"i")/(4 + 3"i") + (5 + 7"i")/(4 - 3"i")`
Write the conjugates of the following complex number:
3 + i
Write the conjugates of the following complex number:
3 – i
Write the conjugates of the following complex number:
`-sqrt(5) - sqrt(7)"i"`
Prove that `(1 + "i")^4 xx (1 + 1/"i")^4` = 16
Answer the following:
Simplify the following and express in the form a + ib:
`(4 + 3"i")/(1 - "i")`
Answer the following:
Solve the following equations for x, y ∈ R:
(x + iy) (5 + 6i) = 2 + 3i
Answer the following:
Find the value of x3 + 2x2 − 3x + 21, if x = 1 + 2i
Answer the following:
Simplify: `("i"^29 + "i"^39 + "i"^49)/("i"^30 + "i"^40 + "i"^50)`
Answer the following:
Simplify: `("i"^238 + "i"^236 + "i"^234 + "i"^232 + "i"^230)/("i"^228 + "i"^226 + "i"^224 + "i"^222 + "i"^220)`
If z ≠ 1 and `"z"^2/("z - 1")` is real, then the point represented by the complex number z lies ______.
Find the value of 2x4 + 5x3 + 7x2 – x + 41, when x = `-2 - sqrt(3)"i"`.
State true or false for the following:
The argument of the complex number z = `(1 + i sqrt(3))(1 + i)(cos theta + i sin theta)` is `(7pi)/12 + theta`.
What is the reciprocal of `3 + sqrt(7)i`.
What is the locus of z, if amplitude of z – 2 – 3i is `pi/4`?
If z = x + iy, then show that `z barz + 2(z + barz) + b` = 0, where b ∈ R, represents a circle.
If |z1| = |z2| = ... = |zn| = 1, then show that |z1 + z2 + z3 + ... + zn| = `|1/z_1 + 1/z_2 + 1/z_3 + ... + 1/z_n|`.
The value of `sqrt(-25) xx sqrt(-9)` is ______.
If |z + 4| ≤ 3, then the greatest and least values of |z + 1| are ______ and ______.
The real value of α for which the expression `(1 - i sin alpha)/(1 + 2i sin alpha)` is purely real is ______.
If a + ib = c + id, then ______.
Let |z| = |z – 3| = |z – 4i|, then the value |2z| is ______.
If `(3 + i)(z + barz) - (2 + i)(z - barz) + 14i` = 0, then `barzz` is equal to ______.
Find the value of `(i^592+i^590+i^588+i^586+i^584)/(i^582+i^580+i^578+i^576+i^574)`
Simplify the following and express in the form a + ib.
`(3i^5 + 2i^7 + i^9)/(i^6 + 2i^8 + 3i^18)`
Simplify the following and express in the form a+ib.
`(3i^5 + 2i^7 + i^9)/(i^6 + 2i^8 + 3i^18`
Find the value of `sqrt(-3) xx sqrt(-6)`
Simplify the following and express in the form a+ib.
`(3i^5 + 2i^7 + i^9)/(i^6 + 2i^8 + 3i^18)`
