Advertisements
Advertisements
Question
If `((1+i)/(1-i))^m` = 1, then find the least positive integral value of m.
Advertisements
Solution
`((1+i)/(1-i))^m` = 1,
⇒ `((1+i)/(1-i) xx (1 + i)/(1 + i))^m` = 1,
⇒ `((1+ i)^2/(1^2 + 1^2))^m = 1`
⇒ `((1^2 + i^2 + 2i)/2)^2 = 1`
⇒ `((1 - 1 + 2i)/2)^2 = 1`
⇒ `((2i)/2)^m = 1`
⇒ `i^m = 1`
∴ m = 4k, where k is an integral
Therefore, the smallest positive integral is 1
Therefore, the least positive integral value of m is 4 (4 x 1).
APPEARS IN
RELATED QUESTIONS
Simplify the following and express in the form a + ib:
(2 + 3i)(1 – 4i)
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:
`(sqrt(5) + sqrt(3)"i")/(sqrt(5) - sqrt(3)"i")`
Write the conjugates of the following complex number:
`-sqrt(-5)`
Write the conjugates of the following complex number:
`sqrt(5) - "i"`
Is (1 + i14 + i18 + i22) a real number? Justify your answer
Evaluate: `("i"^37 + 1/"i"^67)`
If `("a" + 3"i")/(2+ "ib")` = 1 − i, show that (5a − 7b) = 0
If x + iy = `sqrt(("a" + "ib")/("c" + "id")`, prove that (x2 + y2)2 = `("a"^2 + "b"^2)/("c"^2 + "d"^2)`
Answer the following:
Simplify the following and express in the form a + ib:
`3 + sqrt(-64)`
Answer the following:
Simplify the following and express in the form a + ib:
`(1 + 2/"i")(3 + 4/"i")(5 + "i")^-1`
Answer the following:
Simplify the following and express in the form a + ib:
`(5 + 7"i")/(4 + 3"i") + (5 + 7"i")/(4 - 3"i")`
Answer the following:
show that `((1 + "i")/sqrt(2))^8 + ((1 - "i")/sqrt(2))^8` = 2
Answer the following:
If x + iy = `("a" + "ib")/("a" - "ib")`, prove that x2 + y2 = 1
Answer the following:
Simplify: `("i"^29 + "i"^39 + "i"^49)/("i"^30 + "i"^40 + "i"^50)`
The argument of the complex number `(4 + 9i)/(13 + 5i)` is ______
If z1, z2, z3 are complex numbers such that `|z_1| = |z_2| = |z_3| = |1/z_1 + 1/z_2 + 1/z_3|` = 1, then find the value of |z1 + z2 + z3|.
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 value of `(i^(4n + 1) -i^(4n - 1))/2`?
What is the smallest positive integer n, for which (1 + i)2n = (1 – i)2n?
If z = x + iy, then show that `z barz + 2(z + barz) + b` = 0, where b ∈ R, represents a circle.
Solve the equation |z| = z + 1 + 2i.
Multiplicative inverse of 1 + i is ______.
If z1 and z2 are complex numbers such that z1 + z2 is a real number, then z2 = ______.
State True or False for the following:
For any complex number z the minimum value of |z| + |z – 1| is 1.
Which of the following is correct for any two complex numbers z1 and z2?
The point represented by the complex number 2 – i is rotated about origin through an angle `pi/2` in the clockwise direction, the new position of point is ______.
The complex number z which satisfies the condition `|(i + z)/(i - z)|` = 1 lies on ______.
If the least and the largest real values of α, for which the equation z + α|z – 1| + 2i = 0 `("z" ∈ "C" and "i" = sqrt(-1))` has a solution, are p and q respectively; then 4(p2 + q2) is equal to ______.
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 ______.
`((1 + cosθ + isinθ)/(1 + cosθ - isinθ))^n` = ______.
If α and β are the roots of the equation x2 + 2x + 4 = 0, then `1/α^3 + 1/β^3` is equal to ______.
If α, β, γ and a, b, c are complex numbers such that `α/a + β/b + γ/c` = 1 + i and `a/α + b/β + c/γ` = 0, then the value of `α^2/a^2 + β^2/b^2 + γ^2/c^2` is equal to ______.
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)`
Simplify the following and express in the form a+ib:
`(3i^5 + 2i^7 + i^9)/(i^6 + 2i^8 + 3i^18)`
