Advertisements
Advertisements
प्रश्न
What is the smallest positive integer n, for which (1 + i)2n = (1 – i)2n?
उत्तर
n = 2, Because (1 + i)2n = (1 – i)2n ⇒ `((1 + i)/(1 - i))^(2n)` = 1
⇒ (i)2n = 1 Which is possible if n = 2 ......(∴ i4 = 1)
APPEARS IN
संबंधित प्रश्न
Simplify the following and express in the form a + ib:
`(3"i"^5 + 2"i"^7 + "i"^9)/("i"^6 + 2"i"^8 + 3"i"^18)`
Write the conjugates of the following complex number:
3 + i
Write the conjugates of the following complex number:
5i
Write the conjugates of the following complex number:
cosθ + i sinθ
Simplify : `("i"^592 + "i"^590 + "i"^588 + "i"^586 + "i"^584)/("i"^582 + "i"^580 + "i"^578 + "i"^576 + "i"^574)`
Find the value of 1 + i2 + i4 + i6 + i8 + ... + i20
Prove that `(1 + "i")^4 xx (1 + 1/"i")^4` = 16
Find the value of `("i"^6 + "i"^7 + "i"^8 + "i"^9)/("i"^2 + "i"^3)`
Solve the following equation for x, y ∈ R:
2x + i9y (2 + i) = xi7 + 10i16
Answer the following:
Show that `(1 - 2"i")/(3 - 4"i") + (1 + 2"i")/(3 + 4"i")` is real
The value of (2 + i)3 × (2 – i)3 is ______.
Evaluate: (1 + i)6 + (1 – i)3
State true or false for the following:
Multiplication of a non-zero complex number by i rotates it through a right angle in the anti-clockwise direction.
What is the locus of z, if amplitude of z – 2 – 3i is `pi/4`?
Number of solutions of the equation z2 + |z|2 = 0 is ______.
If z = x + iy, then show that `z barz + 2(z + barz) + b` = 0, where b ∈ R, represents a circle.
Find the complex number satisfying the equation `z + sqrt(2) |(z + 1)| + i` = 0.
The sum of the series i + i2 + i3 + ... upto 1000 terms is ______.
Where does z lie, if `|(z - 5i)/(z + 5i)|` = 1.
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 ______.
If a + ib = c + id, then ______.
The complex number z which satisfies the condition `|(i + z)/(i - z)|` = 1 lies on ______.
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 ______.
A complex number z is moving on `arg((z - 1)/(z + 1)) = π/2`. If the probability that `arg((z^3 -1)/(z^3 + 1)) = π/2` is `m/n`, where m, n ∈ prime, then (m + n) is equal to ______.
`((1 + cosθ + isinθ)/(1 + cosθ - isinθ))^n` = ______.
Let z be a complex number such that `|(z - i)/(z + 2i)|` = 1 and |z| = `5/2`. Then the value of |z + 3i| is ______.
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 ______.
Show that `(-1 + sqrt3 i)^3` is a real number.