Advertisements
Advertisements
प्रश्न
What is the smallest positive integer n, for which (1 + i)2n = (1 – i)2n?
Advertisements
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
Find the multiplicative inverse of the complex number:
4 – 3i
Show that 1 + i10 + i20 + i30 is a real number.
Find the value of i + i2 + i3 + i4
Find the value of: x3 – x2 + x + 46, if x = 2 + 3i
Simplify the following and express in the form a + ib:
`3 + sqrt(-64)`
Simplify the following and express in the form a + ib:
`5/2"i"(- 4 - 3 "i")`
Find the value of : x3 + 2x2 – 3x + 21, if x = 1 + 2i
If a = `(-1 + sqrt(3)"i")/2`, b = `(-1 - sqrt(3)"i")/2` then show that a2 = b and b2 = a
Find the value of x and y which satisfy the following equation (x, y ∈ R).
`((x + "i"y))/(2 + 3"i") + (2 + "i")/(2 - 3"i") = 9/13(1 + "i")`
Find the value of x and y which satisfy the following equation (x, y∈R).
If x + 2i + 15i6y = 7x + i3 (y + 4), find x + y
Answer the following:
Simplify the following and express in the form a + ib:
(2i3)2
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:
`(sqrt(5) + sqrt(3)"i")/(sqrt(5) - sqrt(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)`
If `(x + iy)^(1/3)` = a + ib, where x, y, a, b ∈ R, show that `x/a - y/b` = –2(a2 + b2)
Locate the points for which 3 < |z| < 4.
If (2 + i) (2 + 2i) (2 + 3i) ... (2 + ni) = x + iy, then 5.8.13 ... (4 + n2) = ______.
State true or false for the following:
If three complex numbers z1, z2 and z3 are in A.P., then they lie on a circle in the complex plane.
What is the principal value of amplitude of 1 – i?
What is the locus of z, if amplitude of z – 2 – 3i is `pi/4`?
The area of the triangle on the complex plane formed by the complex numbers z, –iz and z + iz is ______.
Number of solutions of the equation z2 + |z|2 = 0 is ______.
Evaluate `sum_(n = 1)^13 (i^n + i^(n + 1))`, where n ∈ N.
If `(1 + i)^2/(2 - i)` = x + iy, then find the value of x + y.
State True or False for the following:
The inequality |z – 4| < |z – 2| represents the region given by x > 3.
The value of `(z + 3)(barz + 3)` is equivalent to ______.
A real value of x satisfies the equation `((3 - 4ix)/(3 + 4ix))` = α − iβ (α, β ∈ R) if α2 + β2 = ______.
Which of the following is correct for any two complex numbers z1 and z2?
The complex number z which satisfies the condition `|(i + z)/(i - z)|` = 1 lies on ______.
If z is a complex number, then ______.
`((1 + cosθ + isinθ)/(1 + cosθ - isinθ))^n` = ______.
The smallest positive integer n for which `((1 + i)/(1 - i))^n` = –1 is ______.
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 `|(6i, -3i, 1),(4, 3i, -1),(20, 3, i)|` = x + iy, then ______.
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)`
i2 + i3 + ... + i4000 =
