Advertisements
Advertisements
Question
Which of the following is correct for any two complex numbers z1 and z2?
Options
|z1z2| = |z1||z2|
arg(z1z2) = arg(z1).arg(z2)
|z1 + z2| = |z1| + |z2|
|z1 + z2| ≥ |z1| – |z2|
Advertisements
Solution
|z1z2| = |z1||z2|
Explanation:
Let z1 = r1(cosθ1 + isinθ1)
∴ |z2| = r1
And z2 = r2(cosθ2 + isinθ2)
∴ |z2| = r2
z1z2 = r1(cosθ1 + isinθ1).r2(cosθ2 + isinθ2)
= r1r2(cosθ1 + isinθ1).(cosθ2 + isinθ2)
= r1r2(cosθ1 cosθ2 + isinθ2 cosθ1 + isinθ1 cosθ2 + i2sinθ1 sinθ2)
= r1r2 [(cosθ1 cosθ2 – sinθ1 sinθ2) + i(sinθ1 cosθ2 + cosθ1 sinθ2)]
= r1r2 [cos(θ1 + θ2) + isin(θ1 + θ2)]
∴ |z1z2| = |z1||z2|
APPEARS IN
RELATED QUESTIONS
Find the number of non-zero integral solutions of the equation `|1-i|^x = 2^x`.
Simplify the following and express in the form a + ib:
`(4 + 3"i")/(1 - "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:
3 – i
Simplify:
`(i^592 + i^590 + i^588 + i^586 + i^584)/(i^582 + i^580 + i^578 + i^576 + i^574)`
If x + iy = `sqrt(("a" + "ib")/("c" + "id")`, prove that (x2 + y2)2 = `("a"^2 + "b"^2)/("c"^2 + "d"^2)`
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:
`3 + sqrt(-64)`
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`
Solve the following equation for x, y ∈ R:
(4 − 5i)x + (2 + 3i)y = 10 − 7i
Answer the following:
Find the value of x3 + 2x2 − 3x + 21, if x = 1 + 2i
If |z1| = 1, |z2| = 2, |z3| = 3 and |9z1z2 + 4z1z3 + z2z3| = 12, then the value of |z1 + z2 + z3| is
The argument of the complex number `(4 + 9i)/(13 + 5i)` is ______
Evaluate: (1 + i)6 + (1 – i)3
State true or false for the following:
If n is a positive integer, then the value of in + (i)n+1 + (i)n+2 + (i)n+3 is 0.
What is the smallest positive integer n, for which (1 + i)2n = (1 – i)2n?
The equation |z + 1 – i| = |z – 1 + i| represents a ______.
For a positive integer n, find the value of `(1 - i)^n (1 - 1/i)^"n"`
If (1 + i)z = `(1 - i)barz`, then show that z = `-ibarz`.
If z = x + iy, then show that `z barz + 2(z + barz) + b` = 0, where b ∈ R, represents a circle.
If |z1| = 1(z1 ≠ –1) and z2 = `(z_1 - 1)/(z_1 + 1)`, then show that the real part of z2 is zero.
Find the complex number satisfying the equation `z + sqrt(2) |(z + 1)| + i` = 0.
The number `(1 - i)^3/(1 - i^2)` is equal to ______.
State True or False for the following:
Multiplication of a non-zero complex number by –i rotates the point about origin through a right angle in the anti-clockwise direction.
State True or False for the following:
The inequality |z – 4| < |z – 2| represents the region given by x > 3.
If `((1 + i)/(1 - i))^x` = 1, then ______.
If a + ib = c + id, then ______.
The smallest positive integer n for which `((1 + i)/(1 - i))^n` = –1 is ______.
If a complex number z satisfies the equation `z + sqrt(2)|z + 1| + i` = 0, then |z| 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)`
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.
`(3"i"^5 + 2"i"^7 + "i"^9)/("i"^6 + 2"i"^8 + 3"i"^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 =
