Advertisements
Advertisements
प्रश्न
If z is a complex number, then ______.
पर्याय
|z2| > |z|2
|z2| = |z|2
|z2| < |z|2
|z2| ≥ |z|2
उत्तर
If z is a complex number, then |z2| = |z|2.
Explanation:
Let z = x + yi
|z| = |z + yi| and |z|2 = |x + yi|2
⇒ |z|2 = x2 + y2 ......(i)
Now z2 = x2 + y2i2 + 2xyi
z2 = x2 – y2 + 2xyi
|z|2 = `sqrt((x^2 - y^2)^2 + (xy)^2)`
= `sqrt(x^4 + y^4 - 2x^2 y^2 + 4x^2 y^2)`
= `sqrt(x^4 + y^4 + 2x^2 y^2)`
= `sqrt((x^2 + y^2)^2`
So |z2| = x2 + y2 = |z|2
So |z2| = |z|2
APPEARS IN
संबंधित प्रश्न
Find the multiplicative inverse of the complex number:
4 – 3i
Find the value of i + i2 + i3 + i4
Simplify the following and express in the form a + ib:
`3 + sqrt(-64)`
Simplify the following and express in the form a + ib:
`(1 + 2/"i")(3 + 4/"i")(5 + "i")^-1`
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:
cosθ + i sinθ
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:
Solve the following equation for x, y ∈ R:
(4 − 5i)x + (2 + 3i)y = 10 − 7i
Answer the following:
Show that `(1 - 2"i")/(3 - 4"i") + (1 + 2"i")/(3 + 4"i")` is real
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 `(x + iy)^(1/3)` = a + ib, where x, y, a, b ∈ R, show that `x/a - y/b` = –2(a2 + b2)
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|.
Locate the points for which 3 < |z| < 4.
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.
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`.
1 + i2 + i4 + i6 + ... + i2n is ______.
If `(z - 1)/(z + 1)` is purely imaginary number (z ≠ – 1), then find the value of |z|.
Find the complex number satisfying the equation `z + sqrt(2) |(z + 1)| + i` = 0.
The value of `sqrt(-25) xx sqrt(-9)` is ______.
State True or False for the following:
The inequality |z – 4| < |z – 2| represents the region given by x > 3.
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 ______.
Let x, y ∈ R, then x + iy is a non-real complex number if ______.
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 ______.
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 ______.
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)`
Show that `(-1 + sqrt3 i)^3` is a real number.
