Advertisements
Advertisements
Question
If z1 and z2 are complex numbers such that z1 + z2 is a real number, then z2 = ______.
Advertisements
Solution
If z1 and z2 are complex numbers such that z1 + z2 is a real number, then z2 = `underlinebb(barz_1)`.
Explanation:
Let z1 = x1 + iy1 and z2 = x2 + iy2
z1 + z2 = (x1 + iy2) + (x2 + iy2)
z1 + z2 = (x1 + x2) + (y1 + y2)i
If z1 + z2 is real then,
y1 + y2 = 0
⇒ y1 = –y2
∴ z2 = x2 – iy1
z2 = x1 – iy1 ......(When x1 = x2)
So z2 = `barz_1`
APPEARS IN
RELATED QUESTIONS
Find the multiplicative inverse of the complex number:
4 – 3i
Reduce `(1/(1-4i) - 2/(1+i))((3-4i)/(5+i))` to the standard form.
Find the number of non-zero integral solutions of the equation `|1-i|^x = 2^x`.
Find the value of i49 + i68 + i89 + i110
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: x3 – 5x2 + 4x + 8, if x = `10/(3 - "i")`.
Write the conjugates of the following complex number:
`-sqrt(5) - sqrt(7)"i"`
Show that 1 + i10 + i100 − i1000 = 0
Is (1 + i14 + i18 + i22) a real number? Justify your answer
If `("a" + 3"i")/(2+ "ib")` = 1 − i, show that (5a − 7b) = 0
If (a + ib) = `(1 + "i")/(1 - "i")`, then prove that (a2 + b2) = 1
Show that `((sqrt(7) + "i"sqrt(3))/(sqrt(7) - "i"sqrt(3)) + (sqrt(7) - "i"sqrt(3))/(sqrt(7) + "i"sqrt(3)))` is real
Find the value of x and y which satisfy the following equation (x, y∈R).
(x + 2y) + (2x − 3y)i + 4i = 5
Answer the following:
Simplify the following and express in the form a + ib:
`(4 + 3"i")/(1 - "i")`
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:
Evaluate: (1 − i + i2)−15
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 z ≠ 1 and `"z"^2/("z - 1")` is real, then the point represented by the complex number z lies ______.
The value of (2 + i)3 × (2 – i)3 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|.
Find the value of 2x4 + 5x3 + 7x2 – x + 41, when x = `-2 - sqrt(3)"i"`.
The value of `(- sqrt(-1))^(4"n" - 3)`, where n ∈ N, is ______.
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 value of `(i^(4n + 1) -i^(4n - 1))/2`?
What is the locus of z, if amplitude of z – 2 – 3i is `pi/4`?
The equation |z + 1 – i| = |z – 1 + i| represents a ______.
If `((1 + i)/(1 - i))^3 - ((1 - i)/(1 + i))^3` = x + iy, then find (x, y).
Solve the equation |z| = z + 1 + 2i.
State True or False for the following:
The locus represented by |z – 1| = |z – i| is a line perpendicular to the join of (1, 0) and (0, 1).
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 = ______.
`((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 ______.
The complex number z = x + iy which satisfy the equation `|(z - 5i)/(z + 5i)|` = 1, lie on ______.
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 =
