Advertisements
Advertisements
Question
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.
Options
True
False
Advertisements
Solution
This statement is False.
Explanation:
Because if `barz_1`, z2 and z3 are in A.P.
Then z2 = `(z_1 + z_3)/2`
⇒ z2 is the midpoint of z1 and z3, which implies that the points z1, z2, z3 are collinear.
APPEARS IN
RELATED QUESTIONS
If `((1+i)/(1-i))^m` = 1, then find the least positive integral value of m.
Find the value of: 2x3 – 11x2 + 44x + 27, if x = `25/(3 - 4"i")`
Simplify the following and express in the form a + ib:
`(4 + 3"i")/(1 - "i")`
Write the conjugates of the following complex number:
3 + i
Write the conjugates of the following complex number:
3 – i
Write the conjugates of the following complex number:
`sqrt(2) + sqrt(3)"i"`
Is (1 + i14 + i18 + i22) a real number? Justify your answer
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).
(x + 2y) + (2x − 3y)i + 4i = 5
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:
`(3"i"^5 + 2"i"^7 + "i"^9)/("i"^6 + 2"i"^8 + 3"i"^18)`
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 equations for x, y ∈ R:
(x + iy) (5 + 6i) = 2 + 3i
Answer the following:
Show that `(1 - 2"i")/(3 - 4"i") + (1 + 2"i")/(3 + 4"i")` is real
Locate the points for which 3 < |z| < 4.
The real value of ‘a’ for which 3i3 – 2ai2 + (1 – a)i + 5 is real is ______.
State true or false for the following:
The points representing the complex number z for which |z + 1| < |z − 1| lies in the interior of a circle.
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?
What is the locus of z, if amplitude of z – 2 – 3i is `pi/4`?
If the complex number z = x + iy satisfies the condition |z + 1| = 1, then z lies on ______.
The equation |z + 1 – i| = |z – 1 + i| represents a ______.
For any two complex numbers z1, z2 and any real numbers a, b, |az1 – bz2|2 + |bz1 + az2|2 = ______.
The value of `sqrt(-25) xx sqrt(-9)` is ______.
The sum of the series i + i2 + i3 + ... upto 1000 terms is ______.
State True or False for the following:
For any complex number z the minimum value of |z| + |z – 1| is 1.
Find `|(1 + i) ((2 + i))/((3 + i))|`.
A real value of x satisfies the equation `((3 - 4ix)/(3 + 4ix))` = α − iβ (α, β ∈ R) if α2 + β2 = ______.
If a + ib = c + id, then ______.
Let |z| = |z – 3| = |z – 4i|, then the value |2z| is ______.
The smallest positive integer n for which `((1 + i)/(1 - i))^n` = –1 is ______.
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 ______.
If `|(6i, -3i, 1),(4, 3i, -1),(20, 3, i)|` = x + iy, then ______.
Let `(-2 - 1/3i)^2 = (x + iy)/9 (i = sqrt(-1))`, where x and y are real numbers, then x – y equals to ______.
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)`
