Advertisements
Advertisements
Question
State True or False for the following:
For any complex number z the minimum value of |z| + |z – 1| is 1.
Options
True
False
Advertisements
Solution
This statement is True.
Explanation:
Let z = x + yi
∴ |z| + |z – 1| = `sqrt(x^2 + y^2) + sqrt((x - 1)^2 + y^2)`
The value of |z| + |z – 1| is minimum, When x = 0, y = 0 i.e., 1.
APPEARS IN
RELATED QUESTIONS
Find the multiplicative inverse of the complex number.
–i
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:
`(1 + 2/i)(3 + 4/i)(5 + i)^-1`
Find the value of : x3 + 2x2 – 3x + 21, if x = 1 + 2i
Find the value of: x3 – 3x2 + 19x – 20, if x = 1 – 4i
Write the conjugates of the following complex number:
`sqrt(5) - "i"`
Find the value of i + i2 + i3 + i4
Find the value of 1 + i2 + i4 + i6 + i8 + ... + i20
If x + iy = (a + ib)3, show that `x/"a" + y/"b"` = 4(a2 − b2)
If x + iy = `sqrt(("a" + "ib")/("c" + "id")`, prove that (x2 + y2)2 = `("a"^2 + "b"^2)/("c"^2 + "d"^2)`
If (a + ib) = `(1 + "i")/(1 - "i")`, then prove that (a2 + b2) = 1
Find the value of x and y which satisfy the following equation (x, y∈R).
(x + 2y) + (2x − 3y)i + 4i = 5
Select the correct answer from the given alternatives:
`sqrt(-3) sqrt(-6)` is equal to
Answer the following:
Solve the following equation for x, y ∈ R:
`(x + "i"y)/(2 + 3"i")` = 7 – i
Answer the following:
Find the value of x4 + 9x3 + 35x2 − x + 164, if x = −5 + 4i
Answer the following:
Find the real numbers x and y such that `x/(1 + 2"i") + y/(3 + 2"i") = (5 + 6"i")/(-1 + 8"i")`
Show that `(1/sqrt(2) + "i"/sqrt(2))^10 + (1/sqrt(2) - "i"/sqrt(2))^10` = 0
Answer the following:
Show that `(1 - 2"i")/(3 - 4"i") + (1 + 2"i")/(3 + 4"i")` is real
If z ≠ 1 and `"z"^2/("z - 1")` is real, then the point represented by the complex number z lies ______.
The argument of the complex number `(4 + 9i)/(13 + 5i)` is ______
The value of (2 + i)3 × (2 – i)3 is ______.
The value of `(- sqrt(-1))^(4"n" - 3)`, where n ∈ N, is ______.
State true or false for the following:
The complex number cosθ + isinθ can be zero for some θ.
If the complex number z = x + iy satisfies the condition |z + 1| = 1, then z lies on ______.
For a positive integer n, find the value of `(1 - i)^n (1 - 1/i)^"n"`
Evaluate `sum_(n = 1)^13 (i^n + i^(n + 1))`, where n ∈ N.
Solve the equation |z| = z + 1 + 2i.
If |z + 1| = z + 2(1 + i), then find z.
Find the complex number satisfying the equation `z + sqrt(2) |(z + 1)| + i` = 0.
Multiplicative inverse of 1 + i is ______.
If |z + 4| ≤ 3, then the greatest and least values of |z + 1| are ______ and ______.
Find `|(1 + i) ((2 + i))/((3 + i))|`.
Where does z lie, if `|(z - 5i)/(z + 5i)|` = 1.
If `((1 + i)/(1 - i))^x` = 1, then ______.
Let x, y ∈ R, then x + iy is a non-real complex number if ______.
If `(3 + i)(z + barz) - (2 + i)(z - barz) + 14i` = 0, then `barzz` is equal to ______.
If α and β are the roots of the equation x2 + 2x + 4 = 0, then `1/α^3 + 1/β^3` is equal to ______.
Let `(-2 - 1/3i)^2 = (x + iy)/9 (i = sqrt(-1))`, where x and y are real numbers, then x – y equals to ______.
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.
`(3i^5 + 2i^7 + i^9)/(i^6 + 2i^8 + 3i^18`
