Advertisements
Advertisements
प्रश्न
If |z + 4| ≤ 3, then the greatest and least values of |z + 1| are ______ and ______.
Advertisements
उत्तर
If |z + 4| ≤ 3, then the greatest and least values of |z + 1| are 6 and 0.
Explanation:
Given that: |z + 4| ≤ 3
For the greatest value of |z + 1|.
= |z + 4 – 3| ≤ |z + 4| + |–3|
= |z + 4 – 3| ≤ 3 + 3 ......[∵ |z + 4| ≤ 3 and |–3| = 3]
= |z + 4 – 3| ≤ 6
Hence, the greatest value of |z + 1| is 6 and for the least value of |z + 1| = 0. .....[∵ The least value of the modulus of complex number is 0.]
APPEARS IN
संबंधित प्रश्न
Find the multiplicative inverse of the complex number:
4 – 3i
Find the multiplicative inverse of the complex number.
–i
Reduce `(1/(1-4i) - 2/(1+i))((3-4i)/(5+i))` to the standard form.
If (a + ib) (c + id) (e + if) (g + ih) = A + iB, then show that (a2 + b2) (c2 + d2) (e2 + f2) (g2 + h2) = A2 + B2.
If `((1+i)/(1-i))^m` = 1, then find the least positive integral value of m.
Find the value of: x3 – x2 + x + 46, if x = 2 + 3i
Find the value of: 2x3 – 11x2 + 44x + 27, if x = `25/(3 - 4"i")`
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`
Write the conjugates of the following complex number:
3 – i
Write the conjugates of the following complex number:
`sqrt(2) + sqrt(3)"i"`
Find the value of 1 + i2 + i4 + i6 + i8 + ... + i20
Is (1 + i14 + i18 + i22) a real number? Justify your answer
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).
If x + 2i + 15i6y = 7x + i3 (y + 4), find x + y
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:
`(3"i"^5 + 2"i"^7 + "i"^9)/("i"^6 + 2"i"^8 + 3"i"^18)`
Solve the following equation for x, y ∈ R:
(4 − 5i)x + (2 + 3i)y = 10 − 7i
Show that `(1/sqrt(2) + "i"/sqrt(2))^10 + (1/sqrt(2) - "i"/sqrt(2))^10` = 0
Answer the following:
Simplify: `("i"^65 + 1/"i"^145)`
If `(x + iy)^(1/3)` = a + ib, where x, y, a, b ∈ R, show that `x/a - y/b` = –2(a2 + b2)
The real value of ‘a’ for which 3i3 – 2ai2 + (1 – a)i + 5 is real is ______.
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.
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 reciprocal of `3 + sqrt(7)i`.
If `(1 + i)^2/(2 - i)` = x + iy, then find the value of x + y.
If `(z - 1)/(z + 1)` is purely imaginary number (z ≠ – 1), then find the value of |z|.
The value of `sqrt(-25) xx sqrt(-9)` is ______.
If z1 and z2 are complex numbers such that z1 + z2 is a real number, then z2 = ______.
If `(3 + i)(z + barz) - (2 + i)(z - barz) + 14i` = 0, then `barzz` is equal to ______.
The smallest positive integer n for which `((1 + i)/(1 - i))^n` = –1 is ______.
Let z be a complex number such that `|(z - i)/(z + 2i)|` = 1 and |z| = `5/2`. Then the value of |z + 3i| is ______.
If a complex number z satisfies the equation `z + sqrt(2)|z + 1| + i` = 0, then |z| is equal 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)`
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.
`(3"i"^5 + 2"i"^7 + "i"^9)/("i"^6 + 2"i"^8 + 3"i"^18)`
i2 + i3 + ... + i4000 =
