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
If `((1+i)/(1-i))^m` = 1, then find the least positive integral value of m.
Simplify the following and express in the form a + ib:
`3 + sqrt(-64)`
Write the conjugates of the following complex number:
cosθ + i sinθ
Find the value of i + i2 + i3 + i4
Simplify:
`(i^592 + i^590 + i^588 + i^586 + i^584)/(i^582 + i^580 + i^578 + i^576 + i^574)`
Find the value of 1 + i2 + i4 + i6 + i8 + ... + i20
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+ 1)/(1 + "i") + (y - 1)/(1 - "i")` = i
Find the value of x and y which satisfy the following equation (x, y ∈ R).
`((x + "i"y))/(2 + 3"i") + (2 + "i")/(2 - 3"i") = 9/13(1 + "i")`
Select the correct answer from the given alternatives:
If n is an odd positive integer then the value of 1 + (i)2n + (i)4n + (i)6n is :
Answer the following:
Simplify the following and express in the form a + ib:
(2 + 3i)(1 − 4i)
Answer the following:
Simplify the following and express in the form a + ib:
`(1 + 2/"i")(3 + 4/"i")(5 + "i")^-1`
Answer the following:
Solve the following equation for x, y ∈ R:
(4 − 5i)x + (2 + 3i)y = 10 − 7i
Answer the following:
Evaluate: (1 − i + i2)−15
Answer the following:
Evaluate: i131 + i49
Answer the following:
Find the value of x3 + 2x2 − 3x + 21, if x = 1 + 2i
If z1 = 5 + 3i and z2 = 2 - 4i, then z1 + z2 = ______.
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:
The argument of the complex number z = `(1 + i sqrt(3))(1 + i)(cos theta + i sin theta)` is `(7pi)/12 + theta`.
The area of the triangle on the complex plane formed by the complex numbers z, –iz and z + iz is ______.
Number of solutions of the equation z2 + |z|2 = 0 is ______.
Evaluate `sum_(n = 1)^13 (i^n + i^(n + 1))`, where n ∈ N.
If `(1 + i)^2/(2 - i)` = x + iy, then find the value of x + y.
If the real part of `(barz + 2)/(barz - 1)` is 4, then show that the locus of the point representing z in the complex plane is a circle.
If `(z - 1)/(z + 1)` is purely imaginary number (z ≠ – 1), then find the value of |z|.
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 number `(1 - i)^3/(1 - i^2)` is equal to ______.
If z1 and z2 are complex numbers such that z1 + z2 is a real number, then z2 = ______.
State True or False for the following:
Multiplication of a non-zero complex number by –i rotates the point about origin through a right angle in the anti-clockwise direction.
State True or False for the following:
The inequality |z – 4| < |z – 2| represents the region given by x > 3.
Let |z| = |z – 3| = |z – 4i|, then the value |2z| is ______.
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)`
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`
