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 number of non-zero integral solutions of the equation `|1-i|^x = 2^x`.
Show that 1 + i10 + i20 + i30 is a real number.
Simplify the following and express in the form a + ib:
(1 + 3i)2 (3 + i)
Simplify the following and express in the form a + ib:
`(5 + 7"i")/(4 + 3"i") + (5 + 7"i")/(4 - 3"i")`
Write the conjugates of the following complex number:
`sqrt(2) + sqrt(3)"i"`
Find the value of i49 + i68 + i89 + i110
Simplify:
`(i^592 + i^590 + i^588 + i^586 + i^584)/(i^582 + i^580 + i^578 + i^576 + i^574)`
Evaluate: `("i"^37 + 1/"i"^67)`
Find the value of `("i"^6 + "i"^7 + "i"^8 + "i"^9)/("i"^2 + "i"^3)`
If (x + iy)3 = y + vi then show that `(y/x + "v"/y)` = 4(x2 – y2)
Select the correct answer from the given alternatives:
The value of is `("i"^592 + "i"^590 + "i"^588 + "i"^586 + "i"^584)/("i"^582 + "i"^580 + "i"^578 + "i"^576 + "i"^574)` is equal to:
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:
(2 + 3i)(1 − 4i)
Answer the following:
Simplify the following and express in the form a + ib:
`5/2"i"(-4 - 3"i")`
Answer the following:
Simplify the following and express in the form a + ib:
(1 + 3i)2(3 + i)
Solve the following equation for x, y ∈ R:
2x + i9y (2 + i) = xi7 + 10i16
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 z1 = 5 + 3i and z2 = 2 - 4i, then z1 + z2 = ______.
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:
Multiplication of a non-zero complex number by i rotates it through a right angle in the anti-clockwise direction.
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"`
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 + 2(1 + i), then find z.
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.
If a + ib = c + id, then ______.
If the least and the largest real values of α, for which the equation z + α|z – 1| + 2i = 0 `("z" ∈ "C" and "i" = sqrt(-1))` has a solution, are p and q respectively; then 4(p2 + q2) is equal to ______.
If `(x + iy)^(1/5)` = a + ib, and u = `x/a - y/b`, then ______.
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 `|(6i, -3i, 1),(4, 3i, -1),(20, 3, i)|` = x + iy, then ______.
Find the value of `(i^592 + i^590 + i^588 + i^586 + i^584)/ (i^582 + i^580 + i^578 + i^576 + i^574)`
Show that `(-1 + sqrt3 i)^3` is a real number.
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)`
Evaluate the following:
i35
Show that `(-1 + sqrt3i)^3` is a real number.
