Advertisements
Advertisements
Question
If \[\left| z + 4 \right| \leq 3\], then find the greatest and least values of \[\left| z + 1 \right|\].
Advertisements
Solution
\[\left| z + 1 \right| = \left| z + 4 - 3 \right|\]
\[ \leq \left| z + 4 \right| + \left| - 3 \right|\]
\[ \leq 3 + 3\]
\[ = 6\]
\[\text { Also }, \]
\[\left| z + 1 \right| \geq 0\]
\[\text { Thus }, 0 \leq \left| z + 1 \right| \leq 6 .\]
Hence, the greatest and least values of
APPEARS IN
RELATED QUESTIONS
Express the given complex number in the form a + ib: `(5i) (- 3/5 i)`
Express the given complex number in the form a + ib:
`[(1/3 + i 7/3) + (4 + i 1/3)] -(-4/3 + i)`
Evaluate the following:
\[\frac{1}{i^{58}}\]
Evaluate the following:
\[\left( i^{41} + \frac{1}{i^{257}} \right)^9\]
Find the value of the following expression:
i + i2 + i3 + i4
Find the value of the following expression:
i5 + i10 + i15
Find the value of the following expression:
1+ i2 + i4 + i6 + i8 + ... + i20
Express the following complex number in the standard form a + i b:
\[\frac{1}{(2 + i )^2}\]
Express the following complex number in the standard form a + i b:
\[(1 + i)(1 + 2i)\]
Express the following complex number in the standard form a + i b:
\[\frac{(1 + i)(1 + \sqrt{3}i)}{1 - i}\] .
Find the real value of x and y, if
\[(3x - 2iy)(2 + i )^2 = 10(1 + i)\]
Find the real value of x and y, if
\[(1 + i)(x + iy) = 2 - 5i\]
Find the multiplicative inverse of the following complex number:
1 − i
Find the multiplicative inverse of the following complex number:
\[(1 + i\sqrt{3} )^2\]
If \[x + iy = \frac{a + ib}{a - ib}\] prove that x2 + y2 = 1.
Find the smallest positive integer value of m for which \[\frac{(1 + i )^n}{(1 - i )^{n - 2}}\] is a real number.
If \[\frac{\left( 1 + i \right)^2}{2 - i} = x + iy\] find x + y.
For a positive integer n, find the value of \[(1 - i )^n \left( 1 - \frac{1}{i} \right)^n\].
If \[\frac{z - 1}{z + 1}\] is purely imaginary number (\[z \neq - 1\]), find the value of \[\left| z \right|\].
Solve the equation \[\left| z \right| = z + 1 + 2i\].
What is the smallest positive integer n for which \[\left( 1 + i \right)^{2n} = \left( 1 - i \right)^{2n}\] ?
Find the number of solutions of \[z^2 + \left| z \right|^2 = 0\].
If \[\left| z \right| = 2 \text { and } \arg\left( z \right) = \frac{\pi}{4}\],find z.
If z is a non-zero complex number, then \[\left| \frac{\left| z \right|^2}{zz} \right|\] is equal to
The argument of \[\frac{1 - i\sqrt{3}}{1 + i\sqrt{3}}\] is
\[\text { If } z = \frac{1}{(1 - i)(2 + 3i)}, \text { than } \left| z \right| =\]
The argument of \[\frac{1 - i}{1 + i}\] is
The value of \[(1 + i )^4 + (1 - i )^4\] is
If \[f\left( z \right) = \frac{7 - z}{1 - z^2}\] , where \[z = 1 + 2i\] then \[\left| f\left( z \right) \right|\] is
If z is a complex number, then
If the complex number \[z = x + iy\] satisfies the condition \[\left| z + 1 \right| = 1\], then z lies on
Find a and b if `1/("a" + "ib")` = 3 – 2i
Evaluate the following : i403
Evaluate the following : i30 + i40 + i50 + i60
State True or False for the following:
2 is not a complex number.
The real value of θ for which the expression `(1 + i cos theta)/(1 - 2i cos theta)` is a real number is ______.
If w is a complex cube-root of unity, then prove the following
(w2 + w − 1)3 = −8
Show that `(-1+sqrt3i)^3` is a real number.
