Advertisements
Advertisements
Question
\[\text { If }z = 1 - \text { cos }\theta + i \text { sin }\theta, \text { then } \left| z \right| =\]
Options
\[2 \sin\frac{\theta}{2}\]
\[2 \cos\frac{\theta}{2}\]
\[2\left| \sin\frac{\theta}{2} \right|\]
\[2\left| \cos\frac{\theta}{2} \right|\]
Advertisements
Solution
\[2\left| \sin\frac{\theta}{2} \right|\]
\[\because z = 1 - \cos\theta + i \sin\theta\]
\[ \Rightarrow \left| z \right| = \sqrt{\left( 1 - \cos\theta \right)^2 + \sin^2 \theta}\]
\[ \Rightarrow \left| z \right| = \sqrt{1 + \cos^2 \theta - 2\cos\theta + \sin^2 \theta}\]
\[ \Rightarrow \left| z \right| = \sqrt{1 + 1 - 2\cos\theta}\]
\[ \Rightarrow \left| z \right| = \sqrt{2\left( 1 - \cos\theta \right)}\]
\[ \Rightarrow \left| z \right| = \sqrt{4 \sin^2 \frac{\theta}{2}}\]
\[ \Rightarrow \left| z \right|=2\left| \sin\frac{\theta}{2} \right|\]
APPEARS IN
RELATED QUESTIONS
Find the value of the following expression:
\[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}}\]
Find the value of the following expression:
1+ i2 + i4 + i6 + i8 + ... + i20
Find the value of the following expression:
(1 + i)6 + (1 − i)3
Express the following complex number in the standard form a + i b:
\[\frac{1 - i}{1 + i}\]
Express the following complex number in the standard form a + i b:
\[\frac{3 - 4i}{(4 - 2i)(1 + i)}\]
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:
\[(1 + i\sqrt{3} )^2\]
Find the least positive integral value of n for which \[\left( \frac{1 + i}{1 - i} \right)^n\] is real.
Evaluate the following:
\[x^4 + 4 x^3 + 6 x^2 + 4x + 9, \text { when } x = - 1 + i\sqrt{2}\]
If z1, z2, z3 are complex numbers such that \[\left| z_1 \right| = \left| z_2 \right| = \left| z_3 \right| = \left| \frac{1}{z_1} + \frac{1}{z_2} + \frac{1}{z_3} \right| = 1\] then find the value of \[\left| z_1 + z_2 + z_3 \right|\] .
Express the following complex in the form r(cos θ + i sin θ):
1 + i tan α
If π < θ < 2π and z = 1 + cos θ + i sin θ, then write the value of \[\left| z \right|\] .
Write the argument of −i.
Find the principal argument of \[\left( 1 + i\sqrt{3} \right)^2\] .
If i2 = −1, then the sum i + i2 + i3 +... upto 1000 terms is equal to
If \[z = \frac{- 2}{1 + i\sqrt{3}}\],then the value of arg (z) is
If \[z = \left( \frac{1 + i}{1 - i} \right)\] then z4 equals
\[\text { If } z = \frac{1}{(1 - i)(2 + 3i)}, \text { than } \left| z \right| =\]
The value of \[(1 + i )^4 + (1 - i )^4\] is
Which of the following is correct for any two complex numbers z1 and z2?
If the complex number \[z = x + iy\] satisfies the condition \[\left| z + 1 \right| = 1\], then z lies on
Simplify : `sqrt(-16) + 3sqrt(-25) + sqrt(-36) - sqrt(-625)`
Express the following in the form of a + ib, a, b ∈ R, i = `sqrt(−1)`. State the values of a and b:
(1 + 2i)(– 2 + i)
Express the following in the form of a + ib, a, b∈R i = `sqrt(−1)`. State the values of a and b:
(1 + i)−3
Evaluate the following : i116
Evaluate the following : `1/"i"^58`
If z1 and z2 both satisfy `z + barz = 2|z - 1|` arg`(z_1 - z_2) = pi/4`, then find `"Im" (z_1 + z_2)`.
If a = cosθ + isinθ, find the value of `(1 + "a")/(1 - "a")`.
State True or False for the following:
The order relation is defined on the set of complex numbers.
Match the statements of Column A and Column B.
| Column A | Column B |
| (a) The polar form of `i + sqrt(3)` is | (i) Perpendicular bisector of segment joining (–2, 0) and (2, 0). |
| (b) The amplitude of `-1 + sqrt(-3)` is | (ii) On or outside the circle having centre at (0, –4) and radius 3. |
| (c) If |z + 2| = |z − 2|, then locus of z is | (iii) `(2pi)/3` |
| (d) If |z + 2i| = |z − 2i|, then locus of z is | (iv) Perpendicular bisector of segment joining (0, –2) and (0, 2). |
| (e) Region represented by |z + 4i| ≥ 3 is | (v) `2(cos pi/6 + i sin pi/6)` |
| (f) Region represented by |z + 4| ≤ 3 is | (vi) On or inside the circle having centre (–4, 0) and radius 3 units. |
| (g) Conjugate of `(1 + 2i)/(1 - i)` lies in | (vii) First quadrant |
| (h) Reciprocal of 1 – i lies in | (viii) Third quadrant |
The real value of θ for which the expression `(1 + i cos theta)/(1 - 2i cos theta)` is a real number is ______.
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+sqrt3i)^3` is a real number.
