Advertisements
Advertisements
प्रश्न
\[\text { If } z = \frac{1}{(2 + 3i )^2}, \text { than } \left| z \right| =\]
विकल्प
\[\frac{1}{13}\]
\[\frac{1}{5}\]
\[\frac{1}{12}\]
none of these
Advertisements
उत्तर
\[\frac{1}{13}\]
\[\text { Let } z = \frac{1}{\left( 2 + 3i \right)^2}\]
\[ \Rightarrow z = \frac{1}{4 + 9 i^2 + 12i} \]
\[ \Rightarrow z = \frac{1}{4 - 9 + 12i} \]
\[ \Rightarrow z = \frac{1}{- 5 + 12i}\]
\[\Rightarrow z=\frac{1}{- 5 + 12i}\times\frac{- 5 - 12i}{- 5 - 12i}\]
\[\Rightarrow z=\frac{- 5 - 12i}{25 + 144}\]
\[ \Rightarrow z=\frac{- 5}{169}-\frac{12i}{169}\]
\[\Rightarrow\left| z \right|=\sqrt{\frac{25}{{169}^2} + \frac{144}{{169}^2}}\]
\[\Rightarrow \left| z \right|=\frac{1}{\sqrt{169}}\]
\[\Rightarrow \left| z \right| = \frac{1}{13}\]
APPEARS IN
संबंधित प्रश्न
Express the given complex number in the form a + ib: i–39
Express the given complex number in the form a + ib: (1 – i)4
Let z1 = 2 – i, z2 = –2 + i. Find Re`((z_1z_2)/barz_1)`
Let z1 = 2 – i, z2 = –2 + i. Find `"Im"(1/(z_1barz_1))`
Evaluate the following:
(ii) i528
Evaluate the following:
\[( i^{77} + i^{70} + i^{87} + i^{414} )^3\]
Evaluate the following:
\[i^{49} + i^{68} + i^{89} + i^{110}\]
Show that 1 + i10 + i20 + i30 is a real number.
Find the value of the following expression:
i + i2 + i3 + i4
Express the following complex number in the standard form a + i b:
\[\frac{(1 - i )^3}{1 - i^3}\]
Express the following complex number in the standard form a + i b:
\[\frac{3 - 4i}{(4 - 2i)(1 + i)}\]
Express the following complex number in the standard form a + i b:
\[\left( \frac{1}{1 - 4i} - \frac{2}{1 + i} \right)\left( \frac{3 - 4i}{5 + i} \right)\]
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.
If \[\left( \frac{1 - i}{1 + i} \right)^{100} = a + ib\] find (a, b).
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 \[\sin\frac{\pi}{5} + i\left( 1 - \cos\frac{\pi}{5} \right)\] in polar form.
If π < θ < 2π and z = 1 + cos θ + i sin θ, then write the value of \[\left| z \right|\] .
If \[\left| z - 5i \right| = \left| z + 5i \right|\] , then find the locus of z.
Find the real value of a for which \[3 i^3 - 2a i^2 + (1 - a)i + 5\] is real.
If a = cos θ + i sin θ, then \[\frac{1 + a}{1 - a} =\]
\[\text { If } z = \frac{1}{(1 - i)(2 + 3i)}, \text { than } \left| z \right| =\]
If \[z = \frac{1 + 7i}{(2 - i )^2}\] , then
The amplitude of \[\frac{1}{i}\] is equal to
The complex number z which satisfies the condition \[\left| \frac{i + z}{i - z} \right| = 1\] lies on
If z is a complex number, then
Find a and b if `1/("a" + "ib")` = 3 – 2i
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:
`("i"(4 + 3"i"))/((1 - "i"))`
Express the following in the form of a + ib, a, b∈R i = `sqrt(−1)`. State the values of a and b:
`((2 + "i"))/((3 - "i")(1 + 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")/(1 - "i"))^2`
Show that `(-1 + sqrt(3)"i")^3` is a real number
Evaluate the following : i93
Evaluate the following : i403
Answer the following:
Show that z = `5/((1 - "i")(2 - "i")(3 - "i"))` is purely imaginary number.
Match the statements of column A and B.
| Column A | Column B |
| (a) The value of 1 + i2 + i4 + i6 + ... i20 is | (i) purely imaginary complex number |
| (b) The value of `i^(-1097)` is | (ii) purely real complex number |
| (c) Conjugate of 1 + i lies in | (iii) second quadrant |
| (d) `(1 + 2i)/(1 - i)` lies in | (iv) Fourth quadrant |
| (e) If a, b, c ∈ R and b2 – 4ac < 0, then the roots of the equation ax2 + bx + c = 0 are non real (complex) and |
(v) may not occur in conjugate pairs |
| (f) If a, b, c ∈ R and b2 – 4ac > 0, and b2 – 4ac is a perfect square, then the roots of the equation ax2 + bx + c = 0 |
(vi) may occur in conjugate pairs |
If a = cosθ + isinθ, find the value of `(1 + "a")/(1 - "a")`.
Find the value of `(i^(592) + i^(590) + i^(588) + i^(586) + i^(584))/(i^(582) + i^(580) + i^(578) + i^(576) + i^(574))`
