Advertisements
Advertisements
प्रश्न
Evaluate the following:
\[x^4 + 4 x^3 + 6 x^2 + 4x + 9, \text { when } x = - 1 + i\sqrt{2}\]
उत्तर
\[ x = - 1 + \sqrt{2}i\]
\[ \Rightarrow x^2 = \left( - 1 + \sqrt{2}i \right)^2 \]
\[ = 1 + 2 i^2 - 2\sqrt{2}i\]
\[ = - 1 - 2\sqrt{2}i\]
\[ \Rightarrow x^3 = \left( - 1 - 2\sqrt{2}i \right) \times \left( - 1 + \sqrt{2}i \right)\]
\[ = 1 - \sqrt{2}i + 2\sqrt{2}i - 4 i^2 \]
\[ = 5 + \sqrt{2}i\]
\[ \Rightarrow x^4 = \left( - 1 - 2\sqrt{2}i \right)^2 \]
\[ = 1 + 8 i^2 + 4\sqrt{2}i\]
\[ = - 7 + 4\sqrt{2}i\]
\[ \Rightarrow x^4 + 4 x^3 + 6 x^2 + 4x + 9 = - 7 + 4\sqrt{2}i + 4\left( 5 + \sqrt{2}i \right) + 6\left( - 1 - 2\sqrt{2}i \right) + 4\left( - 1 + \sqrt{2}i \right) + 9\]
\[ = - 7 + 4\sqrt{2}i + 20 + 4\sqrt{2}i - 6 - 12\sqrt{2}i - 4 + 4\sqrt{2}i + 9\]
\[ = 12\]
APPEARS IN
संबंधित प्रश्न
Express the given complex number in the form a + ib: `(1/5 + i 2/5) - (4 + i 5/2)`
Evaluate the following:
i457
Evaluate the following:
\[\frac{1}{i^{58}}\]
Evaluate the following:
\[i^{37} + \frac{1}{i^{67}}\].
Evaluate the following:
\[( i^{77} + i^{70} + i^{87} + i^{414} )^3\]
Find the value of the following expression:
i49 + i68 + i89 + i110
Find the value of the following expression:
i + i2 + i3 + i4
Find the value of the following expression:
i5 + i10 + i15
Find the real value of x and y, if
\[(x + iy)(2 - 3i) = 4 + i\]
If \[\left( 1 + i \right)z = \left( 1 - i \right) \bar{z}\],then show that \[z = - i \bar{z}\].
What is the smallest positive integer n for which \[\left( 1 + i \right)^{2n} = \left( 1 - i \right)^{2n}\] ?
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
Express the following complex in the form r(cos θ + i sin θ):
tan α − i
Express the following complex in the form r(cos θ + i sin θ):
1 − sin α + i cos α
If π < θ < 2π and z = 1 + cos θ + i sin θ, then write the value of \[\left| z \right|\] .
Write 1 − i in polar form.
If \[\frac{\left( a^2 + 1 \right)^2}{2a - i} = x + iy\] find the value of \[x^2 + y^2\].
Write the value of \[\sqrt{- 25} \times \sqrt{- 9}\].
If \[\left| z + 4 \right| \leq 3\], then find the greatest and least values of \[\left| z + 1 \right|\].
If \[\left| z \right| = 2 \text { and } \arg\left( z \right) = \frac{\pi}{4}\],find z.
If `(3+2i sintheta)/(1-2 i sin theta)`is a real number and 0 < θ < 2π, then θ =
The amplitude of \[\frac{1 + i\sqrt{3}}{\sqrt{3} + i}\] is
\[\frac{1 + 2i + 3 i^2}{1 - 2i + 3 i^2}\] equals
The value of \[(1 + i )^4 + (1 - i )^4\] is
A real value of x satisfies the equation \[\frac{3 - 4ix}{3 + 4ix} = a - ib (a, b \in \mathbb{R}), if a^2 + b^2 =\]
Simplify : `4sqrt(-4) + 5sqrt(-9) - 3sqrt(-16)`
Find a and b if (a + ib) (1 + i) = 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 + 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:
(2 + 3i)(2 – 3i)
Express the following in the form of a + ib, a, b ∈ R, i = `sqrt(−1)`. State the values of a and b:
`(4"i"^8 - 3"i"^9 + 3)/(3"i"^11 - 4"i"^10 - 2)`
Find the value of `(3 + 2/"i")("i"^6 - "i"^7)(1 + "i"^11)`
Evaluate the following : i888
Evaluate the following : i116
If z1 = 3 – 2i and z2 = –1 + 3i, then Im(z1z2) = ______.
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)`.
State True or False for the following:
The order relation is defined on the set of complex numbers.
