Advertisements
Advertisements
प्रश्न
Evaluate the following:
\[x^4 - 4 x^3 + 4 x^2 + 8x + 44,\text { when } x = 3 + 2i\]
Advertisements
उत्तर
\[x = 3 + 2i\]
\[ \Rightarrow x^2 = \left( 3 + 2i \right)^2 \]
\[ = 9 + 4 i^2 + 12i\]
\[ = 5 + 12i\]
\[ \Rightarrow x^3 = x^2 \times x\]
\[ = \left( 5 + 12i \right) \times \left( 3 + 2i \right)\]
\[ = 15 + 10i + 36i - 24\]
\[ = - 9 + 46i\]
\[ \Rightarrow x^4 = \left( x^2 \right)^2 \]
\[ = \left( 5 + 12i \right)^2 \]
\[ = 25 + 144 i^2 + 120i\]
\[ = - 119 + 120i\]
\[ \Rightarrow x^4 - 4 x^3 + 4 x^2 + 8x + 44 = - 119 + 120i - 4\left( - 9 + 46i \right) + 4\left( 5 + 12i \right) + 8\left( 3 + 2i \right) + 44\]
\[ = - 119 + 120i + 36 - 184i + 20 + 48i + 24 + 16i + 44\]
\[ = 5\]
APPEARS IN
संबंधित प्रश्न
Express the given complex number in the form a + ib: `(5i) (- 3/5 i)`
Evaluate: `[i^18 + (1/i)^25]^3`
Evaluate the following:
\[( i^{77} + i^{70} + i^{87} + i^{414} )^3\]
Evaluate the following:
\[i^{49} + i^{68} + i^{89} + i^{110}\]
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 + i)6 + (1 − i)3
Express the following complex number in the standard form a + i b:
\[\frac{3 + 2i}{- 2 + i}\]
Express the following complex number in the standard form a + ib:
\[\frac{(2 + i )^3}{2 + 3i}\]
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 real value of x and y, if
\[(1 + i)(x + iy) = 2 - 5i\]
Find the multiplicative inverse of the following complex number:
1 − i
If \[z_1 = 2 - i, z_2 = - 2 + i,\] find
Re \[\left( \frac{z_1 z_2}{z_1} \right)\]
If \[z_1 = 2 - i, z_2 = - 2 + i,\] find
Im `(1/(z_1overlinez_1))`
Evaluate the following:
\[2 x^3 + 2 x^2 - 7x + 72, \text { when } x = \frac{3 - 5i}{2}\]
Evaluate the following:
\[x^4 + 4 x^3 + 6 x^2 + 4x + 9, \text { when } x = - 1 + i\sqrt{2}\]
For a positive integer n, find the value of \[(1 - i )^n \left( 1 - \frac{1}{i} \right)^n\].
Solve the system of equations \[\text { Re }\left( z^2 \right) = 0, \left| z \right| = 2\].
Solve the equation \[\left| z \right| = z + 1 + 2i\].
If π < θ < 2π and z = 1 + cos θ + i sin θ, then write the value of \[\left| z \right|\] .
If \[\frac{\left( a^2 + 1 \right)^2}{2a - i} = x + iy\] find the value of \[x^2 + y^2\].
\[\text { If } z = \frac{1}{(1 - i)(2 + 3i)}, \text { than } \left| z \right| =\]
\[\text { If }z = 1 - \text { cos }\theta + i \text { sin }\theta, \text { then } \left| z \right| =\]
If \[x + iy = \frac{3 + 5i}{7 - 6i},\] then y =
\[\frac{1 + 2i + 3 i^2}{1 - 2i + 3 i^2}\] equals
The value of \[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}} - 1\] is
If \[f\left( z \right) = \frac{7 - z}{1 - z^2}\] , where \[z = 1 + 2i\] then \[\left| f\left( z \right) \right|\] is
Find a and b if a + 2b + 2ai = 4 + 6i
Find a and b if (a+b) (2 + i) = b + 1 + (10 + 2a)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`
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 : i35
Evaluate the following : i888
Evaluate the following : i403
If `((1 + "i"sqrt3)/(1 - "i"sqrt3))^"n"` is an integer, then n is ______.
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 |
State True or False for the following:
The order relation is defined on the set of complex numbers.
