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: i–39
Express the given complex number in the form a + ib: (1 – i) – (–1 + i6)
If a + ib = `(x + i)^2/(2x^2 + 1)` prove that a2 + b2 = `(x^2 + 1)^2/(2x + 1)^2`
Evaluate the following:
(ii) i528
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 `((1+i)x-2i)/(3+i) + ((2-3i)y+i)/(3-i) = i, xy ∈ R, i = sqrt-1`
Find the real value of x and y, if
\[(1 + i)(x + iy) = 2 - 5i\]
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:
\[2 x^4 + 5 x^3 + 7 x^2 - x + 41, \text { when } x = - 2 - \sqrt{3}i\]
If z1 is a complex number other than −1 such that \[\left| z_1 \right| = 1\] and \[z_2 = \frac{z_1 - 1}{z_1 + 1}\] then show that the real parts of z2 is zero.
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):
tan α − i
Express \[\sin\frac{\pi}{5} + i\left( 1 - \cos\frac{\pi}{5} \right)\] in polar form.
Write 1 − i in polar form.
Write the least positive integral value of n for which \[\left( \frac{1 + i}{1 - i} \right)^n\] is real.
Write the value of \[\sqrt{- 25} \times \sqrt{- 9}\].
If \[\left| z \right| = 2 \text { and } \arg\left( z \right) = \frac{\pi}{4}\],find z.
If a = cos θ + i sin θ, then \[\frac{1 + a}{1 - a} =\]
\[(\sqrt{- 2})(\sqrt{- 3})\] is equal to
\[\text { If }z = 1 - \text { cos }\theta + i \text { sin }\theta, \text { then } \left| z \right| =\]
If \[z = \frac{1 + 7i}{(2 - i )^2}\] , then
The value of (i5 + i6 + i7 + i8 + i9) / (1 + i) is
Find a and b if (a+b) (2 + i) = b + 1 + (10 + 2a)i
Find a and b if abi = 3a − b + 12i
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:
`(2 + sqrt(-3))/(4 + sqrt(-3))`
Show that `(-1 + sqrt(3)"i")^3` is a real number
Find the value of `(3 + 2/i) (i^6 - i^7) (1 + i^11)`.
Evaluate the following : i403
Evaluate the following : i–888
Show that 1 + i10 + i20 + i30 is a real number
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 |
State True or False for the following:
The order relation is defined on the set of complex numbers.
State True or False for the following:
2 is not a complex number.
Show that `(-1+sqrt3i)^3` is a real number.
