Advertisements
Advertisements
प्रश्न
Find the real value of x and y, if
\[(x + iy)(2 - 3i) = 4 + i\]
Advertisements
उत्तर
\[ \left( x + iy \right)\left( 2 - 3i \right) = 4 + i\]
\[2x - 3ix + 2iy - 3 i^2 y = 4 + i\]
\[2x + 3y + i\left( - 3x + 2y \right) = 4 + i\]
\[\text{Comparing both the sides:} \]
\[2x + 3y = 4 . . . . (1) \]
\[ - 3x + 2y = 1 . . . . (2)\]
\[\text { Multiplying equation (1) by 3 and equation (2) by 2 }: \]
\[ 6x + 9y = 12 . . . (3)\]
\[ - 6x + 4y = 2 . . . (4)\]
\[\text { Adding equations (3) and (4) }: \]
\[13y = 14\]
\[y = \frac{14}{13}\]
\[\text { Substituting the value of y in equation (1):} \]
\[2x + 3 \times \frac{14}{13} = 4\]
\[ \Rightarrow 2x = 4 - \frac{42}{13}\]
\[ \Rightarrow 2x = \frac{10}{13}\]
\[ \Rightarrow x = \frac{5}{13}\]
\[ \therefore x = \frac{5}{13}\text { and } y = \frac{14}{13} \]
APPEARS IN
संबंधित प्रश्न
Express the given complex number in the form a + ib: `(5i) (- 3/5 i)`
Express the given complex number in the form a + ib:
`[(1/3 + i 7/3) + (4 + i 1/3)] -(-4/3 + i)`
Evaluate the following:
i457
Evaluate the following:
\[\left( i^{41} + \frac{1}{i^{257}} \right)^9\]
Express the following complex number in the standard form a + i b:
\[\frac{1}{(2 + i )^2}\]
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:
\[\frac{(1 - i )^3}{1 - i^3}\]
Find the real value of x and y, if
\[(1 + i)(x + iy) = 2 - 5i\]
Find the multiplicative inverse of the following complex number:
\[(1 + i\sqrt{3} )^2\]
If \[z_1 = 2 - i, z_2 = 1 + i,\text { find } \left| \frac{z_1 + z_2 + 1}{z_1 - z_2 + i} \right|\]
Find the least positive integral value of n for which \[\left( \frac{1 + i}{1 - i} \right)^n\] is real.
Evaluate the following:
\[x^4 - 4 x^3 + 4 x^2 + 8x + 44,\text { when } x = 3 + 2i\]
If \[\frac{z - 1}{z + 1}\] is purely imaginary number (\[z \neq - 1\]), find the value of \[\left| z \right|\].
Find the number of solutions of \[z^2 + \left| z \right|^2 = 0\].
Express the following complex in the form r(cos θ + i sin θ):
1 + i tan α
Write the value of \[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}}\] .
Find z, if \[\left| z \right| = 4 \text { and }\arg(z) = \frac{5\pi}{6} .\]
If \[\frac{\left( a^2 + 1 \right)^2}{2a - i} = x + iy\] find the value of \[x^2 + y^2\].
If \[\left| z + 4 \right| \leq 3\], then find the greatest and least values of \[\left| z + 1 \right|\].
For any two complex numbers z1 and z2 and any two real numbers a, b, find the value of \[\left| a z_1 - b z_2 \right|^2 + \left| a z_2 + b z_1 \right|^2\].
If n ∈ \[\mathbb{N}\] then find the value of \[i^n + i^{n + 1} + i^{n + 2} + i^{n + 3}\] .
Find the real value of a for which \[3 i^3 - 2a i^2 + (1 - a)i + 5\] is real.
If \[z = \frac{- 2}{1 + i\sqrt{3}}\],then the value of arg (z) is
The principal value of the amplitude of (1 + i) is
If \[z = \left( \frac{1 + i}{1 - i} \right)\] then z4 equals
\[\text { If } z = \frac{1}{(2 + 3i )^2}, \text { than } \left| z \right| =\]
If \[z = \frac{1}{1 - cos\theta - i sin\theta}\] then Re (z) =
The amplitude of \[\frac{1 + i\sqrt{3}}{\sqrt{3} + i}\] is
\[\frac{1 + 2i + 3 i^2}{1 - 2i + 3 i^2}\] equals
Which of the following is correct for any two complex numbers z1 and z2?
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:
`("i"(4 + 3"i"))/((1 - "i"))`
Evaluate the following : i93
Evaluate the following : i116
Show that 1 + i10 + i20 + i30 is a real number
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 |
