Advertisements
Advertisements
प्रश्न
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`
Advertisements
उत्तर
\[ \frac{\left( 1 + i \right)x - 2i}{3 + i} + \frac{\left( 2 - 3i \right)y + i}{3 - i} = i\]
\[ \Rightarrow \frac{\left( 1 + i \right)\left( 3 - i \right)x - 2i\left( 3 - i \right) + \left( 2 - 3i \right)\left( 3 + i \right)y + i\left( 3 + i \right)}{\left( 3 + i \right)\left( 3 - i \right)} = i\]
\[ \Rightarrow \frac{3x - ix + 3ix - i^2 x - 6i + 2 i^2 + 6y + 2iy - 9iy - 3 i^2 y + 3i + i^2}{9 - i^2} = i\]
\[ \Rightarrow \frac{4x + 2ix - 3i + 9y - 7iy - 3}{10} = i\]
\[ \Rightarrow \left( 4x + 9y - 3 \right) + i\left( 2x - 3 - 7y \right) = 10i\]
\[\text { Comparing both the sides: } \]
\[4x + 9y - 3 = 0\]
\[ \Rightarrow 4x + 9y = 3 . . . (1) \]
\[2x - 3 - 7y = 10\]
\[ \Rightarrow 2x - 7y = 13 . . . (2)\]
\[\text{Multiplying equation (2) by 2:} \]
\[4x - 14y = 26 . . . (3) \]
\[\text { Subtracting equation (3) from (1): } \]
\[ 4x + 9y = 3 \]
\[ 4x - 14y = 26 \]
\[ - + - \]
\[ 23y = - 23\]
\[ \therefore y = - 1\]
\[\text { Substituting the value of y in equation (1) }: \]
\[4x - 9 = 3\]
\[ \Rightarrow 4x = 12\]
\[ \Rightarrow x = 3\]
\[ \therefore x = 3 \text { and y } = - 1\]
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: 3(7 + i7) + i(7 + i7)
Express the given complex number in the form a + ib: (1 – i)4
If a + ib = `(x + i)^2/(2x^2 + 1)` prove that a2 + b2 = `(x^2 + 1)^2/(2x + 1)^2`
Let z1 = 2 – i, z2 = –2 + i. Find Re`((z_1z_2)/barz_1)`
Evaluate the following:
i457
Evaluate the following:
\[i^{37} + \frac{1}{i^{67}}\].
Evaluate the following:
\[\left( i^{41} + \frac{1}{i^{257}} \right)^9\]
Show that 1 + i10 + i20 + i30 is a real number.
Find the value of the following expression:
i49 + i68 + i89 + i110
Find the value of the following expression:
\[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}}\]
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
\[(1 + i)(x + iy) = 2 - 5i\]
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))`
If \[a = \cos\theta + i\sin\theta\], find the value of \[\frac{1 + a}{1 - a}\].
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):
\[\frac{1 - i}{\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}}\]
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 \[i^n + i^{n + 1} + i^{n + 2} + i^{n + 3}\] .
If i2 = −1, then the sum i + i2 + i3 +... upto 1000 terms is equal to
The least positive integer n such that \[\left( \frac{2i}{1 + i} \right)^n\] is a positive integer, is.
\[\text { If } z = \frac{1}{(2 + 3i )^2}, \text { than } \left| z \right| =\]
\[\text { If }z = 1 - \text { cos }\theta + i \text { sin }\theta, \text { then } \left| z \right| =\]
The amplitude of \[\frac{1}{i}\] is equal to
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
The complex number z which satisfies the condition \[\left| \frac{i + z}{i - z} \right| = 1\] lies on
Find a and b if `1/("a" + "ib")` = 3 – 2i
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:
`("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 + 3i)(2 – 3i)
Evaluate the following : i35
Evaluate the following : i403
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)`.
Match the statements of Column A and Column B.
| Column A | Column B |
| (a) The polar form of `i + sqrt(3)` is | (i) Perpendicular bisector of segment joining (–2, 0) and (2, 0). |
| (b) The amplitude of `-1 + sqrt(-3)` is | (ii) On or outside the circle having centre at (0, –4) and radius 3. |
| (c) If |z + 2| = |z − 2|, then locus of z is | (iii) `(2pi)/3` |
| (d) If |z + 2i| = |z − 2i|, then locus of z is | (iv) Perpendicular bisector of segment joining (0, –2) and (0, 2). |
| (e) Region represented by |z + 4i| ≥ 3 is | (v) `2(cos pi/6 + i sin pi/6)` |
| (f) Region represented by |z + 4| ≤ 3 is | (vi) On or inside the circle having centre (–4, 0) and radius 3 units. |
| (g) Conjugate of `(1 + 2i)/(1 - i)` lies in | (vii) First quadrant |
| (h) Reciprocal of 1 – i lies in | (viii) Third quadrant |
Show that `(-1+ sqrt(3)i)^3` is a real number.
Show that `(-1+sqrt3i)^3` is a real number.
