Advertisements
Advertisements
Question
Find the real value of x and y, if
\[(1 + i)(x + iy) = 2 - 5i\]
Advertisements
Solution
\[ \left( 1 + i \right)\left( x + iy \right) = 2 - 5i\]
\[ \Rightarrow x + iy + ix + i^2 y = 2 - 5i\]
\[ \Rightarrow x + iy + ix - y = 2 - 5i\]
\[ \Rightarrow \left( x - y \right) + i\left( y + x \right) = 2 - 5i\]
\[\text { Comparing both the sides }, \]
\[x - y = 2 . . . (1) \]
\[x + y = - 5 . . . (2)\]
\[\text { Adding equations (1) and (2) }, \]
\[2x = - 3\]
\[ \Rightarrow x = \frac{- 3}{2}\]
\[\text { Substituting the value of x in equation (1) }, \]
\[\frac{- 3}{2} - y = 2\]
\[ \Rightarrow y = \frac{- 3}{2} - 2\]
\[ \Rightarrow y = \frac{- 7}{2}\]
\[ \therefore x = \frac{- 3}{2} \text { and y } = \frac{- 7}{2}\]
APPEARS IN
RELATED QUESTIONS
Express the given complex number in the form a + ib: 3(7 + i7) + i(7 + i7)
Evaluate the following:
\[i^{37} + \frac{1}{i^{67}}\].
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 + i b:
\[\frac{(1 - i )^3}{1 - i^3}\]
Express the following complex number in the standard form a + i b:
\[\frac{3 - 4i}{(4 - 2i)(1 + i)}\]
Express the following complex number in the standard form a + i b:
\[\left( \frac{1}{1 - 4i} - \frac{2}{1 + i} \right)\left( \frac{3 - 4i}{5 + i} \right)\]
If \[z_1 = 2 - i, z_2 = - 2 + i,\] find
Re \[\left( \frac{z_1 z_2}{z_1} \right)\]
Find the least positive integral value of n for which \[\left( \frac{1 + i}{1 - i} \right)^n\] is real.
Evaluate the following:
\[x^4 + 4 x^3 + 6 x^2 + 4x + 9, \text { when } x = - 1 + i\sqrt{2}\]
Write (i25)3 in polar form.
Express the following complex in the form r(cos θ + i sin θ):
1 − sin α + i cos α
If z1 and z2 are two complex numbers such that \[\left| z_1 \right| = \left| z_2 \right|\] and arg(z1) + arg(z2) = \[\pi\] then show that \[z_1 = - \bar{{z_2}}\].
If n is any positive integer, write the value of \[\frac{i^{4n + 1} - i^{4n - 1}}{2}\].
Write the least positive integral value of n for which \[\left( \frac{1 + i}{1 - i} \right)^n\] is real.
If \[\left| z - 5i \right| = \left| z + 5i \right|\] , then find the locus of z.
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 \[\left| a z_1 - b z_2 \right|^2 + \left| a z_2 + b z_1 \right|^2\].
The polar form of (i25)3 is
The least positive integer n such that \[\left( \frac{2i}{1 + i} \right)^n\] is a positive integer, is.
If \[x + iy = \frac{3 + 5i}{7 - 6i},\] then y =
If θ is the amplitude of \[\frac{a + ib}{a - ib}\] , than tan θ =
The value of \[(1 + i )^4 + (1 - i )^4\] is
Which of the following is correct for any two complex numbers z1 and z2?
If the complex number \[z = x + iy\] satisfies the condition \[\left| z + 1 \right| = 1\], then z lies on
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))`
Evaluate the following : i93
Evaluate the following : i403
Evaluate the following : i30 + i40 + i50 + i60
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)`.
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 |
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 |
If w is a complex cube-root of unity, then prove the following
(w2 + w − 1)3 = −8
