Advertisements
Advertisements
प्रश्न
Find the real value of x and y, if
\[(3x - 2iy)(2 + i )^2 = 10(1 + i)\]
Advertisements
उत्तर
\[ \left( 3x - 2iy \right) \left( 2 + i \right)^2 = 10 \left( 1 + i \right)\]
\[ \Rightarrow \left( 3x - 2iy \right)\left( 4 + i^2 + 4i \right) = 10\left( 1 + i \right)\]
\[ \Rightarrow \left( 3x - 2iy \right)\left( 3 + 4i \right) = 10\left( 1 + i \right)\]
\[ \Rightarrow 9x + 12xi - 6iy - 8 i^2 y = 10 + 10i\]
\[ \Rightarrow 9x + 8y + i\left( 12x - 6y \right) = 10 + 10i\]
\[\text{Comparing both the sides:} \]
\[9x + 8y = 10 . . . . (1)\]
\[12x - 6y = 10\]
\[or, 6x - 3y = 5 . . . (2)\]
\[\text { Multiplying equation (1) by 3 and equation (2) by 8 }, \]
\[27x + 24y = 30 . . . . (3) \]
\[48x - 24y = 40 . . . . (4)\]
\[\text {Adding equations (3) and (4):} \]
\[75x = 70\]
\[ \therefore x = \frac{14}{15}\]
\[\text { Substituting the value of x in equation (1): } \]
\[9 \times \frac{14}{15} + 8y = 10\]
\[ \Rightarrow \frac{126}{15} + 8y = 10\]
\[ \Rightarrow 8y = 10 - \frac{126}{15}\]
\[ \Rightarrow 8y = \frac{24}{15}\]
\[ \Rightarrow y = \frac{1}{5}\]
APPEARS IN
संबंधित प्रश्न
Evaluate: `[i^18 + (1/i)^25]^3`
Let z1 = 2 – i, z2 = –2 + i. Find Re`((z_1z_2)/barz_1)`
Evaluate the following:
\[\frac{1}{i^{58}}\]
Evaluate the following:
\[i^{37} + \frac{1}{i^{67}}\].
Evaluate the following:
\[i^{30} + i^{40} + i^{60}\]
Evaluate the following:
\[i^{49} + i^{68} + i^{89} + i^{110}\]
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 multiplicative inverse of the following complex number:
\[(1 + i\sqrt{3} )^2\]
Find the real values of θ for which the complex number \[\frac{1 + i cos\theta}{1 - 2i cos\theta}\] is purely real.
If \[a = \cos\theta + i\sin\theta\], find the value of \[\frac{1 + a}{1 - a}\].
Evaluate the following:
\[x^4 + 4 x^3 + 6 x^2 + 4x + 9, \text { when } x = - 1 + i\sqrt{2}\]
Evaluate the following:
\[2 x^4 + 5 x^3 + 7 x^2 - x + 41, \text { when } x = - 2 - \sqrt{3}i\]
If \[\left( 1 + i \right)z = \left( 1 - i \right) \bar{z}\],then show that \[z = - i \bar{z}\].
Solve the system of equations \[\text { Re }\left( z^2 \right) = 0, \left| z \right| = 2\].
Write (i25)3 in polar form.
Express the following complex in the form r(cos θ + i sin θ):
tan α − i
Express the following complex in the form r(cos θ + i sin θ):
1 − sin α + i cos α
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 \[\arg\left( z \right) + \arg\left( \bar{z} \right)\].
The least positive integer n such that \[\left( \frac{2i}{1 + i} \right)^n\] is a positive integer, is.
If (x + iy)1/3 = a + ib, then \[\frac{x}{a} + \frac{y}{b} =\]
If \[z = \frac{1 + 2i}{1 - (1 - i )^2}\], then arg (z) equal
\[\text { If }z = 1 - \text { cos }\theta + i \text { sin }\theta, \text { then } \left| z \right| =\]
If \[z = \frac{1}{1 - cos\theta - i sin\theta}\] then Re (z) =
If \[z = \frac{1 + 7i}{(2 - i )^2}\] , then
Find a and b if `1/("a" + "ib")` = 3 – 2i
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"))`
Find the value of `(3 + 2/i) (i^6 - i^7) (1 + i^11)`.
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 |
The real value of θ for which the expression `(1 + i cos theta)/(1 - 2i cos theta)` is a real number is ______.
Show that `(-1+ sqrt(3)i)^3` is a real number.
Find the value of `(i^(592) + i^(590) + i^(588) + i^(586) + i^(584))/(i^(582) + i^(580) + i^(578) + i^(576) + i^(574))`
