Advertisements
Advertisements
प्रश्न
Find the value of x and y which satisfy the following equation (x, y ∈ R).
`((x + "i"y))/(2 + 3"i") + (2 + "i")/(2 - 3"i") = 9/13(1 + "i")`
Advertisements
उत्तर
`((x + "i"y))/(2 + 3"i") + (2 + "i")/(2 - 3"i") = 9/13(1 + "i")`
∴ `((x + "i"y)(2 - 3"i") + (2 + "i")(2 + 3"i"))/((2 + 3"i")(2 - 3"i")) = 9/13(1 + "i")`
∴ `(2x - 3"i"x + 2"i"y - 3y"i"^2 + 4 + 6"i" + 2"i" + 3"i"^2)/(4 - 9"i"^2) = 9/13(1 + "i")`
∴ `(2x - 3"i"x + 2"i"y + 3y + 4 + 6"i" + 2"i" - 3)/(4 + 9) = 9/13(1 + "i")` ...[∵ i2 = – 1]
∴ `((2x + 3y + 1) + (-3"i"x + 2"i"y + 8"i"))/13 = 9/13(1 + "i")`
Equating the real and imaginary parts separately, we get,
2x + 3y + 1 = 9 and – 3x + 2y + 8 = 9
∴ 2x + 3y = 8 ...(1)
and – 3x + 2y = 1 ...(2)
Multiplying equation (1) by 3 and equation (2) by 2, we get,
6x + 9y = 24
and – 6x + 4y = 2
On adding, we get,
13y = 26
∴ y = 2
∴ from (1), 2x + 3(2) = 8
∴ 2x + 6 = 8
∴ 2x = 2
∴ x = 1
Hence, x = 1 and y = 2
APPEARS IN
संबंधित प्रश्न
Find the multiplicative inverse of the complex number:
4 – 3i
If `x – iy = sqrt((a-ib)/(c - id))` prove that `(x^2 + y^2) = (a^2 + b^2)/(c^2 + d^2)`
If (a + ib) (c + id) (e + if) (g + ih) = A + iB, then show that (a2 + b2) (c2 + d2) (e2 + f2) (g2 + h2) = A2 + B2.
Find the value of i49 + i68 + i89 + i110
Simplify the following and express in the form a + ib:
`(4 + 3"i")/(1 - "i")`
Simplify the following and express in the form a + ib:
`(sqrt(5) + sqrt(3)"i")/(sqrt(5) - sqrt(3)"i")`
Simplify the following and express in the form a + ib:
`(3i^5 + 2i^7 + i^9)/(i^6 + 2i^8 + 3i^18)`
Find the value of : x3 + 2x2 – 3x + 21, if x = 1 + 2i
Find the value of: x3 – 5x2 + 4x + 8, if x = `10/(3 - "i")`.
Evaluate: `("i"^37 + 1/"i"^67)`
Find the value of x and y which satisfy the following equation (x, y∈R).
If x + 2i + 15i6y = 7x + i3 (y + 4), find x + y
Select the correct answer from the given alternatives:
If n is an odd positive integer then the value of 1 + (i)2n + (i)4n + (i)6n is :
Select the correct answer from the given alternatives:
The value of is `("i"^592 + "i"^590 + "i"^588 + "i"^586 + "i"^584)/("i"^582 + "i"^580 + "i"^578 + "i"^576 + "i"^574)` is equal to:
Solve the following equation for x, y ∈ R:
2x + i9y (2 + i) = xi7 + 10i16
If z1 = 5 + 3i and z2 = 2 - 4i, then z1 + z2 = ______.
Evaluate: (1 + i)6 + (1 – i)3
The value of `(- sqrt(-1))^(4"n" - 3)`, where n ∈ N, is ______.
What is the reciprocal of `3 + sqrt(7)i`.
Evaluate `sum_(n = 1)^13 (i^n + i^(n + 1))`, where n ∈ N.
If (1 + i)z = `(1 - i)barz`, then show that z = `-ibarz`.
If `(z - 1)/(z + 1)` is purely imaginary number (z ≠ – 1), then find the value of |z|.
For any two complex numbers z1, z2 and any real numbers a, b, |az1 – bz2|2 + |bz1 + az2|2 = ______.
The number `(1 - i)^3/(1 - i^2)` is equal to ______.
A real value of x satisfies the equation `((3 - 4ix)/(3 + 4ix))` = α − iβ (α, β ∈ R) if α2 + β2 = ______.
Which of the following is correct for any two complex numbers z1 and z2?
Let x, y ∈ R, then x + iy is a non-real complex number if ______.
If `(3 + i)(z + barz) - (2 + i)(z - barz) + 14i` = 0, then `barzz` is equal to ______.
`((1 + cosθ + isinθ)/(1 + cosθ - isinθ))^n` = ______.
Let z be a complex number such that `|(z - i)/(z + 2i)|` = 1 and |z| = `5/2`. Then the value of |z + 3i| is ______.
If a complex number z satisfies the equation `z + sqrt(2)|z + 1| + i` = 0, then |z| is equal to ______.
If α and β are the roots of the equation x2 + 2x + 4 = 0, then `1/α^3 + 1/β^3` is equal to ______.
If `|(6i, -3i, 1),(4, 3i, -1),(20, 3, i)|` = x + iy, then ______.
Simplify the following and express in the form a + ib.
`(3i^5 + 2i^7 + i^9)/(i^6 + 2i^8 + 3i^18)`
Show that `(-1 + sqrt3 i)^3` is a real number.
Simplify the following and express in the form a+ib.
`(3i^5 + 2i^7 + i^9)/(i^6 + 2i^8 + 3i^18`
Simplify the following and express in the form a + ib.
`(3i^5 + 2i^7 + i^9)/(i^6 + 2i^8 + 3i^18)`
