Advertisements
Advertisements
Question
Answer the following:
Find the real numbers x and y such that `x/(1 + 2"i") + y/(3 + 2"i") = (5 + 6"i")/(-1 + 8"i")`
Advertisements
Solution
`x/(1 + 2"i") + y/(3 + 2"i") = (5 + 6"i")/(-1 + 8"i")`
∴ `((3 + 2"i")x + (1 + 2"i")y)/((1 + 2"i")(3 + 2"i")) = (5 + 6"i")/(-1 + 8"i")`
∴ `((3 + 2"i")x + (1 + 2"i")y)/(3 + 2"i" + 6"i" + 4"i"^2) = (5 + 6"i")/(-1 + 8"i")`
∴ `((3 + 2"i")x + (1 + 2"i")y)/(3 + 8"i" - 4) = (5 + 6"i")/(-1 + 8"i")` ...[∵ i2 = – 1]
∴ `(3x + 2"i"x + y + 2"i"y)/(-1 + 8"i") = (5 + 6"i")/(-1 + 8"i")`
∴ (3x + y) + (2x + 2y)i = 5 + 6i
Equating the real and imaginary parts separately, we get,
3x + y = 5 ...(1)
and 2x + 2y = 6
i.e., x + y = 3 ...(2)
On subtracting, we get,
2x = 2
∴ x = 1
∴ from (2), 1 + y = 3
∴ y = 2
Hence, x = 1, y = 2.
APPEARS IN
RELATED QUESTIONS
Express the following expression in the form of a + ib.
`((3 + sqrt5)(3 - isqrt5))/((sqrt3 + sqrt2i)-(sqrt3 - isqrt2))`
Find the value of i + i2 + i3 + i4
Simplify the following and express in the form a + ib:
`3 + sqrt(-64)`
Simplify the following and express in the form a + ib:
(2i3)2
Simplify the following and express in the form a + ib:
`5/2"i"(- 4 - 3 "i")`
Simplify the following and express in the form a + ib:
`(sqrt(5) + sqrt(3)"i")/(sqrt(5) - sqrt(3)"i")`
Write the conjugates of the following complex number:
3 – i
Simplify:
`(i^592 + i^590 + i^588 + i^586 + i^584)/(i^582 + i^580 + i^578 + i^576 + i^574)`
Find the value of 1 + i2 + i4 + i6 + i8 + ... + i20
Evaluate: `("i"^37 + 1/"i"^67)`
Prove that `(1 + "i")^4 xx (1 + 1/"i")^4` = 16
If a = `(-1 + sqrt(3)"i")/2`, b = `(-1 - sqrt(3)"i")/2` then show that a2 = b and b2 = a
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:
Answer the following:
Simplify the following and express in the form a + ib:
`3 + sqrt(-64)`
Answer the following:
Simplify the following and express in the form a + ib:
`5/2"i"(-4 - 3"i")`
Answer the following:
Solve the following equations for x, y ∈ R:
(x + iy) (5 + 6i) = 2 + 3i
Show that `(1/sqrt(2) + "i"/sqrt(2))^10 + (1/sqrt(2) - "i"/sqrt(2))^10` = 0
Answer the following:
Simplify: `("i"^65 + 1/"i"^145)`
Answer the following:
Simplify `[1/(1 - 2"i") + 3/(1 + "i")] [(3 + 4"i")/(2 - 4"i")]`
The value of `(- sqrt(-1))^(4"n" - 3)`, where n ∈ N, is ______.
State true or false for the following:
The complex number cosθ + isinθ can be zero for some θ.
What is the reciprocal of `3 + sqrt(7)i`.
Number of solutions of the equation z2 + |z|2 = 0 is ______.
Evaluate `sum_(n = 1)^13 (i^n + i^(n + 1))`, where n ∈ N.
Multiplicative inverse of 1 + i is ______.
If z1 and z2 are complex numbers such that z1 + z2 is a real number, then z2 = ______.
State True or False for the following:
Multiplication of a non-zero complex number by –i rotates the point about origin through a right angle in the anti-clockwise direction.
Find `|(1 + i) ((2 + i))/((3 + i))|`.
The real value of α for which the expression `(1 - i sin alpha)/(1 + 2i sin alpha)` is purely real is ______.
The value of `(z + 3)(barz + 3)` is equivalent to ______.
Which of the following is correct for any two complex numbers z1 and z2?
If `(x + iy)^(1/5)` = a + ib, and u = `x/a - y/b`, then ______.
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)`
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)`
Show that `(-1 + sqrt3i)^3` is a real number.
i2 + i3 + ... + i4000 =
