Advertisements
Advertisements
प्रश्न
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
उत्तर
`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
संबंधित प्रश्न
Express the following expression in the form of a + ib.
`((3 + sqrt5)(3 - isqrt5))/((sqrt3 + sqrt2i)-(sqrt3 - isqrt2))`
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:
(1 + 3i)2 (3 + i)
Simplify the following and express in the form a + ib:
`(4 + 3"i")/(1 - "i")`
Find the value of : x3 + 2x2 – 3x + 21, if x = 1 + 2i
Write the conjugates of the following complex number:
3 + i
Write the conjugates of the following complex number:
3 – i
Is (1 + i14 + i18 + i22) a real number? Justify your answer
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
If x + iy = (a + ib)3, show that `x/"a" + y/"b"` = 4(a2 − b2)
If x + iy = `sqrt(("a" + "ib")/("c" + "id")`, prove that (x2 + y2)2 = `("a"^2 + "b"^2)/("c"^2 + "d"^2)`
Find the value of x and y which satisfy the following equation (x, y∈R).
`(x+ 1)/(1 + "i") + (y - 1)/(1 - "i")` = i
Find the value of x and y which satisfy the following equation (x, y∈R).
If x(1 + 3i) + y(2 − i) − 5 + i3 = 0, 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 :
Answer the following:
Simplify the following and express in the form a + ib:
`5/2"i"(-4 - 3"i")`
Answer the following:
Evaluate: i131 + i49
Answer the following:
If x + iy = `("a" + "ib")/("a" - "ib")`, prove that x2 + y2 = 1
Answer the following:
Show that `(1 - 2"i")/(3 - 4"i") + (1 + 2"i")/(3 + 4"i")` is real
Answer the following:
Simplify: `("i"^238 + "i"^236 + "i"^234 + "i"^232 + "i"^230)/("i"^228 + "i"^226 + "i"^224 + "i"^222 + "i"^220)`
The value of (2 + i)3 × (2 – i)3 is ______.
If z1, z2, z3 are complex numbers such that `|z_1| = |z_2| = |z_3| = |1/z_1 + 1/z_2 + 1/z_3|` = 1, then find the value of |z1 + z2 + z3|.
The real value of ‘a’ for which 3i3 – 2ai2 + (1 – a)i + 5 is real is ______.
State true or false for the following:
Multiplication of a non-zero complex number by i rotates it through a right angle in the anti-clockwise direction.
State true or false for the following:
The complex number cosθ + isinθ can be zero for some θ.
What is the principal value of amplitude of 1 – i?
If |z1| = 1(z1 ≠ –1) and z2 = `(z_1 - 1)/(z_1 + 1)`, then show that the real part of z2 is zero.
Find `|(1 + i) ((2 + i))/((3 + i))|`.
If the least and the largest real values of α, for which the equation z + α|z – 1| + 2i = 0 `("z" ∈ "C" and "i" = sqrt(-1))` has a solution, are p and q respectively; then 4(p2 + q2) is equal to ______.
`((1 + cosθ + isinθ)/(1 + cosθ - isinθ))^n` = ______.
Let `(-2 - 1/3i)^2 = (x + iy)/9 (i = sqrt(-1))`, where x and y are real numbers, then x – y equals to ______.
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.
