Advertisements
Advertisements
प्रश्न
Solve the following equation for x, y ∈ R:
2x + i9y (2 + i) = xi7 + 10i16
Advertisements
उत्तर
2x + i9y (2 + i) = xi7 +10i16
i9 = i8 x i = (i2)4i = (– 1)4i = i
i7 = i6 x i = (i2)3i = (–1)3i = – i
i16 = (i2)8 = (– 1)8 = 1
∴ given equation becomes
2x + iy(2 + i) = – xi + 10
∴ 2x + 2iy + i2y = – xi + 10
∴ 2x + 2iy – y = – xi + 10 ...[∵ i2 = – 1]
∴ 2x + 2iy – y + xi – 10 = 0
∴ (2x – y – 10) + (x + 2y)i = 0
∴ (2x – y) + (x + 2y)i = 10 + 0i
Equating the real and imaginary parts separately, we get,
2x – y = 10 ...(1)
and x + 2y = 0 ...(2)
Multiplying equation (1) by 2, we get, 4x – 2y = 20
Adding this equation with equation (2), we get,
5x = 20
∴ x = 4
∴ from (2), 4 + 2y = 0
∴ 2y = – 4
∴ y = – 2
Hence, x = 4, y = – 2.
APPEARS IN
संबंधित प्रश्न
Show that 1 + i10 + i20 + i30 is a real number.
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:
`(3"i"^5 + 2"i"^7 + "i"^9)/("i"^6 + 2"i"^8 + 3"i"^18)`
Write the conjugates of the following complex number:
`-sqrt(5) - sqrt(7)"i"`
Write the conjugates of the following complex number:
`-sqrt(-5)`
Write the conjugates of the following complex number:
cosθ + i sinθ
Simplify:
`(i^592 + i^590 + i^588 + i^586 + i^584)/(i^582 + i^580 + i^578 + i^576 + i^574)`
Evaluate: `("i"^37 + 1/"i"^67)`
Prove that `(1 + "i")^4 xx (1 + 1/"i")^4` = 16
Find the value of `("i"^6 + "i"^7 + "i"^8 + "i"^9)/("i"^2 + "i"^3)`
If a = `(-1 + sqrt(3)"i")/2`, b = `(-1 - sqrt(3)"i")/2` then show that a2 = b and b2 = a
Find the value of x and y which satisfy the following equation (x, y∈R).
(x + 2y) + (2x − 3y)i + 4i = 5
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:
Simplify the following and express in the form a + ib:
`(3"i"^5 + 2"i"^7 + "i"^9)/("i"^6 + 2"i"^8 + 3"i"^18)`
Answer the following:
Show that `(1 - 2"i")/(3 - 4"i") + (1 + 2"i")/(3 + 4"i")` is real
If |z1| = 1, |z2| = 2, |z3| = 3 and |9z1z2 + 4z1z3 + z2z3| = 12, then the value of |z1 + z2 + z3| is
If `(x + iy)^(1/3)` = a + ib, where x, y, a, b ∈ R, show that `x/a - y/b` = –2(a2 + b2)
Locate the points for which 3 < |z| < 4.
State true or false for the following:
The points representing the complex number z for which |z + 1| < |z − 1| lies in the interior of a circle.
What is the reciprocal of `3 + sqrt(7)i`.
What is the principal value of amplitude of 1 – i?
If (1 + i)z = `(1 - i)barz`, then show that z = `-ibarz`.
The number `(1 - i)^3/(1 - i^2)` is equal to ______.
If z1 and z2 are complex numbers such that z1 + z2 is a real number, then z2 = ______.
State True or False for the following:
The inequality |z – 4| < |z – 2| represents the region given by x > 3.
Find `|(1 + i) ((2 + i))/((3 + i))|`.
If `((1 + i)/(1 - i))^x` = 1, then ______.
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?
The complex number z which satisfies the condition `|(i + z)/(i - z)|` = 1 lies on ______.
Let |z| = |z – 3| = |z – 4i|, then the value |2z| is ______.
If α and β are the roots of the equation x2 + 2x + 4 = 0, then `1/α^3 + 1/β^3` is equal to ______.
The complex number z = x + iy which satisfy the equation `|(z - 5i)/(z + 5i)|` = 1, lie on ______.
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`
Find the value of `sqrt(-3) xx sqrt(-6)`
