Advertisements
Advertisements
Question
If `("a" + 3"i")/(2+ "ib")` = 1 − i, show that (5a − 7b) = 0
Advertisements
Solution
`("a" + 3"i")/(2+ "ib")` = 1 − i
∴ a + 3i = (1 − i)(2 + ib) = 2 + bi − 2i − bi2
= 2 + (b − 2)i − b(−1) ...[∵ i2 = − 1]
∴ a + 3i = (2 + b) + (b − 2)i
Equating real and imaginary parts, we get
a = 2 + b and 3 = b − 2
∴ a = 2 + b and b = 5
∴ a = 2 + 5 = 7
∴ 5a − 7b = 5(7) − 7(5)
= 35 − 35
= 0
APPEARS IN
RELATED QUESTIONS
Find the number of non-zero integral solutions of the equation `|1-i|^x = 2^x`.
If `((1+i)/(1-i))^m` = 1, then find the least positive integral value of m.
Find the value of i49 + i68 + i89 + i110
Simplify the following and express in the form a + ib:
`3 + sqrt(-64)`
Simplify the following and express in the form a + ib:
`(1 + 2/i)(3 + 4/i)(5 + i)^-1`
Find the value of: x3 – 5x2 + 4x + 8, if x = `10/(3 - "i")`.
Write the conjugates of the following complex number:
`sqrt(5) - "i"`
Evaluate: `("i"^37 + 1/"i"^67)`
Find the value of `("i"^6 + "i"^7 + "i"^8 + "i"^9)/("i"^2 + "i"^3)`
If x + iy = `sqrt(("a" + "ib")/("c" + "id")`, prove that (x2 + y2)2 = `("a"^2 + "b"^2)/("c"^2 + "d"^2)`
If (a + ib) = `(1 + "i")/(1 - "i")`, then prove that (a2 + b2) = 1
Show that `((sqrt(7) + "i"sqrt(3))/(sqrt(7) - "i"sqrt(3)) + (sqrt(7) - "i"sqrt(3))/(sqrt(7) + "i"sqrt(3)))` is real
Find the value of x and y which satisfy the following equation (x, y∈R).
(x + 2y) + (2x − 3y)i + 4i = 5
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 + 2i + 15i6y = 7x + i3 (y + 4), find x + y
Answer the following:
Simplify the following and express in the form a + ib:
(1 + 3i)2(3 + i)
Solve the following equation for x, y ∈ R:
2x + i9y (2 + i) = xi7 + 10i16
Answer the following:
Evaluate: (1 − i + i2)−15
Answer the following:
Find the value of x3 + 2x2 − 3x + 21, if x = 1 + 2i
Answer the following:
Simplify: `("i"^65 + 1/"i"^145)`
If z1 = 5 + 3i and z2 = 2 - 4i, then z1 + z2 = ______.
State true or false for the following:
The complex number cosθ + isinθ can be zero for some θ.
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.
1 + i2 + i4 + i6 + ... + i2n is ______.
Number of solutions of the equation z2 + |z|2 = 0 is ______.
If `((1 + i)/(1 - i))^3 - ((1 - i)/(1 + i))^3` = x + iy, then find (x, y).
Solve the equation |z| = z + 1 + 2i.
Find the complex number satisfying the equation `z + sqrt(2) |(z + 1)| + i` = 0.
State True or False for the following:
The inequality |z – 4| < |z – 2| represents the region given by x > 3.
The real value of α for which the expression `(1 - i sin alpha)/(1 + 2i sin alpha)` is purely real is ______.
A real value of x satisfies the equation `((3 - 4ix)/(3 + 4ix))` = α − iβ (α, β ∈ R) if α2 + β2 = ______.
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 ______.
The smallest positive integer n for which `((1 + i)/(1 - i))^n` = –1 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 ______.
Find the value of `(i^592+i^590+i^588+i^586+i^584)/(i^582+i^580+i^578+i^576+i^574)`
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)`
