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 multiplicative inverse of the complex number:
4 – 3i
Find the number of non-zero integral solutions of the equation `|1-i|^x = 2^x`.
If (a + ib) (c + id) (e + if) (g + ih) = A + iB, then show that (a2 + b2) (c2 + d2) (e2 + f2) (g2 + h2) = A2 + B2.
Simplify the following and express in the form a + ib:
`3 + sqrt(-64)`
Simplify the following and express in the form a + ib:
(1 + 3i)2 (3 + i)
Write the conjugates of the following complex number:
`-sqrt(-5)`
Write the conjugates of the following complex number:
`sqrt(5) - "i"`
Find the value of 1 + i2 + i4 + i6 + i8 + ... + i20
If x + iy = (a + ib)3, show that `x/"a" + y/"b"` = 4(a2 − b2)
If (a + ib) = `(1 + "i")/(1 - "i")`, then prove that (a2 + b2) = 1
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")`
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:
(2 + 3i)(1 − 4i)
Answer the following:
Simplify the following and express in the form a + ib:
`5/2"i"(-4 - 3"i")`
Answer the following:
show that `((1 + "i")/sqrt(2))^8 + ((1 - "i")/sqrt(2))^8` = 2
Answer the following:
If x + iy = `("a" + "ib")/("a" - "ib")`, prove that x2 + y2 = 1
Answer the following:
Simplify: `("i"^29 + "i"^39 + "i"^49)/("i"^30 + "i"^40 + "i"^50)`
What is the smallest positive integer n, for which (1 + i)2n = (1 – i)2n?
1 + i2 + i4 + i6 + ... + i2n is ______.
The equation |z + 1 – i| = |z – 1 + i| represents a ______.
Solve the equation |z| = z + 1 + 2i.
If |z1| = 1(z1 ≠ –1) and z2 = `(z_1 - 1)/(z_1 + 1)`, then show that the real part of z2 is zero.
Find the complex number satisfying the equation `z + sqrt(2) |(z + 1)| + i` = 0.
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 = ______.
The point represented by the complex number 2 – i is rotated about origin through an angle `pi/2` in the clockwise direction, the new position of point is ______.
Let x, y ∈ R, then x + iy is a non-real complex number if ______.
If `(x + iy)^(1/5)` = a + ib, and u = `x/a - y/b`, then ______.
If a complex number z satisfies the equation `z + sqrt(2)|z + 1| + i` = 0, then |z| is equal to ______.
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 `(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)`
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)`
Show that `(-1 + sqrt3i)^3` is a real number.
