Advertisements
Advertisements
प्रश्न
If `("a" + 3"i")/(2+ "ib")` = 1 − i, show that (5a − 7b) = 0
Advertisements
उत्तर
`("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
संबंधित प्रश्न
Find the multiplicative inverse of the complex number:
4 – 3i
If `x – iy = sqrt((a-ib)/(c - id))` prove that `(x^2 + y^2) = (a^2 + b^2)/(c^2 + d^2)`
If `((1+i)/(1-i))^m` = 1, then find the least positive integral value of m.
Find the value of: 2x3 – 11x2 + 44x + 27, if x = `25/(3 - 4"i")`
Simplify the following and express in the form a + ib:
`(4 + 3"i")/(1 - "i")`
Simplify the following and express in the form a + ib:
`(1 + 2/i)(3 + 4/i)(5 + i)^-1`
Write the conjugates of the following complex number:
5i
Write the conjugates of the following complex number:
`sqrt(2) + sqrt(3)"i"`
Find the value of 1 + i2 + i4 + i6 + i8 + ... + i20
Evaluate: `("i"^37 + 1/"i"^67)`
If (x + iy)3 = y + vi then show that `(y/x + "v"/y)` = 4(x2 – y2)
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:
`sqrt(-3) sqrt(-6)` 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:
(2i3)2
Answer the following:
Simplify the following and express in the form a + ib:
`(4 + 3"i")/(1 - "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:
Evaluate: (1 − i + i2)−15
Answer the following:
show that `((1 + "i")/sqrt(2))^8 + ((1 - "i")/sqrt(2))^8` = 2
Answer the following:
Show that z = `((-1 + sqrt(-3))/2)^3` is a rational number
Answer the following:
Simplify `[1/(1 - 2"i") + 3/(1 + "i")] [(3 + 4"i")/(2 - 4"i")]`
If z1 = 5 + 3i and z2 = 2 - 4i, then z1 + z2 = ______.
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|.
Find the value of k if for the complex numbers z1 and z2, `|1 - barz_1z_2|^2 - |z_1 - z_2|^2 = k(1 - |z_1|^2)(1 - |"z"_2|^2)`
The real value of ‘a’ for which 3i3 – 2ai2 + (1 – a)i + 5 is real is ______.
The value of `(- sqrt(-1))^(4"n" - 3)`, where n ∈ N, is ______.
State true or false for the following:
If n is a positive integer, then the value of in + (i)n+1 + (i)n+2 + (i)n+3 is 0.
What is the value of `(i^(4n + 1) -i^(4n - 1))/2`?
The area of the triangle on the complex plane formed by the complex numbers z, –iz and z + iz is ______.
For a positive integer n, find the value of `(1 - i)^n (1 - 1/i)^"n"`
Evaluate `sum_(n = 1)^13 (i^n + i^(n + 1))`, where n ∈ N.
If `(1 + i)^2/(2 - i)` = x + iy, then find the value of x + y.
If (1 + i)z = `(1 - i)barz`, then show that z = `-ibarz`.
If |z + 1| = z + 2(1 + i), then find z.
The number `(1 - i)^3/(1 - i^2)` is equal to ______.
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 ______.
If `|(6i, -3i, 1),(4, 3i, -1),(20, 3, i)|` = x + iy, then ______.
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.
