Advertisements
Advertisements
Question
Answer the following:
Show that `(1 - 2"i")/(3 - 4"i") + (1 + 2"i")/(3 + 4"i")` is real
Advertisements
Solution
`(1 - 2"i")/(3 - 4"i") + (1 + 2"i")/(3 + 4"i")`
= `((1 - 2"i")(3 + 4"i") + (3 - 4"i")(1 + 2"i"))/((3 + 4"i")(3 - 4"i"))`
= `(3 + 4"i" - 6"i" - 8"i"^2 + 3 + 6"i" - 4"i" - 8"i"^2)/(9 - 16"i"^2)`
= `(6 - 16"i"^2)/(9 - 16(-1))`
= `(6 - 16(-1))/(9 + 16)` ...[∵ i2 = – 1]
= `22/25`, which is a real number.
APPEARS IN
RELATED QUESTIONS
Show that 1 + i10 + i20 + i30 is a real number.
Find the value of: x3 – x2 + x + 46, if x = 2 + 3i
Find the value of: 2x3 – 11x2 + 44x + 27, if x = `25/(3 - 4"i")`
Simplify the following and express in the form a + ib:
(2 + 3i)(1 – 4i)
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:
`(sqrt(5) + sqrt(3)"i")/(sqrt(5) - sqrt(3)"i")`
Find the value of : x3 + 2x2 – 3x + 21, if x = 1 + 2i
Write the conjugates of the following complex number:
3 + i
Show that 1 + i10 + i100 − i1000 = 0
If `("a" + 3"i")/(2+ "ib")` = 1 − i, show that (5a − 7b) = 0
If (a + ib) = `(1 + "i")/(1 - "i")`, then prove that (a2 + b2) = 1
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:
(1 + 3i)2(3 + i)
Answer the following:
Simplify the following and express in the form a + ib:
`(5 + 7"i")/(4 + 3"i") + (5 + 7"i")/(4 - 3"i")`
Answer the following:
Solve the following equations for x, y ∈ R:
(x + iy) (5 + 6i) = 2 + 3i
Answer the following:
If x + iy = `("a" + "ib")/("a" - "ib")`, prove that x2 + y2 = 1
Answer the following:
Simplify `[1/(1 - 2"i") + 3/(1 + "i")] [(3 + 4"i")/(2 - 4"i")]`
If z ≠ 1 and `"z"^2/("z - 1")` is real, then the point represented by the complex number z lies ______.
The argument of the complex number `(4 + 9i)/(13 + 5i)` is ______
If z1 = 2 – 4i and z2 = 1 + 2i, then `bar"z"_1 + bar"z"_2` = ______.
The value of `(- sqrt(-1))^(4"n" - 3)`, where n ∈ N, is ______.
What is the principal value of amplitude of 1 – i?
The area of the triangle on the complex plane formed by the complex numbers z, –iz and z + iz is ______.
The equation |z + 1 – i| = |z – 1 + i| represents a ______.
If `((1 + i)/(1 - i))^3 - ((1 - i)/(1 + i))^3` = x + iy, then find (x, y).
If (1 + i)z = `(1 - i)barz`, then show that z = `-ibarz`.
The number `(1 - i)^3/(1 - i^2)` is equal to ______.
Multiplicative inverse of 1 + i is ______.
Where does z lie, if `|(z - 5i)/(z + 5i)|` = 1.
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` = ______.
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)`
Show that `(-1 + sqrt3i)^3` is a real number.
