Advertisements
Advertisements
Question
Answer the following:
Simplify the following and express in the form a + ib:
`(1 + 2/"i")(3 + 4/"i")(5 + "i")^-1`
Advertisements
Solution
`(1 + 2/"i")(3 + 4/"i")(5 + "i")^-1`
= `(("i" + 2))/"i".((3"i" + 4))/"i". 1/(5 + "i")`
= `(3"i"^2 + 4"i" + 6"i" + 8)/("i"^2(5 + "i"))`
= `(-3 + 10"i" + 8)/(-1(5 + "i"))` ...[∵ i2 = – 1]
= `((5 + 10"i"))/(-(5 + "i")`
= `((5 + 10"i")(5 - "i"))/(-(5 + "i")(5 - "i"))`
= `(25 - 5"i" + 50"i" - 10"i"^2)/(-(25 - "i"^2)`
= `(25 + 45"i" - 10(-1))/(-[25 - (-1)]`
= `(35 + 45"i")/(-26)`
= `(-35)/26 - 45/26"i"`
APPEARS IN
RELATED QUESTIONS
Find the number of non-zero integral solutions of the equation `|1-i|^x = 2^x`.
Find the value of i49 + i68 + i89 + i110
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:
`3 + sqrt(-64)`
Simplify the following and express in the form a + ib:
(2i3)2
Simplify the following and express in the form a + ib:
`(5 + 7"i")/(4 + 3"i") + (5 + 7"i")/(4 - 3"i")`
Write the conjugates of the following complex number:
3 – i
Write the conjugates of the following complex number:
`sqrt(2) + sqrt(3)"i"`
Find the value of i49 + i68 + i89 + i110
Find the value of i + i2 + i3 + i4
Prove that `(1 + "i")^4 xx (1 + 1/"i")^4` = 16
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
Answer the following:
If x + iy = `("a" + "ib")/("a" - "ib")`, prove that x2 + y2 = 1
Answer the following:
Show that `(1 - 2"i")/(3 - 4"i") + (1 + 2"i")/(3 + 4"i")` is real
Answer the following:
Simplify: `("i"^29 + "i"^39 + "i"^49)/("i"^30 + "i"^40 + "i"^50)`
The argument of the complex number `(4 + 9i)/(13 + 5i)` is ______
Locate the points for which 3 < |z| < 4.
Find the value of 2x4 + 5x3 + 7x2 – x + 41, when x = `-2 - sqrt(3)"i"`.
The real value of ‘a’ for which 3i3 – 2ai2 + (1 – a)i + 5 is real is ______.
If (2 + i) (2 + 2i) (2 + 3i) ... (2 + ni) = x + iy, then 5.8.13 ... (4 + n2) = ______.
State true or false for the following:
The argument of the complex number z = `(1 + i sqrt(3))(1 + i)(cos theta + i sin theta)` is `(7pi)/12 + theta`.
State true or false for the following:
If three complex numbers z1, z2 and z3 are in A.P., then they lie on a circle in the complex plane.
1 + i2 + i4 + i6 + ... + i2n is ______.
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 = x + iy, then show that `z barz + 2(z + barz) + b` = 0, where b ∈ R, represents a circle.
Solve the equation |z| = z + 1 + 2i.
The value of `sqrt(-25) xx sqrt(-9)` is ______.
If |z + 4| ≤ 3, then the greatest and least values of |z + 1| are ______ and ______.
State True or False for the following:
For any complex number z the minimum value of |z| + |z – 1| is 1.
State True or False for the following:
The locus represented by |z – 1| = |z – i| is a line perpendicular to the join of (1, 0) and (0, 1).
Where does z lie, if `|(z - 5i)/(z + 5i)|` = 1.
A real value of x satisfies the equation `((3 - 4ix)/(3 + 4ix))` = α − iβ (α, β ∈ R) if α2 + β2 = ______.
`((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)`
Show that `(-1 + sqrt3 i)^3` is a real number.
