Advertisements
Advertisements
प्रश्न
If a = `(-1 + sqrt(3)"i")/2`, b = `(-1 - sqrt(3)"i")/2` then show that a2 = b and b2 = a
Advertisements
उत्तर
a = `(-1 + sqrt(3)"i")/2`, b = `(-1 - sqrt(3)"i")/2`
∴ a2 = `((-1 + sqrt(3)"i")/2)^2 = (1 - 2sqrt(3)"i" + 3"i"^2)/4`
= `(1 - 2sqrt(3)"i" + 3(-1))/4` ...[∵ i2 = – 1]
= `(-2 - 2sqrt(3)"i")/4`
= `(-1 - sqrt(3)"i")/2` = b
and b2 = `((-1 - sqrt(3)"i")/2)^2 = (1 + 2sqrt(3)"i" + 3"i"^2)/4`
= `(1 + 2sqrt(3)"i" + 3(-1))/4` . ...[∵ i2 = – 1]
= `(-2 + 2sqrt(3)"i")/4`
= `(-1 + sqrt(3)"i")/2` = a
APPEARS IN
संबंधित प्रश्न
Find the multiplicative inverse of the complex number.
–i
If `x – iy = sqrt((a-ib)/(c - id))` prove that `(x^2 + y^2) = (a^2 + b^2)/(c^2 + d^2)`
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`
Write the conjugates of the following complex number:
`-sqrt(-5)`
Write the conjugates of the following complex number:
`sqrt(5) - "i"`
Write the conjugates of the following complex number:
`sqrt(2) + sqrt(3)"i"`
Is (1 + i14 + i18 + i22) a real number? Justify your answer
Prove that `(1 + "i")^4 xx (1 + 1/"i")^4` = 16
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 + 2/"i")(3 + 4/"i")(5 + "i")^-1`
Answer the following:
Evaluate: i131 + i49
Answer the following:
Find the value of x4 + 9x3 + 35x2 − x + 164, if x = −5 + 4i
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
If z1 = 2 – 4i and z2 = 1 + 2i, then `bar"z"_1 + bar"z"_2` = ______.
Find the value of 2x4 + 5x3 + 7x2 – x + 41, when x = `-2 - sqrt(3)"i"`.
What is the smallest positive integer n, for which (1 + i)2n = (1 – i)2n?
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`.
If z = x + iy, then show that `z barz + 2(z + barz) + b` = 0, where b ∈ R, represents a circle.
Find `|(1 + i) ((2 + i))/((3 + i))|`.
Where does z lie, if `|(z - 5i)/(z + 5i)|` = 1.
The real value of α for which the expression `(1 - i sin alpha)/(1 + 2i sin alpha)` is purely real is ______.
Let x, y ∈ R, then x + iy is a non-real complex number if ______.
If z is a complex number, then ______.
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 `(3 + i)(z + barz) - (2 + i)(z - barz) + 14i` = 0, then `barzz` is equal to ______.
The smallest positive integer n for which `((1 + i)/(1 - i))^n` = –1 is ______.
If a complex number z satisfies the equation `z + sqrt(2)|z + 1| + i` = 0, then |z| is equal to ______.
If α, β, γ and a, b, c are complex numbers such that `α/a + β/b + γ/c` = 1 + i and `a/α + b/β + c/γ` = 0, then the value of `α^2/a^2 + β^2/b^2 + γ^2/c^2` is equal to ______.
The complex number z = x + iy which satisfy the equation `|(z - 5i)/(z + 5i)|` = 1, lie on ______.
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.
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:
`(3i^5 + 2i^7 + i^9)/(i^6 + 2i^8 + 3i^18)`
If z = 2 + i, then (z − 1) `(barz − 5) + (barz − 1)` (z − 5) is equal to ______.
