Advertisements
Advertisements
प्रश्न
If (a + ib) = `(1 + "i")/(1 - "i")`, then prove that (a2 + b2) = 1
Advertisements
उत्तर
(a + ib) = `(1 + "i")/(1 - "i") = (1 + "i")^2/((1 - "i")(1 + "i"))`
∴ a + bi = `(1 + 2"i" + "i"^2)/(1 - "i"^2)`
= `(1 + 2"i" - 1)/(1 - (- 1))` ...[∵ i2 = – 1]
= `(2"i")/2`
= i
∴ a + bi = 0 + i
Equating real and imaginary parts, we get
a = 0 and b = 1
∴ a2 + b2 = 02 + 12 = 1
APPEARS IN
संबंधित प्रश्न
Find the multiplicative inverse of the complex number:
4 – 3i
Reduce `(1/(1-4i) - 2/(1+i))((3-4i)/(5+i))` to the standard form.
If `x – iy = sqrt((a-ib)/(c - id))` prove that `(x^2 + y^2) = (a^2 + b^2)/(c^2 + d^2)`
If (a + ib) (c + id) (e + if) (g + ih) = A + iB, then show that (a2 + b2) (c2 + d2) (e2 + f2) (g2 + h2) = A2 + B2.
Find the value of i49 + i68 + i89 + i110
Find the value of: x3 – x2 + x + 46, if x = 2 + 3i
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:
`(3i^5 + 2i^7 + i^9)/(i^6 + 2i^8 + 3i^18)`
Write the conjugates of the following complex number:
3 + i
Find the value of 1 + i2 + i4 + i6 + i8 + ... + i20
Prove that `(1 + "i")^4 xx (1 + 1/"i")^4` = 16
Find the value of `("i"^6 + "i"^7 + "i"^8 + "i"^9)/("i"^2 + "i"^3)`
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:
`5/2"i"(-4 - 3"i")`
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:
`(1 + 2/"i")(3 + 4/"i")(5 + "i")^-1`
Solve the following equation for x, y ∈ R:
2x + i9y (2 + i) = xi7 + 10i16
The argument of the complex number `(4 + 9i)/(13 + 5i)` is ______
If `(x + iy)^(1/3)` = a + ib, where x, y, a, b ∈ R, show that `x/a - y/b` = –2(a2 + b2)
Locate the points for which 3 < |z| < 4.
Find the value of 2x4 + 5x3 + 7x2 – x + 41, when x = `-2 - sqrt(3)"i"`.
If (2 + i) (2 + 2i) (2 + 3i) ... (2 + ni) = x + iy, then 5.8.13 ... (4 + n2) = ______.
State true or false for the following:
Multiplication of a non-zero complex number by i rotates it through a right angle in the anti-clockwise direction.
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`?
If `(1 + i)^2/(2 - i)` = x + iy, then find the value of x + y.
Find the complex number satisfying the equation `z + sqrt(2) |(z + 1)| + i` = 0.
The value of `(z + 3)(barz + 3)` is equivalent to ______.
If `((1 + i)/(1 - i))^x` = 1, then ______.
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 ______.
The complex number z which satisfies the condition `|(i + z)/(i - z)|` = 1 lies on ______.
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 ______.
Let |z| = |z – 3| = |z – 4i|, then the value |2z| is ______.
The smallest positive integer n for which `((1 + i)/(1 - i))^n` = –1 is ______.
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)`
