Advertisements
Advertisements
Question
For a positive integer n, find the value of `(1 - i)^n (1 - 1/i)^"n"`
Advertisements
Solution
We have `(1 - i)^n (1 - 1/i)^"n"`
= `[(1 - i)(1 - 1/i)]^n`
= `[(1 - i) (1 - 1/i xx i/i)]^n`
= `[(1 - i)(1 - i/i^2)]^n`
= `[(1 - i)(1 + i)]^n` .....`[because i^2 = -1]`
= `[1 - i^2]^n`
= `[1 + 1]^"n"`
= 2n
Hence, `(1 - i)^n (1 - 1/i)^n` = 2n.
APPEARS IN
RELATED QUESTIONS
Find the multiplicative inverse of the complex number.
–i
If α and β are different complex numbers with |β| = 1, then find `|(beta - alpha)/(1-baralphabeta)|`
Find the number of non-zero integral solutions of the equation `|1-i|^x = 2^x`.
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 i + i2 + i3 + i4
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:
`(3i^5 + 2i^7 + i^9)/(i^6 + 2i^8 + 3i^18)`
Write the conjugates of the following complex number:
`-sqrt(5) - sqrt(7)"i"`
Find the value of i49 + i68 + i89 + i110
If a = `(-1 + sqrt(3)"i")/2`, b = `(-1 - sqrt(3)"i")/2` then show that a2 = b and b2 = a
If `("a" + 3"i")/(2+ "ib")` = 1 − i, show that (5a − 7b) = 0
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:
(2 + 3i)(1 − 4i)
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:
Solve the following equation for x, y ∈ R:
`(x + "i"y)/(2 + 3"i")` = 7 – i
Answer the following:
Solve the following equations for x, y ∈ R:
(x + iy) (5 + 6i) = 2 + 3i
Answer the following:
Evaluate: (1 − i + i2)−15
If z ≠ 1 and `"z"^2/("z - 1")` is real, then the point represented by the complex number z lies ______.
If |z1| = 1, |z2| = 2, |z3| = 3 and |9z1z2 + 4z1z3 + z2z3| = 12, then the value of |z1 + z2 + z3| is
If `(x + iy)^(1/3)` = a + ib, where x, y, a, b ∈ R, show that `x/a - y/b` = –2(a2 + b2)
The real value of ‘a’ for which 3i3 – 2ai2 + (1 – a)i + 5 is real is ______.
State true or false for the following:
The points representing the complex number z for which |z + 1| < |z − 1| lies in the interior of a circle.
What is the value of `(i^(4n + 1) -i^(4n - 1))/2`?
Number of solutions of the equation z2 + |z|2 = 0 is ______.
If (1 + i)z = `(1 - i)barz`, then show that z = `-ibarz`.
If |z1| = |z2| = ... = |zn| = 1, then show that |z1 + z2 + z3 + ... + zn| = `|1/z_1 + 1/z_2 + 1/z_3 + ... + 1/z_n|`.
The number `(1 - i)^3/(1 - i^2)` is equal to ______.
If z1 and z2 are complex numbers such that z1 + z2 is a real number, then z2 = ______.
State True or False for the following:
Multiplication of a non-zero complex number by –i rotates the point about origin through a right angle in the anti-clockwise direction.
The real value of α for which the expression `(1 - i sin alpha)/(1 + 2i sin alpha)` is purely real is ______.
Which of the following is correct for any two complex numbers z1 and z2?
If a + ib = c + id, 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 z1, z2, z3 are complex numbers such that |z1| = |z2| = |z3| = `|1/z_1 + 1/z_2 + 1/z_3|` = 1, then |z1 + z2 + z3| is ______.
Let z be a complex number such that `|(z - i)/(z + 2i)|` = 1 and |z| = `5/2`. Then the value of |z + 3i| is ______.
Let `(-2 - 1/3i)^2 = (x + iy)/9 (i = sqrt(-1))`, where x and y are real numbers, then x – y equals to ______.
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.
