Advertisements
Advertisements
प्रश्न
Show that 1 + i10 + i20 + i30 is a real number.
Advertisements
उत्तर
1 + i10 + i20 + i30
= 1 + (i4)2 .i2 + (i4)5 + (i4)7 .i2
= 1 + (1)2 (– 1 ) + (1)5 + (1)7 (– 1) ...[∵ i4 = 1, i2 = –1]
= 1 – 1 + 1 –1
= 0, which is a real number.
APPEARS IN
संबंधित प्रश्न
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 : x3 + 2x2 – 3x + 21, if x = 1 + 2i
Find the value of 1 + i2 + i4 + i6 + i8 + ... + i20
If a = `(-1 + sqrt(3)"i")/2`, b = `(-1 - sqrt(3)"i")/2` then show that a2 = b and b2 = a
Find the value of x and y which satisfy the following equation (x, y∈R).
If x(1 + 3i) + y(2 − i) − 5 + i3 = 0, find x + y
Select the correct answer from the given alternatives:
The value of is `("i"^592 + "i"^590 + "i"^588 + "i"^586 + "i"^584)/("i"^582 + "i"^580 + "i"^578 + "i"^576 + "i"^574)` is equal to:
Answer the following:
Simplify the following and express in the form a + ib:
`(4 + 3"i")/(1 - "i")`
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:
Simplify the following and express in the form a + ib:
`(sqrt(5) + sqrt(3)"i")/(sqrt(5) - sqrt(3)"i")`
Answer the following:
Evaluate: i131 + i49
If (1 + i)z = `(1 - i)barz`, then show that z = `-ibarz`.
If `(z - 1)/(z + 1)` is purely imaginary number (z ≠ – 1), then find the value of |z|.
The value of `sqrt(-25) xx sqrt(-9)` is ______.
If z1 and z2 are complex numbers such that z1 + z2 is a real number, then z2 = ______.
If |z + 4| ≤ 3, then the greatest and least values of |z + 1| are ______ and ______.
`((1 + cosθ + isinθ)/(1 + cosθ - isinθ))^n` = ______.
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.
`(3"i"^5 + 2"i"^7 + "i"^9)/("i"^6 + 2"i"^8 + 3"i"^18)`
Simplify the following and express in the form a + ib.
`(3i^5 + 2i^7 + i^9) / (i^6 + 2i^8 + 3i^18)`
i2 + i3 + ... + i4000 =
