Advertisements
Advertisements
प्रश्न
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.
विकल्प
True
False
Advertisements
उत्तर
This statement is True.
Explanation:
Because in + (i)n+1 + (i)n+2 + (i)n+3
= in(1 + i + i2 + i3)
= in(1 + i – 1 – i)
= in(0)
= 0
APPEARS IN
संबंधित प्रश्न
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: 2x3 – 11x2 + 44x + 27, if x = `25/(3 - 4"i")`
Simplify the following and express in the form a + ib:
(1 + 3i)2 (3 + i)
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:
3 + i
Write the conjugates of the following complex number:
3 – i
Is (1 + i14 + i18 + i22) a real number? Justify your answer
Prove that `(1 + "i")^4 xx (1 + 1/"i")^4` = 16
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:
(1 + 3i)2(3 + i)
Answer the following:
Solve the following equation for x, y ∈ R:
`(x + "i"y)/(2 + 3"i")` = 7 – i
Answer the following:
Show that z = `((-1 + sqrt(-3))/2)^3` is a rational number
Answer the following:
Simplify: `("i"^65 + 1/"i"^145)`
The argument of the complex number `(4 + 9i)/(13 + 5i)` is ______
If z1, z2, z3 are complex numbers such that `|z_1| = |z_2| = |z_3| = |1/z_1 + 1/z_2 + 1/z_3|` = 1, then find the value of |z1 + z2 + z3|.
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?
What is the principal value of amplitude of 1 – i?
1 + i2 + i4 + i6 + ... + i2n is ______.
The value of `sqrt(-25) xx sqrt(-9)` is ______.
Multiplicative inverse of 1 + i 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 ______.
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.
State True or False for the following:
The inequality |z – 4| < |z – 2| represents the region given by x > 3.
If `((1 + i)/(1 - i))^x` = 1, 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 `(x + iy)^(1/5)` = a + ib, and u = `x/a - y/b`, then ______.
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 ______.
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)`
Simplify the following and express in the form a + ib.
`(3i^5+2i^7+i^9)/(i^6+2i^8+3i^18)`
