Search Results for: complex

a-2-i-b-a-1-b-1-i-a-2-b-2-i-Powers-of-complex-numbers-

Question Number 9661 by geovane10math last updated on 23/Dec/16 $a) 2^{i} =$ $b) {(a_{1} + b_{1} i)}^{a_{2} + b_{2} i} =$ $P o w e r s o f c o m p l e x n u m b e r s ? ? ?$ Commented…

If-z-1-and-z-2-are-complex-numbers-such-that-z-2-1-and-z-1-2z-2-2-z-1-z-2-1-then-z-1-is-equal-to-

Question Number 140442 by EnterUsername last updated on 07/May/21 $If z_{1} and z_{2} are complex numbers such that ∣ z_{2} ∣\neq 1 and$ $∣ (z_{1} - 2 z_{2}) / (2 - z_{1} {\bar{z}}_{2}) ∣= 1, then ∣ z_{1} ∣ is equal to_____.$ Answered by…

Passage-If-z-1-z-2-and-z-3-are-three-complex-numbers-representing-the-points-A-B-and-C-respectively-in-the-Argands-plane-and-BAC-then-z-3-z-1-z-2-z-1-AC-

Question Number 140260 by EnterUsername last updated on 05/May/21 $Passage : If z_{1}, z_{2} and z_{3} are three complex numbers$ $representing the points A, B and C, respectively, in the$ $Argands plane and ∠ B A C = α, then$ $\frac{z_{3} - z_{1}}{z_{2} - z_{1}} = (\frac{A C}{A B}) (\cos α + i \sin α)$ $$\left({i}\right)\:\mathrm{If}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation}…

Let-p-and-q-be-positive-integers-having-no-positive-common-divisors-except-unity-Let-z-1-z-2-z-q-be-the-q-values-of-z-p-q-where-z-is-a-fixed-complex-number-Then-the-product-z-1-z-2-z

Question Number 140248 by EnterUsername last updated on 05/May/21 $Let p and q be positive integers having no positive$ $common divisors except unity . Let z_{1}, z_{2}, \dots, z_{q} be the$ $q values of z^{p / q}, where z is a fixed complex number . Then$ $the product z_{1} z_{2} \dots z_{q} is equal to$ $$\left(\mathrm{A}\right)\:{z}^{{p}}…

ABCD-is-a-rhombus-Its-diagonals-AC-and-BD-inter-sect-at-M-and-satisfy-BD-2AC-If-the-points-D-and-M-are-represented-by-the-complex-numbers-1-i-and-2-i-respectively-then-A-is-represented-by-A-3-i

Question Number 140205 by EnterUsername last updated on 05/May/21 $A B C D is a rhombus . Its diagonals A C and B D inter -$ $sect at M and satisfy B D = 2 A C . If the points D and$ $M are represented by the complex numbers 1 + i and$ $2 - i, respectively, then A is represented by$ $(A) 3 - i / 2 (B) 3 + i / 2 (C) 1 + 3 i / 2 (D) 1 - 3 i / 2$ Answered by mr W last…

Let-a-gt-0-and-z-1-z-a-z-0-is-a-complex-number-Then-the-maximum-and-minimum-values-of-z-are-A-a-a-2-4-2-B-2a-a-2-4-2-C-a-2-4-

Question Number 140198 by EnterUsername last updated on 05/May/21 $Let a > 0 and ∣ z + (1 / z) ∣= a (z \neq 0 is a complex number) .$ $Then the maximum and minimum values of ∣ z ∣ are$ $(A) \frac{a + \sqrt{a^{2} + 4}}{2} (B) \frac{2 a + \sqrt{a^{2} + 4}}{2}$ $(C) \frac{\sqrt{a^{2} + 4} - a}{2} (D) \frac{\sqrt{a^{2} + 4} - 2 a}{2}$ Answered by Dwaipayan…

Let-z-1-1-i-z-2-1-i-and-z-3-be-complex-numbers-such-that-z-1-z-2-and-z-3-form-an-equilateral-triangle-Then-z-3-is-equal-to-A-3-1-i-B-3-1-i-C-3

Question Number 140113 by EnterUsername last updated on 04/May/21 $Let z_{1} = 1 + i, z_{2} = - 1 - i and z_{3} be complex numbers$ $such that z_{1}, z_{2} and z_{3} form an equilateral triangle .$ $Then z_{3} is equal to$ $(A) \sqrt{3} (1 + i) (B) \sqrt{3} (1 - i)$ $$\left(\mathrm{C}\right)\:\sqrt{\mathrm{3}}\left({i}−\mathrm{1}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{D}\right)\:\sqrt{\mathrm{3}}\left(−\mathrm{1}−{i}\right)…

If-z-1-z-2-and-z-3-are-distinct-complex-numbers-such-that-z-1-z-2-z-3-1-and-z-1-2-z-2-z-3-z-2-2-z-3-z-1-z-3-2-z-1-z-2-1-then-the-value-o

Question Number 139700 by EnterUsername last updated on 30/Apr/21 $If z_{1}, z_{2} and z_{3} are distinct complex numbers such$ $that ∣ z_{1} ∣=∣ z_{2} ∣=∣ z_{3} ∣= 1 and$ $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{{z}_{\mathrm{1}} ^{\mathrm{2}} }{{z}_{\mathrm{2}} {z}_{\mathrm{3}} }+\frac{{z}_{\mathrm{2}} ^{\mathrm{2}}…

Let-a-b-be-non-zero-complex-numbers-and-z-1-z-2-be-the-roots-of-the-equation-z-2-az-b-0-If-there-exists-4-such-that-a-2-b-then-the-points-z-1-z-2-and-the-origin-A-form-an-equilateral-t

Question Number 139641 by EnterUsername last updated on 30/Apr/21 $Let a, b be non - zero complex numbers and z_{1}, z_{2} be$ $the roots of the equation z^{2} + a z + b = 0 . If there exists$ $λ ⩾ 4 such that a^{2} = λ b, then the points z_{1}, z_{2} and the$ $origin$ $$\left(\mathrm{A}\right)\:\mathrm{form}\:\mathrm{an}\:\mathrm{equilateral}\:\mathrm{triangle} \…

Is-i-0-i-1-i-2-i-n-cyclic-for-any-value-of-n-Determine-the-smallest-such-n-if-it-exists-is-a-complex-cuberoot-of-unity-and-i-1-

Question Number 8301 by Rasheed Soomro last updated on 06/Oct/16 $Is {{(ω + i)}^{0}, {(ω + i)}^{1}, {(ω + i)}^{2}, \dots ., {(ω + i)}^{n}}$ $cyclic for any value of n ?$ $Determine the smallest such n if it exists .$ $ω is a complex cuberoot of unity and$ $i = \sqrt{- 1}$ …