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Given that α= 1+3i is a root of the equation z^3 - pz^2 + 18z - q = 0 where p and q are real, find the other roots, then p and q.

All coefficients of z are real, therefore one root must be the complex conjugate so β = 1-3i.It is known that Σαβ = 18 (the coefficient of z), so we can get an equation in the third root, γ, as follows: Σαβ ...
ZG
5005 Views

find an expression for the sum of the series of 1 + 1/2cosx + 1/4cos2x +1/8cos3x + ......

Firstly we can use the fact that cosx is the real part of e^ix. This means that the series above is the real part of the infinite series 1 + 1/2e^ix + 1/4e^2ix + ...... This means that we have an infinite ge...
NK
5900 Views

Determine if these two vectors are perpendicular. a=[1,4,8], b=[0,6,-3]

Start by determining dot product. a.b = 1x0 + 4x6 + 8*-3 = 0as a.b = |a||b|cos(theta)cos(theta) = 0therefore theta = 90 degTherefore vectors are perpendicular
PD
2806 Views

How would you differentiate x^x?

To start off with, we have the expression y = x^x and we want to differentiate it. A clever way to do this would be to first remember implicit differentiation and start by taking the natural logarithm of bot...
AD
3286 Views

Differentiate arctan(x) with respect to x

We can set arctan(x) = y. Remember that arctan(x) is the inverse of tan(x), so we can take the tangent of both sides to give: tan(arctan(x)) = x = tan(y). Tan(x) has the standard derivative of sec^2(x) (whic...
SE
6077 Views