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To find the gradient of the tangent, we can differentiate to give dy/dx=3x^2+9. We can now put in x=2 to find the gradient at (2,1): 3(2)^2+9=21. Therefore the gradient is 21 at (2,1).

This is an example of an inequalities question from FP2. For this, we will need to use the tools learned in this chapter. To start with, it may be tempting to multiply both sides of the inequality by (x+3...

By definition z*  = a - bj.

We can write z/z* = ((a+bj)/(a-bj))*(a+bj)/(a+bj).

We calculate this to be z/z* = (a^2-b^2)/(a^2+b^2) + j(2ab)/(a^2+b^2).

Therefore, Re(z/z*) = (a^2-b^2)/(...

Multiply by complex conjugate

z = 50 / (3+4i) * (3-4i) / (3-4i)

Rationalise

z = 50 ( 3 - 4i) / 25 = 6 - 8i.

Let's say that T is a transformation of the two dimensional plane. Remember that we have the two standard unit vectors (1,0) and (0,1). These are, respectively, the unit vectors pointing in the positive d...