A-level circle question

Answer will be provided during session (easier to write maths than to type)

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At each point P of a curve for which x > 0 the tangent cuts the y-axis at T, and N is the foot of the perpendicular from P to the y-axis. If T is always 1 unit below N and the curve passes through the point (1,0), find the Cartesian equation of the curve.


Integrate y=x^2 between the limits x=3 and x=1


Why is the derivative of a function its gradient?


Find the values of x for which f(x) is an increasing function given that f(x)=8x-2x^2