a) P is the point on the curve y= 2x^3 +kx -5 where x= 1 and gradient is -2.

(i) The value of the constant k
(ii) The equation of the normal to the curve at P

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  1. y = 2x^3 + kx - 5
    y' = 6x^2 + k
    so, you want
    6+k = -2
    k = -8
    y = 2x^3-8x-5
    since the slope is -2 at x=1,
    y(1) = -11
    The normal has slope m = 1/2, so at P(1,-11)
    y+11 = 1/2 (x-1)

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