C1 2015 8 Edexcel

8

(a)

Factoring out x then using difference of two squares:

9x - 4x^3 = x(9 - 4x^2) = x(3 - 2x)(3 + 2x)

(b)

Note that C intersects the x-axis at x = 0, x = \dfrac 3 2 and x =-\dfrac 3 2. We consider the signs of each factor in the ranges x < -\dfrac 3 2, -\dfrac 3 2 \le x \le 0, 0 < x < \dfrac 3 2 and x \ge \dfrac 3 2.

Note that for x < -\dfrac 3 2 we have 3 - 2x > 0, x < 0 and 3 + 2x < 0, so y > 0.

For -\dfrac 3 2 \le x \le 0 we have 3 - 2x \ge 0, x \le 0 and 3 + 2x \ge 0, so y \le 0.

For 0 < x < \dfrac 3 2 we have 3 - 2x > 0, x > 0 and 3 + 2x > 0, so y > 0.

Finally for x \ge \dfrac 3 2 we have 3 - 2x \le 0, x > 0 and 3 + 2x > 0, so y \le 0.

This gives the graph:

Alternatively observe that since there are 3 distinct roots, none of these roots are double roots, so we can expect a sign change after each root.

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At x = 1 we have y = 9 - 4 = 5 (so B = (1,5)) and at x = -2 we have y = -18 + 4 \times 8 = 14 (so A = (-2, 14)). We then have:

|AB| = \sqrt {(14 - 5)^2 + (-2 - 1)^2} = \sqrt {81 + 9} = \sqrt {90} = \sqrt {9} \sqrt {10} = 3 \sqrt {10}

So in particular k = 3.

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