A curve is defined by the parametric equations x = t - 5, y = t² — 25.

i) Find dy/dx in terms of t.

ii) Find the equation of the tangent to the curve at the point where t = -6.

iii) Express the equation of this curve in the simplest Cartesian form.

The function f(x) = 4(x - 2)²e^x-2 has stationary points at x = 0 and x = 2 as shown in the diagram below.

Draw separate sketches of the following graphs. In each case, label any asymptotes and the coordinates of any turning points.1

i)y =1/f(x)

ii) y² = f(x)

iii) y = f([x]) Starting from a point O, a particle P moves due north such that its displacement (in metres) from O is given by y = 40t, where t is in seconds.Also starting from O, a particle Q moves due east such that its displacement from O is given by x= t².

Let r be the distance between the two particles at a time t. Find the rate at which r is changing,when t = 4.

d)P(x) is an even polynomial of degree 4, i.e. P(-x) = P(x). Two of its zeros are at x = 1and x = 2.

i)Find the other two roots.

ii) If P(0) = 8, write P(x) in factored form.

By using the binomial expansion, show that:(q + p)^n - (q − p)^n = 2² (n1) q^n-1 p + 2 (n3)q^n-3p³ + ...

What is the last term in the expansion when n is odd? What is the last term in the expansion when n is even?