2. (a) Use the identity cos2 θ + sin2 θ = 1 to prove that tan2 θ = sec2 θ – 1.
(b) Solve, for 0 ≤ θ < 360°, the equation
2 tan2 θ + 4 sec θ + sec2 θ = 2
3. Rabbits were introduced onto an island. The number of rabbits, P, t years after they were
introduced is modelled by the equation
P = 80e1/3t, t ∈ ℝ, t ≥ 0
(a) Write down the number of rabbits that were introduced to the island.
(b) Find the number of years it would take for the number of rabbits to first exceed 1000.
(c) Find dP/dt
(d) Find P when dP/dt 50.
4. (i) Differentiate with respect to x
(a) x2 cos 3x
(b) ln(x2 + 1)/(x2 + 1)
(ii) A curve C has the equation
y = √(4x + 1), x > -¼ , y > 0
The point P on the curve has x-coordinate 2. Find an equation of the tangent to C at P in the form ax + by + c = 0, where a, b and c are integers.
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