1.Find the values of a and b for the polynomial f(x) = 2x^3 + ax^2 - 4x + b, given that f(x) is divisible by x+1 and x-3. Write f(x) in a factorised form.
Hint: consider f(-1) and f(3)
2. The population of a country is found to be growing continuously at an annual rate of 1.98% t years after 1 January 1960. The total population t years after 1 January 1960 is given by the formula P(t) = 120 x 10^6 e^kt
How many years after 1 January 1960 would it take for the population to reach 240 x 10^6?
3. The decay of a radioactive substance is modelled by M=Mo(sorry meant to be subscript zero) (10) ^-kt, where Mo is the initial of the substance, M is the mass remaining after t years, and k is a constant. If it takes 150 years for the substance to reach half of its original mass in, show that k = log2/150
This solution is comprised of detailed step-by-step calculations and analysis of the given problems and provides students with a clear perspective of the underlying concepts.