Basically I've got nine questions, mixed subjects, from Heinemann Pure Maths 3, Review Exercise 1, and if anyone can do ANY of them it would be very helpful and I'd be so grateful.
1) f(x)= x^3+ax^2+bx+6
Find, in terms of a and b, the remainder when f(x) is divided by x-2 and x+3. Given that these remainders are equal, express a in terms of b.
2) Expand (1-4x)^1/4 in ascending powers of x up to and including the term x^3, simplifying each coefficient.
3) Find the coordinates of the centre and radius of the circle whose equation is x^2+y^2-16x-12y+96=0
Also find the least and greatest distances of the origin O from the circumference of the circle.
4)a) Given that (x+1) is a factor of the expression (2x^3+ax^2-5x-2), find the value of the constant a. Show that, with this value of a, (x-2) is another factor of this expression and hence, or otherwise, factorise the expression completely.
b) When divided by (x-2) the expression (x^3+x^2+2x+2) leaves a remainder R. Find the value of R.
5) The population, p, of insects on an island, t hours after midday, is given by p=1000e^(kt) where k is a constant. Given that when t=0, the rate of change of the population with respect to time is 100 per hour,
a) Find k
b) Find the population when t=6
6)Differentiate with respect to x:
7) When a metal cube is heated, the length of each edge increases at the rate of 0.03cm/s. Find the rate of increase, in cm^2/s, of the total surface area of the cube, when the length of each edge is 8cm.
8) Given that (x-2) is a factor of f(x) where f(x)=x^3-x^2+Ax+B
find an equation satisfied by the constants A and B.
Given that when f(x) is divided by (x-3) the remainder is 10, find the second equation satisfied by A and B.
Solve your equations to find A and B.
Using your values for A and B, find 3 values of x for which f(x)=0
9) Find the coordinates of the turning points on the curve with the equation
Thank you in advance