1. Given that the coefficient of x^2 in the expansion of
(1+2x+3x^2)(1-kx)^4 is 11, find the
a. possible values of k
b. the respective coefficient of x for both values of
k.
Help for the 2nd part please. Anyway in the 1st segment we were
told to simplify the first 3 terms of the equation (1-kx)^4. I hope
this makes your work easier.
To be frank, I do not understand what the question requires of
me for b.
2. The variables x and y are related by the equation y =
(x-2)/(x^2 +5)
Firstly we were told to find dy/dx.
Then we were asked to
determine the range of values of x for which both y and
dy/dx are positive.
How do you go about doing it?
Thanks.
When you expand out (1+2x+3x^2)(1-kx)^4, you should get :
(1+2x+3x^2)(1 - 4kx + 6k^2x^2 - 4k^3x^3 +k^4x^4 ).
You will realise that the coefficient of x is ( 2 - 4k ).
Since you have already done part a), you will realise that k =
-2/3 or 2.
Now, you will only need to sub these 2 values into ( 2 - 4k ) to
get 14/3 and -6, which are the two respective values of
coefficients of x.
Take a look at y = (x-2)/(x^2 +5).
You will realise that for y to be positive, x must be more than
2.
Now take a look at dy/dx = - [ (x-5) (x+1) ] / (x^2 + 5)^2.
You will realise that for dy/dx to be positive, x must be more
than -1 and less than 5.
From this, you will realise that for both y and dy/dx to be
positive, x must be more than 2 and less than 5.