Mathematics Questions 10 marks

1. For what values of b is the line y=12x+b tangent to x^3
The slope of the tangent line of y=x^3 is 3x^2
Let 3x^2=12
x=2 or -2
when x=2 y=8
So 8=24+b, b=-16
when x=-2, y=-8
so -8=-24+b, b=16
b is 16 or -16
2. P(x) = (x+1)(x-3)Q(x) + a(x+1) + b where Q(x) is a polynomial and a and b are real numbers.
When P(x) is divided by (x+1) the remainder is -11
When P(x) is divided by (x-3) the remainder is 1
(i) What is the value of b?
(ii) What is the remainder when P(x) is divided by (x+1)(x-3) ?
(i)
sub x=-1
P(-1) = b = -11
(ii)
Since when P(x) is divided by (x-3) the remainder is 1
P(3)=4a-11=1
a=3
P(x)=(x+1)(x-3)Q(x) + 3(x+1) - 11
The remainder when P(x) is divided by (x+1)(x-3) is
3(x+1)-11=3x-8
3. Given that sinθ=3/4 and 0≦θ≦180, find all possible values of cosθ in exact form.
Here sinθ=3/4
sin^2θ=9/16
cos^2θ=1-sin^2θ=7/16
cosθ=sqrt(7/16) [when 0≦θ≦90]
or cosθ=-sqrt(7/16) [when 90≦θ≦180]