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(a) Find the set of values of k that satisfy the inequality k^2 - k - 12 < 0. (b) We have a triangle ABC, of lengths AC = 4 and BC = 2. Given that cos B < 1/4 , find the range of possible values for AB:

Solution: Let us consider the function f(k) = k 2 - k -12. Firstly, we want to find out the roots of this function. We calculate the discriminant: Δ = b 2 – 4 a c = 49 hence we get the roots: k 1 = (1-7)/(2 ...
AH
Answered by Alexandru H. • Maths tutor
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Given that y = arcos(x/2) find dy/dx of arccos(x/2) and hence find the integral from 0 to 1 of arcos(x/2)dx

a) First we can rewrite y=arccos(x/2) as cosy = x/2 which allows us to use implicit differentiation to easily differentiate each side. Taking d/dx of each side gives us -siny*(dy/dx) = 1/2 . Rearranging this...
OP
Answered by Olivia P. • Maths tutor
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Three girls and four boys are seated randomly on a straight bench. Find the probability that the girls sit together and the boys sit together.

There is a total of 7! possible ways in which the seven children can be seated on the bench. The number of favourable arrangements when four boys are seated on one side and three girls on the other side is 4...
Answered by • Maths tutor
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Let f (x) = 5x and g(x) = x2 + 1 , for x ∈  . (a) Find f-1(x) . (b) Find ( f ° g) (7) .

a,y=5xfinding the inverse: x=5y - dividing by 5 (1/5)x=y f -1 (x)=(1/5)x b,5x(7 2 +1)=250
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Let (x + 3) be a factor of the polynomial P(x) = x^3 + ax^2 - 7x + 6. Find a and the other two factors.

Firstly, (x - c) is a factor of a polynomial P(x) if and only if there exists a polynomial Q(x) such that P(x) = (x - c)Q(x), where c is a real or complex constant. Therefore, we can write: P(x) = x^3 + ax^2...
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