Mensuration

We need to find the roots (or zeros) of the polynomial, which are the values of x for which the polynomial's value is 0.

We set the polynomial equal to zero and solve for x. We can use the factored form from the previous part.
x² - 12x + 32 = 0 From part (i), we know that the factored form of the polynomial is (x-4)(x-8).
So, the equation is:
(x-4)(x-8) = 0 For the product of two factors to be zero, at least one of the factors must be zero.
So, either:
x - 4 = 0 x = 4 Or:
x - 8 = 0 x = 8 The two values of x for which the polynomial becomes zero are 4 and 8.

The polynomial becomes zero for x=4 and x=8.

We are given the curved surface area (CSA) of a cone. We need to find the CSA of a new cone which has the same radius but its slant height is one-third of the original cone's slant height.

The formula for the Curved Surface Area (CSA) of a cone is given by:
CSA = π r l where r is the radius of the base and l is the slant height.

Let the original cone have radius r and slant height l.
Given, the CSA of the original cone is 60 square centimetres.
CSAₑₓtₒᵣᵢgᵢₙₐl = π r l = 60 cm² Now, consider the new cone.
The radius of the new cone is the same, r' = r.
The slant height of the new cone is one-third of the original slant height, l' = (1)/(3)l.
The CSA of the new cone will be:
CSAₑₓtₙₑw = π r' l' Substituting the new values:
CSAₑₓtₙₑw = π (r) ((1)/(3)l) CSAₑₓtₙₑw = (1)/(3) (π r l) Since we know that π r l = 60, we can substitute this value:
CSAₑₓtₙₑw = (1)/(3) × 60 CSAₑₓtₙₑw = 20 cm² The curved surface area of the new cone is 20 square centimetres.
(Note: The checkmark in the provided image is on (c) 30, which would be correct if the slant height was halved. Based on the question text "one third", the correct answer is 20.)