TMUAmockz Set 1 Paper 2 · Question 1

Let $p$ and $q$ be real constants with $p\ne q$ . Two curves are defined by

C_1:y=(x-p)^2(x-q)\quad\text{and}\quad C_2:y=(x-q)^2(x-p)

Consider the following four statements.

I. The curves $C_1$ and $C_2$ always intersect at exactly two distinct points.
II. The curve $C_1$ always has a local minimum at $x=p$ .
III. The sum of the $y$ -coordinates of all intersection points of $C_1$ and $C_2$ is always zero.
IV. The tangent to $C_1$ at the point where $x=p$ is also tangent to $C_2$ at that point.

Which of the following statements are always true?

AI and III only

BII and IV only

CI and IV only

DI, III and IV only

EII, III and IV only

FI, II, III and IV

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