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# Constructing A Homotopy

TOPO-KCXEJI

Which of the following assignments of values to the variables $p, q$ makes the family of functions $f_t:[0,1]\rightarrow\mathbb R^2$ sending:

$$s\mapsto\left\{\begin{matrix}(1-t)(1/3,0)+t(0,3s)&\text{ if }0\le s\le 1/3\\\ (1-t)(s,0)+t(ps-q,1)\text{ if }1/3\le s\le 2/3\\\ (1-t)(2/3,0)+t(1,3-3s)\text{ if }2/3\le s\le 1\end{matrix}\right.$$

...where $0\le t\le 1$ a homotopy of paths?

A

$$p=2,q=1$$

B

$$p=2,q=2$$

C

$$p=3,q=1$$

D

$$p=3,q=2$$

E

$$p=3,q=3$$