Clavis TMUA Set C Paper 2 · Question 14

In this question $f$ is a polynomial.

Consider the following three conditions:

Which of these conditions is/are necessary and sufficient for

\int_p^q f(x)\,dx=0\quad\text{for all real numbers }p\text{ and }q?

Which row in the following table is correct?

	Condition I is necessary and sufficient	Condition II is necessary and sufficient	Condition III is necessary and sufficient
A	yes	yes	yes
B	yes	yes	no
C	yes	no	yes
D	yes	no	no
E	no	yes	yes
F	no	yes	no
G	no	no	yes
H	no	no	no

ABCDEFGH

examday

	Condition I is necessary and sufficient	Condition II is necessary and sufficient	Condition III is necessary and sufficient
A	yes	yes	yes
B	yes	yes	no
C	yes	no	yes
D	yes	no	no
E	no	yes	yes
F	no	yes	no
G	no	no	yes
H	no	no	no

	Condition I is necessary and sufficient	Condition II is necessary and sufficient	Condition III is necessary and sufficient
A	yes	yes	yes
B	yes	yes	no
C	yes	no	yes
D	yes	no	no
E	no	yes	yes
F	no	yes	no
G	no	no	yes
H	no	no	no

	Condition I is necessary and sufficient	Condition II is necessary and sufficient	Condition III is necessary and sufficient
A	yes	yes	yes
B	yes	yes	no
C	yes	no	yes
D	yes	no	no
E	no	yes	yes
F	no	yes	no
G	no	no	yes
H	no	no	no