A polygon has $n$ vertices, where $n \geq 3$. It has the following properties:

• Every vertex of the polygon lies on the circumference of a circle $C$.
• The centre of the circle $C$ is inside the polygon.
• The radii from the centre of the circle $C$ to the vertices of the polygon cut the polygon into $n$ triangles of equal area.

For which values of $n$ are these properties sufficient to deduce that the polygon is regular?
A no values of $n$
B $n = 3$ only
C $n = 3$ and $n = 4$ only
D $\quad n = 3$ and $n \geq 5$ only
E all values of $n$

Consider the following statement about a pentagon P: (*) If at least one of the interior angles in P is $108 ^ { \circ }$, then all the interior angles in P form an arithmetic sequence.
Which of the following is/are true? I The statement (*) II The contrapositive of (*) III The converse of (*)
A none of them B I only C II only D III only E I and II only F I and III only G II and III only H I, II and III

In this question, $k$ is a positive integer. Consider the following theorem: If $2 ^ { k } + 1$ is a prime, then $k$ is a power of $2 . \quad ( * )$ Which of the following statements, taken individually, is/are equivalent to (*)? I If $k$ is a power of 2 , then $2 ^ { k } + 1$ is prime. II $\quad 2 ^ { k } + 1$ is not prime only if $k$ is not a power of 2 . III A sufficient condition for $k$ to be a power of 2 is that $2 ^ { k } + 1$ is prime.

	Statement I is equivalent to (*)	Statement II is equivalent to (*)	Statement III is equivalent to (*)
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

Let $x$ be a real number. Which one of the following statements is a sufficient condition for exactly three of the other four statements?
A $x \geq 0$ B $x = 1$ C $x = 0$ or $x = 1$ D $x \geq 0$ or $x \leq 1$ E $\quad x \geq 0$ and $x \leq 1$

Let $\mathbf { p } , \mathbf { q }$ and $\mathbf { r }$ be propositions with their negations denoted by $\mathbf { p } ^ { \prime } , \mathbf { q } ^ { \prime } , \mathbf { r } ^ { \prime }$ respectively. Which of the following is equivalent to the proposition
$$p \vee q \Rightarrow q \wedge r$$
?
A) $\mathrm { p } ^ { \prime } \wedge \mathrm { q } ^ { \prime } \Rightarrow \mathrm { q } ^ { \prime } \vee \mathrm { r } ^ { \prime }$
B) $\mathrm { p } ^ { \prime } \wedge \mathrm { q } ^ { \prime } \Rightarrow \mathrm { q } ^ { \prime } \wedge \mathrm { r } ^ { \prime }$
C) $p ^ { \prime } \vee q ^ { \prime } \Rightarrow q ^ { \prime } \wedge r ^ { \prime }$
D) $q ^ { \prime } \wedge r ^ { \prime } \Rightarrow p ^ { \prime } \vee q ^ { \prime }$
E) $q ^ { \prime } \vee r ^ { \prime } \Rightarrow p ^ { \prime } \wedge q ^ { \prime }$