Let $K$ be a compact set of $\mathbb{R}$ and $A$ a subset of $C(K, \mathbb{R}^d)$. Show that a subset $A \subset C(K, \mathbb{R}^d)$ is relatively compact if and only if every sequence $(f_n)_{n \in \mathbb{N}} \in A^{\mathbb{N}}$ admits a subsequence that converges uniformly to a limit $f \in C(K, \mathbb{R}^d)$.

1. A triangle $A B C$ is given, right-angled at $B$. Prove that this triangle is isosceles if and only if the altitude $B H$ relative to the hypotenuse is congruent to half the hypotenuse.

For integers $m$ and $n$, both greater than 1, consider the following three statements : $P : m$ divides $n$, $Q : m$ divides $n ^ { 2 }$, $R : m$ is prime, then
(1) $Q \wedge R \rightarrow P$
(2) $P \wedge Q \rightarrow R$
(3) $Q \rightarrow R$
(4) $Q \rightarrow P$

The contrapositive of the following statement, "If the side of a square doubles, then its area increases four times", is
(1) if the area of a square increases four times, then its side is not doubled.
(2) if the area of a square increases four times, then its side is doubled.
(3) if the area of a square does not increase four times, then its side is not doubled.
(4) if the side of a square is not doubled, then its area does not increase four times.

The proposition $( \sim p ) \vee ( p \wedge \sim q )$ is equivalent to
(1) $p \rightarrow \sim q$
(2) $p \wedge \sim q$
(3) $q \rightarrow p$
(4) none

The Boolean expression $\sim ( p \vee q ) \vee ( \sim p \wedge q )$ is equivalent to
(1) $\sim q$
(2) $\sim p$
(3) $p$
(4) $q$

The Boolean expression $( p \wedge \sim q ) \Rightarrow ( q \vee \sim p )$ is equivalent to:
(1) $q \Rightarrow p$
(2) $p \Rightarrow q$
(3) $\sim q \Rightarrow p$
(4) $p \Rightarrow \sim q$

The negation of the Boolean expression $(\sim q \wedge p) \Rightarrow (\sim p \vee q)$ is logically equivalent to
(1) $p \Rightarrow q$
(2) $q \Rightarrow p$
(3) $\sim p \Rightarrow q$
(4) $\sim q \Rightarrow p$

For $\alpha \in \mathbb{N}$, consider a relation $R$ on $\mathbb{N}$ given by $R = \{(x, y) : 3x + \alpha y \text{ is a multiple of } 7\}$. The relation $R$ is an equivalence relation if and only if
(1) $\alpha = 14$
(2) $\alpha$ is a multiple of 4
(3) 4 is the remainder when $\alpha$ is divided by 10
(4) 4 is the remainder when $\alpha$ is divided by 7

The statement $( p \wedge q ) \Rightarrow ( p \wedge r )$ is equivalent to
(1) $q \Rightarrow ( p \wedge r )$
(2) $p \Rightarrow ( p \wedge r )$
(3) $( p \wedge r ) \Rightarrow ( p \wedge q )$
(4) $( p \wedge q ) \Rightarrow r$

The statement $(p \Rightarrow (q \vee p)) \Rightarrow r$ is NOT equivalent to:
(1) $p \wedge \sim r \Rightarrow q$
(2) $\sim q \Rightarrow (\sim r \vee p)$
(3) $p \Rightarrow (q \vee r)$
(4) $p \wedge \sim q \Rightarrow r$

The compound statement $( \sim ( P \wedge Q ) ) \vee ( ( \sim P ) \wedge Q ) \Rightarrow ( ( \sim P ) \wedge ( \sim Q ) )$ is equivalent to
(1) $( ( \sim P ) \vee Q ) \wedge ( ( \sim Q ) \vee P )$
(2) $( \sim Q ) \vee P$
(3) $( ( \sim P ) \vee Q ) \wedge ( \sim Q )$
(4) $( \sim P ) \vee Q$

Let $p$ and $q$ be two statements. Then $\sim ( p \wedge ( p \rightarrow \sim q ) )$ is equivalent to
(1) $p \vee ( p \wedge ( \sim q ) )$
(2) $p \vee ( ( \sim p ) \wedge q )$
(3) $( \sim p ) \vee q$
(4) $p \wedge q$

The negation of $( p \wedge ( - q ) ) \vee ( - p )$ is equivalent to
(1) $p \wedge ( - q )$
(2) $p \wedge q$
(3) $p \vee ( q \vee ( - p ) )$
(4) $p \wedge ( q \wedge ( - p ) )$

The statement $\sim p \vee ( \sim p \wedge q )$ is equivalent to
(1) $\sim p \wedge q$
(2) $p \wedge q \wedge \sim p$
(3) $\sim p \wedge q \wedge q$
(4) $\sim p \vee q$

The statement $( p \wedge ( \sim q ) ) \vee ( ( \sim p ) \wedge q ) \vee ( ( \sim p ) \wedge ( \sim q ) )$ is equivalent to
(1) $\sim p \vee q$
(2) $\sim p \vee \sim q$
(3) $p \vee \sim q$
(4) $p \vee q$

Given that $f ( x )$ and $g ( x )$ and $h ( x )$ are linear polynomials and $$f ( x ) \cdot g ( x ) \cdot h ( x ) = 0$$ prove that at least one of the following statements must be true; (I) $f ( x ) \cdot g ( x ) = 0$, (II) $g ( x ) \cdot h ( x ) = 0$, (III) $f ( x ) \cdot h ( x ) = 0$. For each of the three statements, give examples of polynomials for which that statement is true and the other two statements are false.

Triangles $A B C$ and $X Y Z$ have the same area.
Which of these extra conditions, taken independently, would imply that they are congruent?
(1) $A B = X Y$ and $B C = Y Z$
(2) $A B = X Y$ and $\angle A B C = \angle X Y Z$
(3) $\angle A B C = \angle X Y Z$ and $\angle B C A = \angle Y Z X$

Consider the following statement:
For any positive integer $N$ there is a positive integer $K$ such that $N ( K m + 1 ) - 1$ is not prime for any positive integer $m$.
Which one of the following is the negation of this statement?

Consider the following statement about the positive integers $a , b$ and $n$ :
(*): $a b$ is divisible by $n$
The condition 'either $a$ or $b$ is divisible by $n$ ' is:

$x$ is a real number and f is a function. Given that exactly one of the following statements is true, which one is it?
A $x \geq 0$ only if $\mathrm { f } ( x ) < 0$
B $x < 0$ if $\mathrm { f } ( x ) \geq 0$
C $\quad x \geq 0$ only if $\mathrm { f } ( x ) \geq 0$
D $\mathrm { f } ( x ) < 0$ if $x < 0$
E $\quad \mathrm { f } ( x ) \geq 0$ only if $x \geq 0$ F $\quad \mathrm { f } ( x ) \geq 0$ if and only if $x < 0$

Consider the following two statements about the polynomial $\mathrm { f } ( x )$ : $P : \quad \mathrm { f } ( x ) = 0$ for exactly three real values of $x$ $Q : \quad \mathrm { f } ^ { \prime } ( x ) = 0$ for exactly two real values of $x$ Which one of the following is correct?
A $P$ is necessary but not sufficient for $Q$.
B $P$ is sufficient but not necessary for $Q$.
C $P$ is necessary and sufficient for $Q$.
D $P$ is not necessary and not sufficient for $Q$.

Consider the following statement about the polynomial $\mathrm { p } ( x )$, where $a$ and $b$ are real numbers with $a < b$ : (*) There exists a number $c$ with $a < c < b$ such that $\mathrm { p } ^ { \prime } ( c ) = 0$.
Which one of the following is true?
A The condition $\mathrm { p } ( a ) = \mathrm { p } ( b )$ is necessary and sufficient for ( $*$ )
B The condition $\mathrm { p } ( a ) = \mathrm { p } ( b )$ is necessary but not sufficient for (*)
C The condition $\mathrm { p } ( a ) = \mathrm { p } ( b )$ is sufficient but not necessary for ( $*$ )
D The condition $\mathrm { p } ( a ) = \mathrm { p } ( b )$ is not necessary and not sufficient for ( $*$ )

Consider the following statements about a polynomial $\mathrm { f } ( x )$ : I $\mathrm { f } ( x ) = p x ^ { 3 } + q x ^ { 2 } + r x + s$, where $p \neq 0$. II There is a real number $t$ for which $\mathrm { f } ^ { \prime } ( t ) = 0$. III There are real numbers $u$ and $v$ for which $\mathrm { f } ( u ) \mathrm { f } ( v ) < 0$. Which of these statements is/are sufficient for the equation $\mathrm { f } ( x ) = 0$ to have a real solution?

	Statement I is sufficient	Statement II is sufficient	Statement III is 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

A straight line $L$ passes through $( 1,2 )$. Let P be the statement if the $y$-intercept of $L$ is negative, then the $x$-intercept of $L$ is positive. Which of the following statements must be true? I P II the converse of P III the contrapositive of P
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