5. Which of the following statements is/are true for real numbers $x , y$ ?
(a) If $x ^ { 2 } = y ^ { 2 }$ then $x = y$.
(b) If $x ^ { 3 } = y ^ { 3 }$ then $x = y$.
(c) If $x < y$ then $x ^ { 2 } < y ^ { 2 }$.
(d) If $x < y$ then $x ^ { 3 } < y ^ { 3 }$.

7. Which of the following inequalities are correct?
(a) $1 + x \leq e ^ { x }$ for all $x \in \mathbb { R }$
(b) $2 ^ { n } \leq 2 n$ ! for all positive integers $n$
(c) $\left( 1 + x ^ { 2 } \right) ^ { n } \leq ( 1 + x ) ^ { 2 n }$ for all $x \in ( 0 , \infty )$
(d) $\sin ( x ) \leq \tan ( x )$ for all $x \in \left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$

16. If $a, b \in \mathbb{R}$ and $ab > 0$, then which of the following inequalities always holds?
(A) $a^2 + b^2 > 2ab$
(B) $a + b \geq 2\sqrt{ab}$
(C) $\frac{1}{a} + \frac{1}{b} > \frac{2}{\sqrt{ab}}$
(D) $\frac{b}{a} + \frac{a}{b} \geq 2$

Which of the following inequalities always holds? ( )
A. $a ^ { 2 } + b ^ { 2 } \leq 2 a b$
B. $a ^ { 2 } + b ^ { 2 } \geq - 2 a b$
C. $a + b \geq - 2 \sqrt { | a b | }$
D. $a + b \leq 2 \sqrt { | a b | }$

19. If $a \hat { I } ( 0 , \Pi / 2 )$ then $\sqrt { } \left( x ^ { 2 } + x \right)$ is always greater than or equal to :
(a) $2 \tan a$
(b) 1
(c) 2
(d) $\quad \sec ^ { 2 } a$

15. For any real numbers $a , b$, and $c$ where $a \geq b$, consider these three statements:
$$\begin{array} { l l } 1 & - b \geq - a \\ 2 & a ^ { 2 } + b ^ { 2 } \geq 2 a b \\ 3 & a c \geq b c \end{array}$$
Which of the statements 1,2 , and 3 must be true?
A none
B 1 only
C 2 only
D 3 only
E 1 and 2 only F 1 and 3 only G 2 and 3 only H 1,2 and 3

Let x and y be real numbers with $-1 < y < 0 < x$. Which of the following statements are always true?
I. $x + y > 0$ II. $x - y > 1$ III. $x \cdot ( y + 1 ) > 0$
A) Only I
B) Only III
C) I and II
D) I and III
E) II and III