2. Which of the following statements are true?\\
(a) Let $f ( x ) = x ^ { 3 } + x - 1$ for $x \in \mathbb { R }$. Then the equation $f ( x ) = 0$ has at least one root in $[ - 1,0 )$.\\
(b) Let $f ( x ) = x ^ { 3 } + x - 1$ for $x \in \mathbb { R }$. Then the equation $f ( x ) = 0$ has at least one root in $[ 0,1 )$.\\
(c) Let $f ( x ) = x ^ { 3 } + x + 1$ for $x \in \mathbb { R }$. Then the equation $f ( x ) = 0$ has at least one root in $[ 0,1 )$.\\
(d) Let $f ( x ) = x ^ { 3 } + x + 1$ for $x \in \mathbb { R }$. Then the equation $f ( x ) = 0$ has at least one root in $[ - 1,0 )$.\\
3. 26 children participated in a chess tournament. A child got two points for winning, zero for losing and one point for a draw. Each child played against every other child. After the tournament was over no child had an odd score. There were no draws in the entire tournament and no two players had the same score. For convenience assume that the children are named $A , B , C , \ldots , Z$, by the 26 letters of the English alphabet in increasing order of scores, so $A$ has lowest score, $Z$ has highest score. Thus we have $A < B < C < \cdots < X < Y < Z$. Pick all statements which are true.\\
(a) K lost to Q\\
(b) K lost to B\\
(c) M lost to N\\
(d) If L lost to M then N lost to M.\\
4. 14 teams participate in a volleyball tournament. Each team plays the other exactly once. There are no draws. Assume the teams are labeled by $j , 1 \leq j \leq 14$. Let $x _ { j }$ denote the number of games team $j$ wins and $y _ { j }$ the number of games that team $j$ lost. Pick the correct alternative(s):\\
(a) $\sum _ { j } x _ { j } ^ { 2 } = \sum _ { j } y _ { j } ^ { 2 }$.\\
(b) $\sum _ { j } x _ { j } ^ { 2 } > \sum _ { j } y _ { j } ^ { 2 }$.\\
(c) $\sum _ { j } x _ { j } = \sum _ { j } y _ { j }$.\\
(d) $\sum _ { j } \left| x _ { j } \right| = \sum _ { j } \left| y _ { j } \right|$.\\