Let $0 < x < \frac { 1 } { 6 }$ be a real number. When a certain biased dice is rolled, a particular face $F$ occurs with probability $\frac { 1 } { 6 } - x$ and and its opposite face occurs with probability $\frac { 1 } { 6 } + x$; the other four faces occur with probability $\frac { 1 } { 6 }$. Recall that opposite faces sum to 7 in any dice. Assume that the probability of obtaining the sum 7 when two such dice are rolled is $\frac { 13 } { 96 }$. Then, the value of $x$ is: (A) $\frac { 1 } { 8 }$ (B) $\frac { 1 } { 12 }$ (C) $\frac { 1 } { 24 }$ (D) $\frac { 1 } { 27 }$.
An office has 8 officers including two who are twins. Two teams, Red and Blue, of 4 officers each are to be formed randomly. What is the probability that the twins would be together in the Red team? (A) $\frac { 1 } { 6 }$ (B) $\frac { 3 } { 7 }$ (C) $\frac { 1 } { 4 }$ (D) $\frac { 3 } { 14 }$
Suppose Roger has 4 identical green tennis balls and 5 identical red tennis balls. In how many ways can Roger arrange these 9 balls in a line so that no two green balls are next to each other and no three red balls are together? (A) 8 (B) 9 (C) 11 (D) 12
Let $f ( x )$ be a degree 4 polynomial with real coefficients. Let $z$ be the number of real zeroes of $f$, and $e$ be the number of local extrema (i.e., local maxima or minima) of $f$. Which of the following is a possible $( z , e )$ pair? (A) $( 4,4 )$ (B) $( 3,3 )$ (C) $( 2,2 )$ (D) $( 0,0 )$
A number is called a palindrome if it reads the same backward or forward. For example, 112211 is a palindrome. How many 6-digit palindromes are divisible by 495? (A) 10 (B) 11 (C) 30 (D) 45
Let $A$ be a square matrix of real numbers such that $A ^ { 4 } = A$. Which of the following is true for every such $A$? (A) $\operatorname { det } ( A ) \neq - 1$ (B) $A$ must be invertible. (C) $A$ can not be invertible. (D) $A ^ { 2 } + A + I = 0$ where $I$ denotes the identity matrix.
Consider the real-valued function $h : \{ 0,1,2 , \ldots , 100 \} \rightarrow \mathbb { R }$ such that $h ( 0 ) = 5 , h ( 100 ) = 20$ and satisfying $h ( i ) = \frac { 1 } { 2 } ( h ( i + 1 ) + h ( i - 1 ) )$, for every $i = 1,2 , \ldots , 99$. Then, the value of $h ( 1 )$ is: (A) 5.15 (B) 5.5 (C) 6 (D) 6.15.
Let $f ( x ) = \frac { 1 } { 2 } x \sin x - ( 1 - \cos x )$. The smallest positive integer $k$ such that $\lim _ { x \rightarrow 0 } \frac { f ( x ) } { x ^ { k } } \neq 0$ is: (A) 3 (B) 4 (C) 5 (D) 6.
Nine students in a class gave a test for 50 marks. Let $S _ { 1 } \leq S _ { 2 } \leq \cdots \leq S _ { 5 } \leq \cdots \leq S _ { 8 } \leq S _ { 9 }$ denote their ordered scores. Given that $S _ { 1 } = 20$ and $\sum _ { i = 1 } ^ { 9 } S _ { i } = 250$, let $m$ be the smallest value that $S _ { 5 }$ can take and $M$ be the largest value that $S _ { 5 }$ can take. Then the pair $( m , M )$ is given by (A) $( 20,35 )$ (B) $( 20,34 )$ (C) $( 25,34 )$ (D) $( 25,50 )$.
Let 10 red balls and 10 white balls be arranged in a straight line such that 10 each are on either side of a central mark. The number of such symmetrical arrangements about the central mark is (A) $\frac { 10 ! } { 5 ! 5 ! }$ (B) $10 !$ (C) $\frac { 10 ! } { 5 ! }$ (D) $2 \cdot 10 !$
If $z = x + i y$ is a complex number such that $\left| \frac { z - i } { z + i } \right| < 1$, then we must have (A) $x > 0$ (B) $x < 0$ (C) $y > 0$ (D) $y < 0$.
Let $S = \left\{ x - y \mid x , y \text{ are real numbers with } x ^ { 2 } + y ^ { 2 } = 1 \right\}$. Then the maximum number in the set $S$ is (A) 1 (B) $\sqrt { 2 }$ (C) $2 \sqrt { 2 }$ (D) $1 + \sqrt { 2 }$.
Let $A B C D$ be a rectangle with its shorter side $a > 0$ units and perimeter $2 s$ units. Let $P Q R S$ be any rectangle such that vertices $A , B , C$ and $D$ respectively lie on the lines $P Q , Q R , R S$ and $S P$. Then the maximum area of such a rectangle $P Q R S$ in square units is given by (A) $s ^ { 2 }$ (B) $2 a ( s - a )$ (C) $\frac { s ^ { 2 } } { 2 }$ (D) $\frac { 5 } { 2 } a ( s - a )$.
Let $p ( n )$ be the number of digits when $8 ^ { n }$ is written in base 6, and let $q ( n )$ be the number of digits when $6 ^ { n }$ is written in base 4. For example, $8 ^ { 2 }$ in base 6 is 144, hence $p ( 2 ) = 3$. Then $\lim _ { n \rightarrow \infty } \frac { p ( n ) q ( n ) } { n ^ { 2 } }$ equals: (A) 1 (B) $\frac { 4 } { 3 }$ (C) $\frac { 3 } { 2 }$ (D) 2.
For a real number $\alpha$, let $S _ { \alpha }$ denote the set of those real numbers $\beta$ that satisfy $\alpha \sin ( \beta ) = \beta \sin ( \alpha )$. Then which of the following statements is true? (A) For any $\alpha , S _ { \alpha }$ is an infinite set. (B) $S _ { \alpha }$ is a finite set if and only if $\alpha$ is not an integer multiple of $\pi$. (C) There are infinitely many numbers $\alpha$ for which $S _ { \alpha }$ is the set of all real numbers. (D) $S _ { \alpha }$ is always finite.
If $A = \left( \begin{array} { l l } 1 & 1 \\ 0 & i \end{array} \right)$ and $A ^ { 2018 } = \left( \begin{array} { l l } a & b \\ c & d \end{array} \right)$, then $a + d$ equals: (A) $1 + i$ (B) 0 (C) 2 (D) 2018.
A bag contains some candies, $\frac { 2 } { 5 }$ of them are made of white chocolate and the remaining $\frac { 3 } { 5 }$ are made of dark chocolate. Out of the white chocolate candies, $\frac { 1 } { 3 }$ are wrapped in red paper, the rest are wrapped in blue paper. Out of the dark chocolate candies, $\frac { 2 } { 3 }$ are wrapped in red paper, the rest are wrapped in blue paper. If a randomly selected candy from the bag is found to be wrapped in red paper, then what is the probability that it is made up of dark chocolate? (A) $\frac { 2 } { 3 }$ (B) $\frac { 3 } { 4 }$ (C) $\frac { 3 } { 5 }$ (D) $\frac { 1 } { 4 }$