Define the collections $\left\{ E _ { 1 } , E _ { 2 } , E _ { 3 } , \ldots \right\}$ of ellipses and $\left\{ R _ { 1 } , R _ { 2 } , R _ { 3 } , \ldots \right\}$ of rectangles as follows: $E _ { 1 } : \frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1 ;$ $R _ { 1 }$ : rectangle of largest area, with sides parallel to the axes, inscribed in $E _ { 1 }$; $E _ { n }$ : ellipse $\frac { x ^ { 2 } } { a _ { n } ^ { 2 } } + \frac { y ^ { 2 } } { b _ { n } ^ { 2 } } = 1$ of largest area inscribed in $R _ { n - 1 } , n > 1$; $R _ { n }$ : rectangle of largest area, with sides parallel to the axes, inscribed in $E _ { n } , n > 1$. Then which of the following options is/are correct?
(A) The eccentricities of $E _ { 18 }$ and $E _ { 19 }$ are NOT equal
(B) $\quad \sum _ { n = 1 } ^ { N } \left( \right.$ area of $\left. R _ { n } \right) < 24$, for each positive integer $N$
(C) The length of latus rectum of $E _ { 9 }$ is $\frac { 1 } { 6 }$
(D) The distance of a focus from the centre in $E _ { 9 }$ is $\frac { \sqrt { 5 } } { 32 }$

Let $$M = \left\{ ( x , y ) \in \mathbb { R } \times \mathbb { R } : x ^ { 2 } + y ^ { 2 } \leq r ^ { 2 } \right\} ,$$ where $r > 0$. Consider the geometric progression $a _ { n } = \frac { 1 } { 2 ^ { n - 1 } } , n = 1,2,3 , \ldots$. Let $S _ { 0 } = 0$ and, for $n \geq 1$, let $S _ { n }$ denote the sum of the first $n$ terms of this progression. For $n \geq 1$, let $C _ { n }$ denote the circle with center ( $S _ { n - 1 } , 0$ ) and radius $a _ { n }$, and $D _ { n }$ denote the circle with center ( $S _ { n - 1 } , S _ { n - 1 }$ ) and radius $a _ { n }$. Consider $M$ with $r = \frac { 1025 } { 513 }$. Let $k$ be the number of all those circles $C _ { n }$ that are inside $M$. Let $l$ be the maximum possible number of circles among these $k$ circles such that no two circles intersect. Then
(A) $k + 2 l = 22$
(B) $2 k + l = 26$
(C) $2 k + 3 l = 34$
(D) $3 k + 2 l = 40$

Let $$M = \left\{ ( x , y ) \in \mathbb { R } \times \mathbb { R } : x ^ { 2 } + y ^ { 2 } \leq r ^ { 2 } \right\} ,$$ where $r > 0$. Consider the geometric progression $a _ { n } = \frac { 1 } { 2 ^ { n - 1 } } , n = 1,2,3 , \ldots$. Let $S _ { 0 } = 0$ and, for $n \geq 1$, let $S _ { n }$ denote the sum of the first $n$ terms of this progression. For $n \geq 1$, let $C _ { n }$ denote the circle with center ( $S _ { n - 1 } , 0$ ) and radius $a _ { n }$, and $D _ { n }$ denote the circle with center ( $S _ { n - 1 } , S _ { n - 1 }$ ) and radius $a _ { n }$. Consider $M$ with $r = \frac { \left( 2 ^ { 199 } - 1 \right) \sqrt { 2 } } { 2 ^ { 198 } }$. The number of all those circles $D _ { n }$ that are inside $M$ is
(A) 198
(B) 199
(C) 200
(D) 201

Let $\alpha = \sum_{k=0}^{n} \frac{\binom{n}{k}^2}{k+1}$ and $\beta = \sum_{k=0}^{n-1} \frac{\binom{n}{k}\binom{n}{k+1}}{k+2}$. If $5\alpha = 6\beta$, then $n$ equals $\underline{\hspace{1cm}}$.