Model Paper 3 — Maths STD 11 Science — Question

$M =\left\{( x , y ) \in R \times R : x ^2+ y ^2 \leq r ^2\right\},$

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 $\left(S_{n-1}, 0\right)$ and radius $a _{ n }$, and $D _{ n }$ denote the circle with center $\left( S _{ n -1}, S _{ n -1}\right)$ and radius $a _{ n }$.

($1$) 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

$(A)$ $k +2 l=22$ $(B)$ $2 k +l=26$ $(C)$ $2 k +3 l=34$ $(D)$ $3 k +2 l=40$

($2$) Consider $M$ with $r =\frac{\left(2^{199}-1\right) \sqrt{2}}{2^{158}}$. The number of all those circles $D _{ a }$ that are inside $M$ is

$(A) 198$ $(B) 199$ $(C) 200$ $(D) 201$

Give the answer or qution ($1$) and ($2$)

$(A)$ $a_x=1 ms ^{-2}$ implies that when the particle is at the origin, $a_y=1 ms ^{-2}$

$(B)$ $a_x=0$ implies $a_y=1 ms ^{-2}$ at all times

$(C)$ at $t=0$, the particle's velocity points in the $x$-direction

$(D)$ $a_x=0$ implies that at $t=1 s$, the angle between the particle's velocity and the $x$ axis is $45^{\circ}$