Embibe Experts Solutions for Chapter: Sequences and Series, Exercise 1: JEE Advanced Paper 1 - 2018

Let $f\left(x\right)=\sin \left(\pi \cos x\right)$ and $g\left(x\right)=\cos \left(2\pi \sin x\right)$ be two functions defined for $x>0.$ Define the following sets whose elements are written in the increasing order:
$X=\left\{x:f\left(x\right)=0\right\}, Y=\left\{x:f^{\text{'}}\left(x\right)=0\right\}$
$Z=\left\{x:g\left(x\right)=0\right\}, W=\left\{x:g^{\text{'}}\left(x\right)=0\right\}$
List- $\mathrm{I}$ contains the sets $X, Y, Z$ and $W.$ List- $\mathrm{I}\mathrm{I}$ contains some information regarding these sets.
 

Let $f\left(x\right)=\sin \left(\pi \cos x\right)$ and $g\left(x\right)=\cos \left(2\pi \sin x\right)$ be two functions defined for $x>0.$ Define the following sets whose elements are written in the increasing order:
$X=\left\{x:f\left(x\right)=0\right\}, Y=\left\{x:f^{\text{'}}\left(x\right)=0\right\}$
$Z=\left\{x:g\left(x\right)=0\right\}, W=\left\{x:g^{\text{'}}\left(x\right)=0\right\}$
List- $I$ contains the sets $X, Y, Z$ and $W.$ List- $\mathrm{I}\mathrm{I}$ contains some information regarding these sets.
 

For any positive integer $n$, let $S_n:(0,\infty )\to R$ be defined by
$S_n\left(x\right)=\sum _{k=1}^n{\cot }^{-1}\left(\frac{1+k(k+1)x^2}{x}\right)$
where for any $x\in R,{\cot }^{-1}\left(x\right)\in (0,\pi )$ and ${\tan }^{-1}\left(x\right)\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$. Then which of the following statements is (are) TRUE ?

Let $m$ be the minimum possible value of ${\log }_3\left(3^{y_1}+3^{y_2}+3^{y_3}\right)$ , where $y_1, y_2, y_3$ are real numbers for which $y_1+y_2+y_3=9$. Let $M$ be the maximum possible value of $\left({\log }_3{x}_1+{\log }_3 x_2+{\log }_3 x_3\right)$, where $x_1, x_2, x_3$ are positive real numbers for which $x_1+x_2+x_3=9$. Then the value of ${\log }_2 \left(m^3\right)+{\log }_3\left(M^2\right)$ is

Let $a_1, a_2, a_3, .........$ be a sequence of positive integers in arithmetic progression with common difference $2$. Also, let $b_1, b_2, b_3, ........$ be a sequence of positive integers in geometric progression with common ratio $2$. If $a_1=b_1=c$, then the number of all possible values of $c$, for which the equality $2 \left(a_1+a_2+........+a_n\right)=b_1+b_2+........+b_n$ holds for some positive integer $n$, is _______

Let $AP \left(a;d\right)$ denote the set of all the terms of an infinite arithmetic progression with first term $a$ and common difference $d>0.$ If $AP\left(1;3\right)\cap AP\left(2;5\right)\cap AP\left(3;7\right)=AP\left(a;d\right)$ then $a+d$ equals ____

Let $X$ be the set consisting of the first $2018$ terms of the arithmetic progression $\mathrm{1,6},11,\dots ,$ and $Y$ be the set consisting of the first $2018$ terms of the arithmetic progression $\mathrm{9,16,23},\dots$ . Then, the number of elements in the set $X\cup Y$ is___.

The number of real solutions of the equation ${\sin }^{-1}\left(\sum _{i=1}^{\infty }{x}^{i+1}-x\sum _{i=1}^{\infty }{\left(\frac{x}{2}\right)}^i\right)=\frac{\pi }{2}-{\cos }^{-1}\left(\sum _{i=1}^{\infty }{\left(-\frac{x}{2}\right)}^i-\sum _{i=1}^{\infty }{\left(-x\right)}^i\right)$ lying in the interval $\left(-\frac{1}{2},\frac{1}{2}\right)$ is____.

(Here, the inverse trigonometric functions ${\sin }^{-1}x \& {\cos }^{-1}x$ assume values in $\left[-\frac{\pi }{2},\frac{\pi }{2}\right] \& \left[0,\pi \right]$

respectively.)