If a positive real $x$ satisfies the following equation $\log _2 x+\log _{\sqrt{2}} x=48$

then the value of $x$ is $\_\_\_\_$

Consider the following statements about four numbers:

(S1) The average of the four numbers is 25

(S2) Each number is at most 40

(S3) Each number is at least 20

Choose the option that is necessarily correct.

Two cars, $P$ and $Q$, start from a point $X$ in India at 10 AM . Car $P$ travels North with a speed of $25 \mathrm{~km} / \mathrm{h}$ and car Q travels East with a speed of $30 \mathrm{~km} / \mathrm{h}$. Car P travels continuously but car Q stops for some time after travelling for one hour. If both the cars are at the same distance from $X$ at 11:30 AM, for how long (in minutes) did car Q stop?

The ceiling function of a real number $x$, denoted by $\operatorname{ce}(x)$, is defined as the smallest integer that is greater than or equal to $x$. Similarly, the floor function, denoted by $f l(x)$, is defined as the largest integer that is smaller than or equal to $x$. Which one of the following statements is NOT correct for all possible values of $x$ ?