The process of caustification is used for the preparation of which of the following?

If the circles $x^2+y^2-2 \lambda x-2 y-7=0$ and $3(x^2+y^2)-8 x+29 y=0$ are orthogonal,then $\lambda$ is equal to

The apparent frequency of a sound wave as heard by an observer is $10\%$ more than the actual frequency. If the velocity of sound in air is $330 \, m/s$,then:
$(i)$ The source may be moving towards the observer with a velocity of $30 \, m/s$
$(ii)$ The source may be moving towards the observer with a velocity of $33 \, m/s$
$(iii)$ The observer may be moving towards the source with a velocity of $30 \, m/s$
$(iv)$ The observer may be moving towards the source with a velocity of $33 \, m/s$

$A$ mass of $1\, kg$ is suspended by a string $A$. Another string $C$ is connected to its lower end (see figure). If a sudden jerk is given to $C$,then

Ethyl chloride reacts with sodium ethoxide to form a compound $A$. Which of the following reactions also yields $A$?

$A$ variable circle passes through the fixed point $A(p, q)$ and touches the $x$-axis. The locus of the other end of the diameter through $A$ is