Compound $A$ $(C_3H_6O)$ undergoes the following reactions to form $B$ and $C$. Identify $A$,$B$,and $C$.

If $p \Rightarrow (q \vee r)$ is false,then the truth values of $p, q, r$ are respectively:

Let $C$ be the centroid of the triangle with vertices $(3, -1)$,$(1, 3)$,and $(2, 4)$. Let $P$ be the point of intersection of the lines $x + 3y - 1 = 0$ and $3x - y + 1 = 0$. Then the line passing through the points $C$ and $P$ also passes through the point

The enolic form of acetone contains:

Three rods $A, B,$ and $C$ of thermal conductivities $K, 2K,$ and $4K$ and equal cross-sectional areas and lengths $2l, l,$ and $l$ respectively are connected as shown in the figure. If the ends of the rods are maintained at temperatures $100\ ^oC, 50\ ^oC,$ and $0\ ^oC$ respectively,then the temperature $\theta$ of the junction is

$\log _4 2 - \log _8 2 + \log _{16} 2 - \dots$ is equal to