$A$ bar magnet of length $3 \, cm$ has points $A$ and $B$ along its axis at distances of $24 \, cm$ and $48 \, cm$ from the center on opposite sides. The ratio of the magnetic fields at these points is:

Consider the following statements regarding the major pigments and stored food in the different groups of algae and select the correct option.
$I.$ In Chlorophyceae, the stored food material is starch and the major pigments are chlorophyll-$a$ and chlorophyll-$b$.
$II.$ In Phaeophyceae, laminarin is the stored food and major pigments are chlorophyll-$a$ and chlorophyll-$c$.
$III.$ In Rhodophyceae, floridean starch is the stored food and major pigments are chlorophyll-$a$, $d$ and phycoerythrin.

The voltage of an $ac$ source varies with time according to the equation $V = 100 \sin(100 \pi t) \cos(100 \pi t)$,where $t$ is in seconds and $V$ is in volts. Then:

The oxoanion which contains all equivalent $M-O$ bonds is:
$(I)$ $CrO_4^{2-}$
$(II)$ $MnO_4^{-}$
$(III)$ $Cr_2O_7^{2-}$

The curve satisfying the differential equation,$ydx - (x + 3y^2) dy = 0$ and passing through the point $(1, 1)$,also passes through the point

If $f: R \rightarrow R$ is defined by $f(x) = [x-3] + |x-4|$ for $x \in R$,then $\lim_{x \rightarrow 3^{-}} f(x)$ is equal to