Match the corresponding entries of Column-$I$ with Column-$II$. [Where $m$ is the magnification produced by the mirror]

Column-$I$	Column-$II$
$1. m = -2$	$a$. Convex mirror
$2. m = -\frac{1}{2}$	$b$. Concave mirror
$3. m = +2$	$c$. Real Image
$4. m = +\frac{1}{2}$	$d$. Virtual image

The most abundant prokaryotes helpful to humans in making curd from milk and in the production of antibiotics are the ones categorized as:

$A$ line segment $l$ of length $a \ cm$ is rotated about a vertical line $L$ keeping the line $l$ in one of the following three positions: $(I)$ $l$ is parallel to $L$ and is at a distance of $r \ cm$ from $L$,$(II)$ $l$ is perpendicular to $L$ and its mid-point is at a distance $r \ cm$ from $L$,$(III)$ $l$ and $L$ are in the same plane and $l$ is inclined to $L$ at an angle $30^{\circ}$ with its mid-point at a distance $r \ cm$ from $L$. Let $A_1, A_2, A_3$ be the areas so generated. If $r > (a / 2)$,then

In humans,which disorder is caused due to the absence of one of the $X$ chromosomes,i.e.,$45$ chromosomes with $XO$ genotype?

If $\alpha, \beta$ and $\gamma$ are the lengths of the altitudes of a $\triangle ABC$ with area $\Delta$,then $\frac{\Delta^2}{R^2}\left(\frac{1}{\alpha^2}+\frac{1}{\beta^2}+\frac{1}{\gamma^2}\right)$ is equal to

If the distance between the foci of an ellipse is $6$ and the length of the minor axis is $8$,then the eccentricity is