An elaborate network of filamentous proteinaceous structures present in the cytoplasm which helps in the maintenance of cell shape is called

The value of '$a$' for which the three given planes
$P_1: (a+1)x - y - z = a$
$P_2: x - (a+1)y - z = a$
$P_3: x - y - (a+1)z = a$
have no point in common,is

$A$ straight line passing through the origin $O$ meets the parallel lines $4x + 2y = 9$ and $2x + y + 6 = 0$ at points $P$ and $Q$ respectively. In what ratio does the point $O$ divide the line segment $PQ$?

The particles emitted during the sequential radioactive decay of ${}^{238}U_{92}$ to ${}^{206}Pb_{82}$ are

$A$ narrow parallel beam of light is incident paraxially on a solid transparent sphere of radius $r$ kept in air. What should be the refractive index if the beam is to be focused at the farther surface of the sphere?

The radius of the circle with the polar equation $r^2-8r(\sqrt{3}\cos\theta+\sin\theta)+15=0$ is