If the tangent at point $(1, 2)$ on the curve $y = ax^2 + bx + \frac{1}{2}$ is parallel to the normal at $(-2, 2)$ on the curve $y = x^2 + 6x + 10$,then:

The angle between the lines represented by $y^2 \sin^2 \theta - xy \sin^2 \theta + x^2(\cos^2 \theta - 1) = 0$ is

The weight of a molecule of the compound $C_{60}H_{122}$ is

$A$ stunt man jumps his car over a crater as shown (neglect air resistance).

$A$ metal wire $PQ$ slides on parallel metallic rails having a separation of $0.25\ m$,each having negligible resistance. There is a $2\,\Omega$ resistor and a $10\ V$ battery as shown in the figure. $A$ uniform magnetic field of magnitude $0.5\ T$ is directed into the plane of the paper. $A$ force of $0.5\ N$ to the left is required to keep the wire $PQ$ moving with a constant speed to the right. With what speed is the wire $PQ$ moving? (Neglect self-inductance of the loop)

An ovule which becomes curved so that the nucellus and embryo sac lie at a right angle to the funicle is called: