$A$ certain vector in the $xy$-plane has an $x$-component of $4 ~m$ and a $y$-component of $10 ~m$. It is then rotated in the $xy$-plane so that its $x$-component is doubled. Then its new $y$-component is (approximately) (in $~m$)

If $P$ is the set of all real numbers $\alpha$ such that the product of the lengths of perpendiculars from $(\alpha, 1)$ to the pair of straight lines $3x^2+7xy+2y^2=0$ is $\frac{\sqrt{2}}{5}$,then the sum of the elements in $P$ is

The $9^{th}$ pair of cranial nerves in a frog is:

The basis of the quantum mechanical model of an atom is

$A$ body of mass $M \ kg$ is at the top point of a smooth hemisphere of radius $5 \ m$. It is released to slide down the surface of the hemisphere. It leaves the surface when its velocity is $5 \ m/s$. At this instant,the angle made by the radius vector of the body with the vertical is: (Acceleration due to gravity $g = 10 \ m/s^2$) (in $^{\circ}$)

The de-Broglie wavelength of a particle moving with a velocity of $2.25 \times 10^8 \ m/s$ is equal to the wavelength of a photon. The ratio of the kinetic energy of the particle to the kinetic energy of the photon is: (in $/8$)