$A$ solid sphere and a disc of same mass $M$ and radius $R$ are kept such that their curved surfaces are in contact and their centers lie along the same horizontal line. The moment of inertia of the two-body system about an axis passing through their point of contact and perpendicular to the plane of the disc is

The orbital angular momentum of a satellite revolving at a distance $r$ from the centre is $L$. If the distance is increased to $4r$,then the new angular momentum will be:

If $x^2+\alpha y^2+2 \beta y=a^2$ represents a pair of perpendicular lines,then $\beta$ equals to

$A$ galvanometer,having a resistance of $50 \Omega$,gives a full-scale deflection for a current of $0.05 \text{ A}$. The length in metres of a resistance wire of cross-sectional area $2.97 \times 10^{-2} \text{ cm}^2$ that can be used to convert the galvanometer into an ammeter which can read a maximum of $5 \text{ A}$ current is: (Specific resistance of the wire $= 5 \times 10^{-7} \Omega\text{-m}$)

The transformed equation of $x^2+y^2=r^2$ when the axes are rotated through an angle $36^{\circ}$ is

$\int_0^3 \frac{3x+1}{x^2+9} dx$ is equal to :