Which of the following statements is correct?

Let $PQ$ and $RS$ be tangents at the extremities of the diameter $PR$ of a circle of radius $r$. If $PS$ and $RQ$ intersect at a point $X$ on the circumference of the circle,then $2r$ equals

$A$ block placed on a rough inclined plane of inclination $\theta=30^{\circ}$ can just be pushed upwards by applying a force $F$ as shown. If the angle of inclination of the inclined plane is increased to $\theta=60^{\circ}$,the same block can just be prevented from sliding down by application of a force of the same magnitude $F$. The coefficient of friction between the block and the inclined plane is:

$\int \frac{3^x dx}{\sqrt{9^x-1}}$ is equal to

Three rods of equal lengths are joined to form an equilateral triangle $ABC$. $D$ is the mid-point of $AB$. The coefficient of linear expansion is $\alpha_1$ for the material of rod $AB$ and $\alpha_2$ for the material of rods $AC$ and $BC$. If the distance $DC$ remains constant for small changes in temperature,then

$A$ parallel plate capacitor is charged to a potential difference of $100\ V$ and disconnected from the source of emf. $A$ slab of dielectric is then inserted between the plates. Which of the following three quantities change?
$(i)$ The potential difference
$(ii)$ The capacitance
$(iii)$ The charge on the plates