If $k=\tan \left(\frac{\pi}{4}+\frac{1}{2} \cos ^{-1}\left(\frac{2}{3}\right)\right)+\tan \left(\frac{1}{2} \sin ^{-1}\left(\frac{2}{3}\right)\right)$, then

the number of solutions of the equation $\sin ^{-1}(k x-1)=\sin ^{-1} x-\cos ^{-1} x$ is $\_\_\_\_$.

$$ \text { The value of } \sum\limits_{r=1}^{20}\left(\left|\sqrt{\pi\left(\int\limits_0^r x|\sin \pi x| d x\right)}\right|\right) \text { is } $$

Let $P Q R$ be a triangle such that $\overrightarrow{P Q}=-2 \hat{i}-\hat{j}+2 \hat{k}$ and $\overrightarrow{\mathrm{PR}}=a \hat{\mathrm{i}}+b \hat{\mathrm{j}}-4 \hat{\mathrm{k}}, a, b \in \mathrm{Z}$. Let S be the point on QR , which is equidistant from the lines PQ and PR . If $|\overrightarrow{\mathrm{PR}}|=9$ and $\overrightarrow{\mathrm{PS}}=\hat{\mathrm{i}}-7 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}$, then the value of $3 a-4 b$ is $\_\_\_\_$

Two circular discs of radius each 10 cm are joined at their centres by a rod of length 30 cm and mass 600 gm as shown in figure.

If the mass of each disc is 600 gm and applied torque between two discs is $43 \times 10^5$ dyne.cm, the angular acceleration of the discs about the given axis $A B$ is $\_\_\_\_$ $\mathrm{rad} / \mathrm{s}^2$.