For the following questions answer them individually
Let PQR be a triangle such that $$\overrightarrow{PQ}=-2\widehat{i}-\widehat{j}+2\widehat{k}$$ and $$\overrightarrow{PR}=a\widehat{i}+b\widehat{j}-4\widehat{k},a,b \in Z$$. Let S be the point on QR, which is equidistant from the lines PQ and PR. If $$|\overrightarrow{PR}|=9$$ and $$\overrightarrow{PS}=\widehat{i}-7\widehat{j}+2\widehat{k}$$, then the value of 3a - 4b is_______
For some $$\theta \in \left(0,\frac{\pi}{2}\right)$$, let the eccentricity and the length of the latus rectum of the hyperbola $$x^{2}-y^{2}\sec^{2}\theta =8$$ be $$e_{1}$$ and $$l_{1}$$,respectively, and let the eccentricity and the length of the latus rectum of the ellipse $$x^{2}\sec^{2}\theta +y^{2}=6$$ be $$e_{2}$$ and $$l_{2}$$.respectively. If $$e_{1}^{2}=e_{2}^{2}\left(\sec^{2}\theta +1\right)$$, then $$\left(\frac{l_{1}l_{2}}{e_{1}e_{2}}\right)\tan^{2}\theta$$ is equal to_____
In a G.P., if the product of the first three terms is 27 and the set of all possible values for the sum of its first three te1ms is R-(a,b), then $$a^{2}+b^{2}$$ is equal to______
The value of $$\sum_{r=1}^{20}\left(\left|\sqrt{\pi\left(\int_{0}^{r}x|\sin \pi x\right)}\right|\right)$$ is ______.
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 $$\sin^{-1}(kx-1)=\sin^{-1} x-\cos^{-1} x$$ 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 AB is_______$$rad/s^{2}$$.
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Two wires A and B made of different materials of lengths 6.0 cm and 5.4 cm, respectively and area of cross sections $$3.0\times 10^{-5}m^{2}\text{ and }4.5\times 10^{-5}m^{2}$$, respectively are stretched by the same magnitude under a given load. The ratio of the Young's modulus of A to that of B is x : 3. The value of x is ___ .
An atom $$^8_{3}X$$ is bombarded by shower of fundamental particles and in 10s this atom absorbed 10 electrons, 10 protons and 9 neutrons. The percentage growth in the smface area of the nucleons is recorded by:
The electric current in the circuit is given as $$i=i_{\circ}(t/T)$$. The r.m.s cm1:ent for the period t = 0 to t = T is______.
In the potentiometer, when the cell in the secondary circuit is shunted with 4Ω resistance, the balance is obtained at the length 120 cm of wire. Now when the same cell is shunted with 12Ω resistance, the balance is shifted to a length of 180 cm. The internal resistance of cell is_________Ω
