A solid horizontal surface is covered with a thin layer of oil. A rectangular block of mass $$m=0.4$$ $$kg$$ is at rest on this surface. An impulse of $$1.0$$ $$Ns$$ is applied to the block at time $$t=0$$ so that it starts moving along the $$x$$-axis with a velocity $$v\left( t \right) = {v_0}{e^{ - t/\tau }},$$ where $${v_0}$$ is a constant and $$\tau = 4s.$$ The displacement of the block, in metres, at $$t = \tau $$ is ______________ Take $${e^{ - 1}} = 0.37.$$

In an experiment to measure the speed of sound by a resonating air column, a tuning fork of frequency $$500$$ $$Hz$$ is used. The length of the air column is varied by changing the level of water in the resonance tube. Two successive resonances are heard at air columns of length $$50.7$$ $$cm$$ and $$83.9$$ $$cm.$$ Which of the following statements is (are) true?

A wire is bent in the shape of a right angled triangle and is placed in front of a concave mirror of focal length $$f,$$ as shown in the figure. Which of the figures shown in the four options qualitatively represent(s) the shape of the image of the bent wire? (These figures are not to scale.)

A moving coil galvanometer has $$50$$ turns and each turn has an area $$2 \times {10^{ - 4}}\,{m^2}.$$ The magnetic field produced by the magnet inside the galvanometer is $$0.02$$ $$T.$$ The torsional constant of the suspension wire is $${10^{ - 4}}\,N\,m\,ra{d^{ - 1}}.$$ When a current flows through the galvanometer, a full scale deflection occurs if the coil rotates by $$0.2$$ $$rad$$. The resistance of the coil of the galvanometer is $$50\Omega .$$ This galvanometer is to be converted into an ammeter capable of measuring current in the range $$0-1.0$$ $$A$$. For this purpose, a shunt resistance is to be added in parallel to the galvanometer. The value of this shunt resistance, in ohms, is _____________.