In a radioactive decay chain, $${}_{90}^{232}Th$$ nucleus decays to $${}_{82}^{212}Pb$$ nucleus. Let $${N_\alpha }$$ and $${N_\beta }$$ be the number of $$\alpha $$ and $${\beta ^ - }$$ particles, respectively, emitted in this decay process. Which of the following statements is (are) true?

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.)