When 16.8 g of white solid X were heated, 4.4 g of acid gas A, that turned lime water milky was driven off together with 1.8 g of a gas B which condensed to a colourless liquid.

The solid that remained, Y, dissolved in water to give an alkaline solution, which with excess barium chloride solution gave a white precipitate Z. The precipitate efferversced with acid giving off carbon dioxide. Identify A, B and Y and write down the equation for the thermal decomposition of X.

Find the area of the region bounded by the $$x$$-axis and the curves defined by $$$y = \tan x, - {\pi \over 3} \le x \le {\pi \over 3};\,\,y = \cot x,{\pi \over 6} \le x \le {{3\pi } \over 2}$$$

Given a function $$f(x)$$ such that
(i) it is integrable over every interval on the real line and
(ii) $$f(t+x)=f(x),$$ for every $$x$$ and a real $$t$$, then show that
the integral $$\int\limits_a^{a + 1} {f\,\,\left( x \right)} \,dx$$ is independent of a.

If $$M$$ and $$N$$ are any two events, the probability that exactly one of them occurs is