If $$E$$ denotes expectation, the variance of a random variable $$X$$ is given by

An examination consists of two papers, paper $$1$$ and paper $$2.$$ The probability of failing in paper $$1$$ is $$0.3$$ and that in paper $$2$$ is $$0.2.$$ Given that a student has failed in paper $$2,$$ the probability of failing in paper $$1$$ is $$0.6.$$ The probability of a student failing in both the papers is

The following plot shows a function $$y$$ which varies linearly with $$x$$. The value of the integral $$\,\,{\rm I} = \int\limits_1^2 {y\,dx\,\,} $$

$$\mathop {Lim}\limits_{\theta \to 0} {{\sin \left( {\theta /2} \right)} \over \theta }\,\,\,$$ is