Two fair dice, each with faces numbered 1, 2, 3, 4, 5 and 6, are rolled together and the sum of the numbers on the faces is observed. This process is repeated till the sum is either a prime number or a perfect square. Suppose the sum turns out to be a perfect square before it turns out to be a prime number. If p is the probability that this perfect square is an odd number, then the value of 14p is ..........

Let the function f : [0, 1] $$ \to $$ R be defined by

$$f(x) = {{{4^x}} \over {{4^x} + 2}}$$

Then the value of $$f\left( {{1 \over {40}}} \right) + f\left( {{2 \over {40}}} \right) + f\left( {{3 \over {40}}} \right) + ... + f\left( {{{39} \over {40}}} \right) - f\left( {{1 \over 2}} \right)$$ is ..........

Let $$f:R \to R$$ be a differentiable function such that its derivative f' is continuous and f($$\pi $$) = $$-$$6.

If $$F:[0,\pi ] \to R$$ is defined by $$F(x) = \int_0^x {f(t)dt} $$, and if $$\int_0^\pi {(f'(x)} + F(x))\cos x\,dx$$ = 2

then the value of f(0) is ...........

Let the function $$f:(0,\pi ) \to R$$ be defined by $$f(\theta ) = {(\sin \theta + \cos \theta )^2} + {(\sin \theta - \cos \theta )^4}$$

Suppose the function f has a local minimum at $$\theta $$ precisely when $$\theta \in \{ {\lambda _1}\pi ,....,{\lambda _r}\pi \} $$, where $$0 < {\lambda _1} < ...{\lambda _r} < 1$$. Then the value of $${\lambda _1} + ... + {\lambda _r}$$ is .............