Let $${J_{n,m}} = \int\limits_0^{{1 \over 2}} {{{{x^n}} \over {{x^m} - 1}}dx} $$, $$\forall$$ n > m and n, m $$\in$$ N. Consider a matrix $$A = {[{a_{ij}}]_{3 \times 3}}$$ where $${a_{ij}} = \left\{ {\matrix{ {{j_{6 + i,3}} - {j_{i + 3,3}},} & {i \le j} \cr {0,} & {i > j} \cr } } \right.$$. Then $$\left| {adj{A^{ - 1}}} \right|$$ is :

The function f(x), that satisfies the condition
$$f(x) = x + \int\limits_0^{\pi /2} {\sin x.\cos y\,f(y)\,dy} $$, is :

If [x] is the greatest integer $$\le$$ x, then

$${\pi ^2}\int\limits_0^2 {\left( {\sin {{\pi x} \over 2}} \right)(x - [x]} {)^{[x]}}dx$$ is equal to :

Let f be a non-negative function in [0, 1] and twice differentiable in (0, 1). If $$\int_0^x {\sqrt {1 - {{(f'(t))}^2}} dt = \int_0^x {f(t)dt} } $$, $$0 \le x \le 1$$ and f(0) = 0, then $$\mathop {\lim }\limits_{x \to 0} {1 \over {{x^2}}}\int_0^x {f(t)dt} $$ :