Let the functions $$f:( - 1,1) \to R$$ and $$g:( - 1,1) \to ( - 1,1)$$ be defined by $$f(x) = |2x - 1| + |2x + 1|$$ and $$g(x) = x - [x]$$, where [x] denotes the greatest integer less than or equal to x. Let $$f\,o\,g:( - 1,1) \to R$$ be the composite function defined by $$(f\,o\,g)(x) = f(g(x))$$. Suppose c is the number of points in the interval ($$-$$1, 1) at which $$f\,o\,g$$ is NOT continuous, and suppose d is the number of points in the interval ($$-$$1, 1) at which $$f\,o\,g$$ is NOT differentiable. Then the value of c + d is ............

The value of the limit

$$\mathop {\lim }\limits_{x \to {\pi \over 2}} {{4\sqrt 2 (\sin 3x + \sin x)} \over {\left( {2\sin 2x\sin {{3x} \over 2} + \cos {{5x} \over 2}} \right) - \left( {\sqrt 2 + \sqrt 2 \cos 2x + \cos {{3x} \over 2}} \right)}}$$

is ...........

Let b be a nonzero real number. Suppose f : R $$ \to $$ R is a differentiable function such that f(0) = 1. If the derivative f' of f satisfies the equation $$f'(x) = {{f(x)} \over {{b^2} + {x^2}}}$$

for all x$$ \in $$R, then which of the following statements is/are TRUE?

Let a and b be positive real numbers such that a > 1 and b < a. Let P be a point in the first quadrant that lies on the hyperbola $${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$. Suppose the tangent to the hyperbola at P passes through the point (1, 0), and suppose the normal to the hyperbola at P cuts off equal intercepts on the coordinate axes. Let $$\Delta $$ denote the area of the triangle formed by the tangent at P, the normal at P and the X-axis. If e denotes the eccentricity of the hyperbola, then which of the following statements is/are TRUE?