Which of the following is not correct for relation R on the set of real numbers ?

If f : R $$\to$$ R is a function defined by f(x)= [x - 1] \cos \left( {{{2x - 1} \over 2}} \right)\pi, where [.] denotes the greatest integer function, then f is :

\mathop {\lim }\limits_{n \to \infty } {\left( {1 + {{1 + {1 \over 2} + ........ + {1 \over n}} \over {{n^2}}}} \right)^n} is equal to :

The value of \mathop {\lim }\limits_{h \to 0} 2\left\{ {{{\sqrt 3 \sin \left( {{\pi \over 6} + h} \right) - \cos \left( {{\pi \over 6} + h} \right)} \over {\sqrt 3 h\left( {\sqrt 3 \cosh - \sinh } \right)}}} \right\} is :

Let f(x) be a differentiable function at x = a with f'(a) = 2 and f(a) = 4. Then \mathop {\lim }\limits_{x \to a} {{xf(a) - af(x)} \over {x - a}} equals :

Let $$f(x)={\sin }^{-1}x$$ and g(x) = {{{x^2} - x - 2} \over {2{x^2} - x - 6}}. If $$g(2)=\underset{x\to 2}{\lim }g(x)$$, then the domain of the function fog is :

Let f : R $$\to$$ R be defined as f(x) = \left\{ \matrix{ 2\sin \left( { - {{\pi x} \over 2}} \right),if\,x 1 \hfill \cr} \right. If f(x) is continuous on R, then a + b equals :

Let {S_k} = \sum\limits_{r = 1}^k {{{\tan }^{ - 1}}\left( {{{{6^r}} \over {{2^{2r + 1}} + {3^{2r + 1}}}}} \right)}. Then $$\underset{k\to \infty }{\lim }{S}_k$$ is equal to :

Let the functions f : R $$\to$$ R and g : R $$\to$$ R be defined as : f(x) = \left\{ {\matrix{ {x + 2,} & {x < 0} \cr {{x^2},} & {x \ge 0} \cr } } \right. and g(x) = \left\{ {\matrix{ {{x^3},} & {x < 1} \cr {3x - 2,} & {x \ge 1} \cr } } \right. Then, the number of points in R where (fog) (x) is NOT differentiable is equal to :

Let f : S $$\to$$ S where S = (0, $$\infty$$) be a twice differentiable function such that f(x + 1) = xf(x). If g : S $$\to$$ R be defined as g(x) = loge f(x), then the value of |g''(5) $$-$$ g''(1)| is equal to :

Let $$\alpha$$ $$\in$$ R be such that the function f(x) = \left\{ {\matrix{ {{{{{\cos }^{ - 1}}(1 - {{\{ x\} }^2}){{\sin }^{ - 1}}(1 - \{ x\} )} \over {\{ x\} - {{\{ x\} }^3}}},} & {x \ne 0} \cr {\alpha ,} & {x = 0} \cr } } \right. is continuous at x = 0, where {x} = x $$-$$ [ x ] is the greatest integer less than or equal to x. Then :

The value of \mathop {\lim }\limits_{x \to {0^ + }} {{{{\cos }^{ - 1}}(x - {{[x]}^2}).{{\sin }^{ - 1}}(x - {{[x]}^2})} \over {x - {x^3}}}, where [ x ] denotes the greatest integer $$\le$$ x is :

The value of the limit \mathop {\lim }\limits_{\theta \to 0} {{\tan (\pi {{\cos }^2}\theta )} \over {\sin (2\pi {{\sin }^2}\theta )}} is equal to :

The value of \mathop {\lim }\limits_{n \to \infty } {{[r] + [2r] + ... + [nr]} \over {{n^2}}}, where r is a non-zero real number and [r] denotes the greatest integer less than or equal to r, is equal to :

If \mathop {\lim }\limits_{x \to 0} {{{{\sin }^{ - 1}}x - {{\tan }^{ - 1}}x} \over {3{x^3}}} is equal to L, then the value of (6L + 1) is

If f(x) = \left\{ {\matrix{ {{1 \over {|x|}}} & {;\,|x|\, \ge 1} \cr {a{x^2} + b} & {;\,|x|\, < 1} \cr } } \right. is differentiable at every point of the domain, then the values of a and b are respectively :

Let f : R $$\to$$ R be a function defined as f(x) = \left\{ \matrix{ {{\sin (a + 1)x + \sin 2x} \over {2x}},if\,x 0 \hfill \cr} \right. If f is continuous at x = 0, then the value of a + b is equal to :

Let a function f : R $$\to$$ R be defined as f(x) = \left\{ {\matrix{ {\sin x - {e^x}} & {if} & {x \le 0} \cr {a + [ - x]} & {if} & {0 < x < 1} \cr {2x - b} & {if} & {x \ge 1} \cr } } \right. where [ x ] is the greatest integer less than or equal to x. If f is continuous on R, then (a + b) is equal to:

If $$f:R\to R$$ is given by $$f(x)=x+1$$, then the value of \mathop {\lim }\limits_{n \to \infty } {1 \over n}\left[ {f(0) + f\left( {{5 \over n}} \right) + f\left( {{{10} \over n}} \right) + ...... + f\left( {{{5(n - 1)} \over n}} \right)} \right] is :

Let f : R $$\to$$ R be defined as f(x) = \left\{ {\matrix{ {{{{x^3}} \over {{{(1 - \cos 2x)}^2}}}{{\log }_e}\left( {{{1 + 2x{e^{ - 2x}}} \over {{{(1 - x{e^{ - x}})}^2}}}} \right),} & {x \ne 0} \cr {\alpha ,} & {x = 0} \cr } } \right. If f is continuous at x = 0, then $$\alpha$$ is equal to :