Match the following

Column $$I$$

(A) $$\sum\limits_{i = 1}^\infty {{{\tan }^{ - 1}}\left( {{1 \over {2{i^2}}}} \right) = t,} $$ then tan $$t=$$

(B) Sides $$a, b, c$$ of a triangle $$ABC$$ are in $$AP$$ and
$$\cos {\theta _1} = {a \over {b + c}},\,\cos {\theta _2} = {b \over {a + c}},\cos {\theta _3} = {c \over {a + b}},$$
then $${\tan ^2}\left( {{{{\theta _1}} \over 2}} \right) + {\tan ^2}\left( {{{{\theta _3}} \over 2}} \right) = $$

(C) A line is perpendicular to $$x + 2y + 2z = 0$$ and
passes through $$(0, 1, 0)$$. The perpendicular distance of this line from the origin is

Column $$II$$

(p) $$1$$

(q) $${{\sqrt 5 } \over 3}$$

(r) $${2 \over 3}$$

$$f(x)$$ is cubic polynomial with $$f(2)=18$$ and $$f(1)=-1$$. Also $$f(x)$$ has local maxima at $$x=-1$$ and $$f'(x)$$ has local minima at $$x=0$$, then

Let $$f\left( x \right) = \left\{ {\matrix{ {{e^x},} & {0 \le x \le 1} \cr {2 - {e^{x - 1}},} & {1 < x \le 2} \cr {x - e,} & {2 < x \le 3} \cr } } \right.$$ and $$g\left( x \right) = \int\limits_0^x {f\left( t \right)dt,x \in \left[ {1,3} \right]} $$
then $$g(x)$$ has

For a twice differentiable function $$f(x),g(x)$$ is defined as $$4\sqrt {65} g\left( x \right) = \left( {f'{{\left( x \right)}^2} + f''\left( x \right)} \right)\,\,f\left( x \right)$$ on $$\,\,\,\left[ {a,\,\,\,e} \right].$$ If for $$a < b < c < d < e,\,f\left( a \right) = 0,f\left( b \right) = 2,f\left( c \right) = - 1,f\left( d \right) = 2,f\left( e \right) = 0$$ then find the minimum number of zeros of $$g(x)$$.