$Let f_{n} (x) = \frac{1}{x^{n}}, for all x ε R - {0}, where n ε N and g (x) = \max . \{f_{2} (x), f_{3} (x), \frac{1}{2}\}, for$ $all x ε R - {0} . Then match the following$
COLUMN –I
COLUMN -II
A) The possible value (s) of $\sum_{k = 1}^{\infty} f_{2 k} (\cos e c θ) + \sum_{k = 1}^{\infty} f_{2 k} (\sec θ)$ where $θ \neq \frac{k π}{2}, k \in N$ is/are P) 1
B) The value (s) of n for which $f_{n} (x) S g n (x)$ is even is/are Q) 2
C) Number of values of x for which y = g ( $X$ ) , for all $x \in [\frac{1}{2}, 2]$ is non differentiable is/are R) 3
D) The values of two consecutive integers between which $\int_{\frac{1}{2}}^{2} g (x) d x$ lies are S) 4

Difficult

Solution

$a) G \cdot E = (\sin^{2} θ + \sin^{4} θ + \sin^{6} θ + . . . .) + (\cos^{2} θ + \cos^{4} θ + \cos^{6} θ + . . . .) = \frac{\sin^{2} θ}{1 - \sin^{2} θ} + \frac{\cos^{2} θ}{1 - \cos^{2} θ} = \tan^{2} θ + \cot^{2} θ \geq 2 (\sin c e A . M \geq G . M)$
$b) if n is odd t a k e n = 1 \Rightarrow f_{1} (x) S g n (x) = \frac{1}{x} S g n (x) = \frac{1}{x} i f x > 0$
$= \frac{- 1}{x} i f x < 0$ , $w h i c h i s a n e v e n f u n c t i o n .$
$(c) g (x) = \{\begin{cases} \frac{1}{x^{3}}, \frac{1}{2} \leq x \leq 1 \\ \frac{1}{x^{2}}, 1 \leq x \leq \sqrt{2} \\ \frac{1}{2}, \sqrt{2} \leq x \leq 2 \end{cases}$
$g (x) i s n o t d i f f e r e n t i a b l e a t x = 1, \sqrt{2}$
$d) \int_{\frac{1}{2}}^{2} g (x) d x = \int_{\frac{1}{2}}^{1} \frac{1}{x^{3}} d x + \int_{1}^{\sqrt{2}} \frac{1}{x^{2}} d x + \int_{\sqrt{2}}^{2} \frac{1}{2} d x = \frac{7}{2} - \sqrt{2} lies between 2, 3$

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COLUMN –I	COLUMN -II
A) The possible value (s) of $\sum_{k = 1}^{\infty} f_{2 k} (\cos e c θ) + \sum_{k = 1}^{\infty} f_{2 k} (\sec θ)$ where $θ \neq \frac{k π}{2}, k \in N$ is/are	P) 1
B) The value (s) of n for which $f_{n} (x) S g n (x)$ is even is/are	Q) 2
C) Number of values of x for which y = g ( $X$ ) , for all $x \in [\frac{1}{2}, 2]$ is non differentiable is/are	R) 3
D) The values of two consecutive integers between which $\int_{\frac{1}{2}}^{2} g (x) d x$ lies are	S) 4