Let f1 - - f2 - -2- -2 - f3 - -1- e-2 - 2 - and f4 - - be functions defined byi f1x - sin -1 - e-x2ii f2x - -sin x-tan-1x if x- 0 1 if x - 0- where the inverse trigonometric function tan-1x assumes values in -2- -2 -iii f3x - -sin logex - 2- where for t - -t- denotes the greatest integer less than or equal to t-iv f4x - x2sin 1x if x- 0 0 if x - 0-List - IList - IIP- The function f1 is 1- NOT continuous at x - 0Q- The function f2 is 2- continuous at x - 0 and NOT differentiable at x - 0R- The function f3 is 3- differentiable at x - 0 and its derivative is NOT continuous at x - 0S- The function f4 is 4- diffferentiable at x - 0 and its derivative is continuous at x - 0The correct option is

LIST - I	LIST - II
P. The range of $f$ is	$1. (- \infty, \frac{1}{1 - e}] \cup [\frac{e}{e - 1}, \infty)$
Q. The range of $g$ contins	$2. (0, 1)$
R. The domain of $f$ contains	$3. [- \frac{1}{2}, \frac{1}{2}]$
S. The domain of $g$ is	$4. (- \infty, 0) \cup (0, \infty)$
	$5. (- \infty, \frac{e}{e - 1}]$
	$6. (- \infty, 0) \cup (\frac{1}{2}, \frac{e}{e - 1}]$

The correct option is

Q. Let

f_{1} : R \to R,

f_{2} : [0, \infty) \to R,

f_{3} : R \to R

and

f_{4} : R \to [0, \infty)

be defined by

f_{1} (x) = {\begin{matrix} | x | if x < 0, e^{x} if x \geq 0; \end{matrix}

f_{2} (x) = x^{2};

f_{3} (x) = {\begin{matrix} sin x if x < 0, x if x \geq 0; \end{matrix}

and

f_{4} (x) = {\begin{matrix} f_{2} (f_{1} (x)) if x < 0, f_{2} (f_{1} (x)) - 1 if x \geq 0. \end{matrix}

\begin{matrix} L i s t I & L i s t I I P. f_{4} is & 1. onto but not one-one Q. f_{3} is & 2. neither continuous nor one-one R. f_{2} \circ f_{1} is & 3. differentiable but not one-one S. f_{2} is & 4. continuous and not one-one \end{matrix}

which of the following option is correct ?

Q. Each of the following function is defined to be zero at x = 0

f_{1} (x) = x^{2} s g n (x)

f_{2} (x) = \int_{0}^{x} t^{2} s i n (\frac{1}{t}) d t

f_{3} (x) = x^{- \frac{1}{3}} [s i n x]

f_{4} (x) = x^{3} [- x]

[where sgn(x) denotes signum function and [.] denotes greatest integer function]

\begin{matrix} List - I & List - II I & The function f_{1} & (P) & continuous but not differentiable at x = 0 II & The function f_{2} is & (Q) & first derivative exists at x = 0 but second derivative does not exist III & The function f_{3} is & (R) & second derivative exists at x = 0, but it is not continuous there IV & The function f_{4} is & (S) & second derivative exists at x = 0 and is continuous at x = 0. \end{matrix}

Which of the following is the only CORRECT combination?

Let $f_{1} : R \to R; f_{2} : [0, \infty) \to f_{3} : R \to R$ and $f_{4} : R \to [0, \infty)$ be defined by
$f_{1} (x) = | x | i f x < 0 = e^{x} i f x \geq 0 f_{2} (x) = x^{2} f_{3} (x) = s i n x i f x < 0 = x i f x \geq 0$
and $f_{4} (x) = f_{2} (f_{1} (x)), i f x < 0 f_{2} (f_{1} (x)) - 1, i f x \geq 0$

$\begin{matrix} L i s t - I & L i s t - I I P . & f_{2} i s & 1. & Onto but not one-one Q . & f_{3} i s & 2. & Neither continuous nor one-one R . & f_{4} i s & 3. & Differentiable but not one-one S . & f_{2} o f_{1} i s & 4. & Continuous and one-one \end{matrix}$

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List - I	List - II
P. The function $f_{1}$ is	$1.$ NOT continuous at $x = 0$
Q. The function $f_{2}$ is	$2.$ continuous at $x = 0$ and NOT differentiable at $x = 0$
R. The function $f_{3}$ is	$3.$ differentiable at $x = 0$ and its derivative is NOT continuous at $x = 0$
S. The function $f_{4}$ is	$4.$ diffferentiable at $x = 0$ and its derivative is continuous at $x = 0$