Let $x_0$ be the real number such that $e^{x_0} + x_0 = 0$. For a given real number $\alpha$, define

$g(x) = \frac{3x e^x + 3x - \alpha e^x - \alpha x}{3(e^x + 1)}$

for all real numbers $x$.

Then which one of the following statements is TRUE?

Let $\mathbb{R}$ denote the set of all real numbers. Define the function $f : \mathbb{R} \to \mathbb{R}$ by

$f(x)=\left\{\begin{array}{cc}2-2 x^2-x^2 \sin \frac{1}{x} & \text { if } x \neq 0, \\ 2 & \text { if } x=0 .\end{array}\right.$

Then which one of the following statements is TRUE?

Let $\mathbb{R}$ denote the set of all real numbers. For a real number $x$, let [ x ] denote the greatest integer less than or equal to $x$. Let $n$ denote a natural number.

Match each entry in List-I to the correct entry in List-II and choose the correct option.

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function defined by

$ f(x)=\left\{\begin{array}{cc} x^2 \sin \left(\frac{\pi}{x^2}\right), & \text { if } x \neq 0, \\ 0, & \text { if } x=0 . \end{array}\right. $

Then which of the following statements is TRUE?

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ be functions defined by

$ f(x)=\left\{\begin{array}{ll} x|x| \sin \left(\frac{1}{x}\right), & x \neq 0, \\ 0, & x=0, \end{array} \quad \text { and } g(x)= \begin{cases}1-2 x, & 0 \leq x \leq \frac{1}{2}, \\ 0, & \text { otherwise } .\end{cases}\right. $

Let $a, b, c, d \in \mathbb{R}$. Define the function $h: \mathbb{R} \rightarrow \mathbb{R}$ by

$ h(x)=a f(x)+b\left(g(x)+g\left(\frac{1}{2}-x\right)\right)+c(x-g(x))+d g(x), x \in \mathbb{R} . $

Match each entry in List-I to the correct entry in List-II.

Let $S$ be the set of all $(\alpha, \beta) \in \mathbb{R} \times \mathbb{R}$ such that

$ \lim\limits_{x \rightarrow \infty} \frac{\sin \left(x^2\right)\left(\log _e x\right)^\alpha \sin \left(\frac{1}{x^2}\right)}{x^{\alpha \beta}\left(\log _e(1+x)\right)^\beta}=0 . $

Then which of the following is (are) correct?

Let a, b $\in$ R and f : R $\to$ R be defined by $f(x) = a\cos (|{x^3} - x|) + b|x|\sin (|{x^3} + x|)$. Then f is

Let $f:\left[ { - {1 \over 2},2} \right] \to R$ and $g:\left[ { - {1 \over 2},2} \right] \to R$ be function defined by $f(x) = [{x^2} - 3]$ and $g(x) = |x|f(x) + |4x - 7|f(x)$, where [y] denotes the greatest integer less than or equal to y for $y \in R$. Then

Let $g:R \to R$ be a differentiable function with $g(0) = 0$, $g'(0) = 0$ and $g'(1) \ne 0$. Let

$f(x) = \left\{ {\matrix{ {{x \over {|x|}}g(x),} & {x \ne 0} \cr {0,} & {x = 0} \cr } } \right.$

and $h(x) = {e^{|x|}}$ for all $x \in R$. Let $(f\, \circ \,h)(x)$ denote $f(h(x))$ and $(h\, \circ \,f)(x)$ denote $f(f(x))$. Then which of the following is (are) true?

$a \in R$ (the set of all real numbers), a $\ne$ $-$1,

$\mathop {\lim }\limits_{n \to \infty } {{({1^a} + {2^a} + ... + {n^a})} \over {{{(n + 1)}^{a - 1}}[(na + 1) + (na + 2) + ... + (na + n)]}} = {1 \over {60}}$, Then a = ?

If $\mathop {\lim }\limits_{x \to \infty } \left( {{{{x^2} + x + 1} \over {x + 1}} - ax - b} \right) = 4$, then

Let $f(x) = \left\{ {\matrix{ {{x^2}\left| {\cos {\pi \over x}} \right|,} & {x \ne 0} \cr {0,} & {x = 0} \cr } } \right.$

x$\in$R, then f is

For every integer n, let a_n and b_n be real numbers. Let function f : R $\to$ R be given by

$f(x) = \left\{ {\matrix{ {{a_n} + \sin \pi x,} & {for\,x \in [2n,2n + 1]} \cr {{b_n} + \cos \pi x,} & {for\,x \in (2n - 1,2n)} \cr } } \right.$, for all integers n. If f is continuous, then which of the following hold(s) for all n ?

If $\mathop {\lim }\limits_{x \to 0} {[1 + x\ln (1 + {b^2})]^{1/x}} = 2b{\sin ^2}\theta $, $b > 0$ and $\theta \in ( - \pi ,\pi ]$, then the value of $\theta$ is

Let f : R $\to$ R be a function such that $f(x + y) = f(x) + f(y),\,\forall x,y \in R$. If f(x) is differentiable at x = 0, then

If $f(x) = \left\{ {\matrix{ { - x - {\pi \over 2},} & {x \le - {\pi \over 2}} \cr { - \cos x} & { - {\pi \over 2} < x \le 0} \cr {x - 1} & {0 < x \le 1} \cr {\ln x} & {x > 1} \cr } } \right.$, then

Let $L = \mathop {\lim }\limits_{x \to 0} {{a - \sqrt {{a^2} - {x^2}} - {{{x^2}} \over 4}} \over {{x^4}}},a > 0$. If L is finite, then

Which of the following is true?

Let $g(x) = {{{{(x - 1)}^n}} \over {\log {{\cos }^m}(x - 1)}};0 < x < 2,m$ and $n$ are integers, $m \ne 0,n > 0$, and let $p$ be the left hand derivative of $|x - 1|$ at $x = 1$. If $\mathop {\lim }\limits_{x \to {1^ + }} g(x) = p$, then

Let $f(x)=2+\cos x$ for all real $x$.

STATEMENT - 1 : For each real $t$, there exists a point $c$ in $[t, t+\pi]$ such that $f^{\prime}(C)=0$.

STATEMENT - 2 : $f(t)=f(t+2 \pi)$ for each real $t$.

The line $y=x$ meets $y=k e^{\mathrm{x}}$ for $k \leq 0$ at

The positive value of $k$ for which $k e^{x}-x=0$ has only one root is

For $k > 0$, the set of all values of $k$ for which $k e^{x}-x=0$ has two distinct roots is

Let $f(x) = {{{x^2} - 6x + 5} \over {{x^2} - 5x + 6}}$.

Match the conditions/expressions in Column I with statements in Column II.

In the following [x] denotes the greatest integer less than or equal to x.

Match the functions in Column I with the properties Column II.

For $x>0, \mathop {\lim }\limits_{x \to 0}\left((\sin x)^{1 / x}+(1 / x)^{\sin x}\right)$ is :

If $f(x)=\min \left\{1, x^2, x^3\right\}$, then

If $f(x-y)=f(x) \circ g(y)-f(y) \circ g(x)$ And $g(x-y) =g(x) \circ g(y)+f(x) \circ f(y)$ for all $x, y \in \mathrm{R}$. If right-hand derivative at $x=0$ exists for $f(x)$, find the derivative of $g(x)$ at $x=0$