Question 1 :
Let $f(x) = [x]$ and $g(x) = \left\{\begin{matrix}0, & x \in Z\\ x^2, & x \in R - Z \end{matrix}\right.$. Then which of the following is not true ([.] represents the greatest integer function)?
Question 3 :
Discuss the continuity and differentiability of the function f(x) in(0,3) where $\displaystyle f\left ( x \right )=\left\{\begin{matrix}\left | 2x-3 \right |\left [ x \right ], &0\leq x\leq 2 \\\dfrac{x^{2}}{2}, &2< x\leq 3 \end{matrix}\right.$ <br>
Question 4 :
If $ \displaystyle f\left ( x \right )=\left\{\begin{matrix}x\sin \left ( 1/x \right ) & for x\neq 0\\ 0& for x=0\end{matrix}\right. $ then <br>
Question 5 :
The function $f\left( x \right) = \,{\sin ^{ - 1}}\left( {\cos \,x} \right)\,is\,: - $
Question 6 :
Consider $f(x)=\begin{cases} \left[ \cfrac { 2\left( \sin { x } -\sin ^{ 3 }{ x } \right) +\left| \sin { x } -\sin ^{ 3 }{ x } \right| }{ 2\left( \sin { x } -\sin ^{ 3 }{ x } \right) -\left| \sin { x } -\sin ^{ 3 }{ x } \right| } \right] ,\quad x\neq \cfrac { \pi }{ 2 } \quad for\quad x\in \left( 0,\pi \right) \\ 3\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad x=\cfrac { \pi }{ 2 } \end{cases}$, where $[]$ denotes the greatest integer function, then-
Question 7 :
The function $y=f\left( x \right)$, defined parametrically as $x=2t-\left| t-1 \right| $ and $y=2{ t }^{ 2 }+t\left| t \right| $, is
Question 8 :
If the function $\displaystyle f(x)=\left [ \frac{(x-5)^{3}}{A} \right ]\sin (x-5)+a\cos (x-2)$ where $[.]$ denotes the greatest integer function,is continuous and differentiable in (7,9),then:
Question 9 :
Let $f : R\rightarrow R$ be a function such that $|f(x)| \leq x^2$, for all $x\epsilon R$. Then, at $x=0$, f is :<br>
Question 10 :
If f (x + y) = 2 f(x) f(y) all x, y $\in$ R where f' (0) = 3 and f (4) =2, then f'(4) is equal to
Question 11 :
Let $\mathrm{f}:\mathrm{R}\rightarrow \mathrm{R}$ be any function. Define $\mathrm{g}:\mathrm{R}\rightarrow \mathrm{R}$ by $g(x)=|f(x)| $for all $x$. Then $g$ is<br/>
Question 12 :
For real number y, let $ \displaystyle \left [ y \right ] $ denote the greatest integer less than or equal to y. Then $ \displaystyle f\left ( x \right )=\frac{\tan \left ( \pi \left [ x-\pi \right ] \right )}{1+\left [ x \right ]^{2}} $ is <br>
Question 13 :
<u></u>Let $f : R \rightarrow R$ be a continuous function such that $f(x^2) = f(x^3)$ for all $ x \in R$. Consider the following statements.<br>I. f is an odd function.<br>II. f is an even function.<br>III. f is differentiable everywhere
Question 14 :
The function $f(x)=\left[ { x }^{ 2 } \right] +{ \left[ -x \right] }^{ 2 }$, where $[]$ denotes the greatest integer function is
Question 15 :
Let $\displaystyle f\left ( x \right )=x^{3}-x^{2}+x+1$ <br> $\displaystyle g\left ( x \right )=\begin{cases} max\left \{ f\left ( t \right ),\>0\leq t\leq x \right \},0\leq x\leq1 \\ 3-x,\>1< x\leq 2\end{cases}$<br>Discuss the continuity and differentiability of the function $g(x)$ in the interval $(0,2)$.<br>
Question 17 :
The function $f(x)=\begin{cases} ax(x-1)+b\quad \quad when\quad x<1 \\ x-1\quad \quad \quad when\quad \quad 1\le x\le 3 \\ p{ x }^{ 2 }+qx+2\quad \quad when\quad x>3 \end{cases}$ Find the values of the constants $a,b,p,q$ so that $(i)f(x)$ is continuous for all $x$ $(ii)f'(1)$ does not exist $(iii)f'(x)$ is continuous at $x=3$
Question 18 :
Given $f(x)=\cos ^{ -1 }{ \left( sgn\left( \cfrac { 2\left| x \right| }{ 3x-\left| x \right| } \right) \right) } $ where $sgn(.)$ denotes the signum function $\left[ \cdot \right] $ denotes the greatest integer function. Discuss the continuity and differentiablity at $x=\pm 1$
Question 19 :
If $f(x)=\cos \pi \left( \left| x \right| +\left[ x \right] \right) $, then $f(x)$ is/are (where $\left[ \cdot \right] $ denotes greatest integer function)
Question 20 :
If $f(x) =\begin{cases} 2x^2+12x+16&, -4\le x\le -2\\2-|x|& , -2 < x \le 1\\4x-x^2-2& , 1< x \le 3 \end{cases}$. Then $f(x)$ is
Question 21 :
Let [x] denote the greatest integer less than or equal to x.<br>If $f\left ( x \right )=\left [ x\sin \pi x \right ]$, then f(x) is<br>
Question 23 :
Let $f:\left[ {0,2} \right] \to R$ a function which is continuous on $\left[ {0,2} \right]$ and is differentiable on $\left( {0,2} \right)$ with $f\left( 0 \right) = 1$. Let $F\left( x \right) = \int\limits_0^{{x^2}} {f\left( {\sqrt t } \right)} dt,$ for $x \in \left[ {0,2} \right]$, if $F'\left( x \right) = f'\left( x \right),\forall x \in \left( {0,2} \right),$ then $F(2)$ equals to
Question 24 :
Let $f(x)=\left [ \tan ^{2}x \right ]$, where [.] denotes the greatest integer function. Then
Question 25 :
Examine for continuity and differentiability at the points $x=1, x=2$, the function $f$ defined by $f(x)=\begin{cases} x\left[ x \right] ,\quad \quad \quad \quad 0\le x<2 \\ (x-1)\left[ x \right] ,\quad 2\le x\le 3 \end{cases}$ where $\left[ x \right] =$ greatest integer less than or equal to $x$
Question 27 :
Discuss the continuity and differentiability of the following function $\displaystyle f\left ( x \right )=\begin{cases} x^{2} & x< -2 \\ 4 & -2\leq x\leq 2 \\ x^{2} & x> 2 \end{cases}$<br>
Question 28 :
Let $f_{1} : \mathbb {R}\rightarrow \mathbb {R}, f_{2} : \left (-\dfrac {\pi}{2}, \dfrac {\pi}{2}\right ) \rightarrow \mathbb {R}, f_{3} : \left (-1, e^{\frac {\pi}{2}} - 2\right ) \rightarrow \mathbb {R}$ and $f_{4} : \mathbb{R}\rightarrow \mathbb {R}$ be functions defined by<br/>(i) $f_{1}(x) = \sin (\sqrt {1 - e^{-x^{2}}})$<br/>(ii) $f_{2}(x) = \left\{\begin{matrix}\dfrac {|\sin x|}{\tan^{-1}(x)}\ & if & x\neq 0\\ 1\ &if & x = 0\end{matrix}\right.$<div> where the inverse trigonometric function $\tan^{-1}(x)$ assumes values in $\left (-\dfrac {\pi}{2}, \dfrac {\pi}{2}\right )$.<br/>(iii) $f_{3}(x) = [\sin (\log_{e}(x + 2)]$, where for $t\ \epsilon\ \mathbb {R}, [t]$ denotes the greatest integer less than or equal to $t$,<br/>(iv) $f_{4}(x) = \left\{\begin{matrix} x^{2} \sin \left (\dfrac {1}{x}\right )& if\ x\neq 0 \\ 0 & if\ x = 0\end{matrix}\right.$<br/><table class="wysiwyg-table"><tbody><tr><td>List - I</td><td>List - II</td></tr><tr><td>P. The function $f_{1}$ is</td><td>$1.$ NOT continuous at $x = 0$</td></tr><tr><td>Q. The function $f_{2}$ is</td><td>$2.$ continuous at $x = 0$ and NOT differentiable at $x = 0$</td></tr><tr><td>R. The function $f_{3}$ is</td><td>$3.$ differentiable at $x = 0$ and its derivative is NOT continuous at $x = 0$</td></tr><tr><td>S. The function $f_{4}$ is</td><td>$4.$ diffferentiable at $x = 0$ and its derivative is continuous at $x = 0$</td></tr></tbody></table>The correct option is</div>
Question 29 :
<p>The function $f(x) = \displaystyle \frac{{\tan (\pi [x - \pi ])}}{{1 + {{[x]}^2}}}$, where [x] is the greatest integer function,</p>
Question 30 :
If $f(x) = \sqrt{1-e^{-x^2}}$, then at $x = 0$, $f(x)$ is<br>
Question 31 :
Let $f(x)=[x]^{2}+\sqrt{\{x\}}$ where [] & {}respectively denotes the greatest integer and fractional part functions, <span>then which of the following is correct?</span>
Question 33 :
$f(x)={ \left( \sin ^{ -1 }{ x } \right) }^{ 2 }.\cos { \left( 1/x \right) } $ if $x\ne 0, f(0)=0, f(x)$ is:
Question 34 :
If $f(x) = \left\{\begin{matrix} x\sin \left (\dfrac {1}{x}\right ),& x\neq 0\\ 0, & x = 0\end{matrix}\right.$, then at $x = 0$ the function $f(x)$ is<br>
Question 35 :
The function $f(x)=\left\{\begin{matrix}<br>1-2x+3x^2-4x^3+....to \infty, & x \neq -1\\ <br>1 & ,x=-1<br>\end{matrix}\right.$ is<br>
Question 36 :
Let $f(x) = \left\{\begin{matrix}2a - x, &if\ -a < x < a \\ 3x - 2a, &if\ a \leq x \end{matrix}\right.$. Then, which of the following is true?<br>
Question 37 :
If $\displaystyle f(x)=\left (\tan\left (\frac{\pi}{4}+\ln x \right ) \right )^{\frac{1}{2}\log_{x}e}$ is to be made continuous at $x=0$, then $[f(1)]$ should be equal to (where $[.]$denote greatest integer function) :
Question 38 :
If $f(x)=\left | x \right |+\left | \cos x \right |$, then
Question 39 :
Let $f:(-1,1)\rightarrow R$ be a differentiable function satisfying <br> $(f'(x))^4=16(f(x))^2$ for all $x\in (-1,1)$<br> $f(0)=0$<br>The number of such functions is
Question 40 :
$f(x) = \left ( \left [ x \right ]-\left [ -x \right ] \right ) sin^{-1} |x-1|$.<br> Which of the following statements is/are correct ? <br>(Note : $[.]$ denotes the greatest integer function)<br>
Question 42 :
If $f(x)=\left\{ \begin{matrix} \dfrac { |x+2| }{ tan^{ -1 }(x+2) } & x\neq -2 \\ 2, & x=-2 \end{matrix} \right. $ then, $f(x)$ is: