Graph of [x]

Q.

What is the value of $e$?

Q. If $f\left(x\right)=\{\begin{array}{cc}|x|+1,& x<0\\ 0,& x=0\\ |x|-1,& x>0\end{array}$ For what value(s) of $a$ does $\underset{x\to a}{lim}f\left(x\right)$ exists ?

Q. Let $f:[a,b]\to [1,\infty )$ be a continuous function and let $g:R\to R$ be defined as
$g\left(x\right)=\{\begin{array}{c}0\text{if}x<a,\\ \underset{a}{\overset{x}{\int }}f\left(t\right)dt\text{if}a\le x\le b,\\ \underset{a}{\overset{b}{\int }}f\left(t\right)dt\text{if}x>b.\end{array}$
Then

1. $g\left(x\right)$ is continuous but not differentiable at $a$
2. $g\left(x\right)$ is differentiable on $R$
3. $g\left(x\right)$ is continuous but not differentiable at $b$
4. $g\left(x\right)$ is continuous and differentiable at either $a$ or $b$ but not both

Q.

Let $a_1,a_2,\cdots ,a_n$ be fixed real numbers and define a function $f\left(x\right)=(x-a_1)(x-a_2)\dots (x-a_n)$
What is $\underset{x\to a_1}{lim}f\left(x\right)?$ For some $a\ne a_1,a_2\dots .a_n$ compute $\underset{x\to a}{lim}f\left(x\right)$.

Q. Let $f\left(x\right)=\{\begin{array}{cc}{x}^2-[x{]}^2,& x<2\\ a,& x=2\\ x^2-[x{]}^2+3,& x>2\end{array}$. If $f\left(x\right)$ is continuous at $x=2$, then the value of $a$ is
(where $[.]$ represents the greatest integer function and $a\in R$)

1. $0$
2. $1$
3. $2$
4. $3$

Q. Let $f$ and $g$ be continuous functions in $[a,b]$ and $[b,c]$ respectively. A function $h\left(x\right)$ is defined as $h\left(x\right)=\{\begin{array}{cc}f\left(x\right),& x\in [a,b)\\ g\left(x\right),& x\in (b,c]\end{array}.$ If $f\left(b\right)=g\left(b\right),$ then which of the following is/are CORRECT?

1. $h\left(x\right)$ has a removable discontinuity at $x=b$
2. $h\left(x\right)$ is continuous in $[a,c]$
3. $h\left(b^{-}\right)=g\left(b^{+}\right)$ and $h\left(b^{+}\right)=f\left(b^{-}\right)$
4. $h\left(b^{+}\right)=g\left(b^{-}\right)$ and $h\left(b^{-}\right)=f\left(b^{+}\right)$

Q. Let $g\left(x\right)=\left[x\right]$, where $[.]$ represents greatest integer function. Then the function $f\left(x\right)=\left(g\right(x){)}^2-g(x)$ is discontinuous at :

1. $x\in R$
2. $x\in Z-\left\{1\right\}$
3. $x\in Z$
4. $x\in Z-\{0,1,-1\}$

Q. If the area of the domain of the function $f(x,y)=\sqrt{16-x^2-y^2}-\sqrt{|x|-y}$ is $k\pi$ sq. units, then the value of $k$ is

Q. The function $f\left(x\right)$ is defined as $|\left[x\right]x|$ for $-1<x\le 2$. The number of points where $f\left(x\right)$ is non differentiable is

Q. Which of the following is/are periodic?

1. $f\left(x\right)=\{\begin{array}{cc}1,& x\text{is rational}\\ 0,& x\text{is irrational}\end{array}$
2. $f\left(x\right)=\{\begin{array}{cc}x-\left[x\right];& 2n\le x\le 2n+1\\ \frac{1}{2};& 2n+1\le x\le 2n+2\end{array},n\in Z$
3. $f\left(x\right)=(-1{)}^{\left[\frac{2x}{\pi }\right]}$
4. $f\left(x\right)=x-[x+3]+\tan \left(\frac{nx}{2}\right)$

Q. Find the value of k, so that the following function is continuous at x = 2.
$f\left(x\right)=\{\begin{array}{c}f\left(x\right)=\frac{x^3+x^2-16x+20}{{(x-2)}^2},x\ne 2\\ k,x=2\end{array}$

Q.

Show that the Signum function $f:R\to R$, given by

$f\left(x\right)=\{\begin{array}{c}1,ifx>0\\ 0,ifx=0\\ -1,ifx<0\end{array}$
is neither one-one nor onto.

Q. The set of points where the function $f\left(x\right)=x|x|$ is differentiable is

1. $(-\infty ,\infty )$
2. $(-\infty ,0)\cup (0,\infty )$
3. $[0,\infty ]$
4. $(0,\infty )$

Q. The function $f\left(x\right)$ is defined as $|\left[x\right]x|$ for $-1<x\le 2$.
The area under the curve $y=f\left(x\right)$ is

1. $1$
2. $2$
3. $3$
4. $4$

Q. Let $f\left(x\right)=\{\begin{array}{cc}4a-bx,& x<1\\ 3,& x=1\\ 4x-b{x}^2,& x>1\end{array}$. If $f\left(x\right)$ is continuous at $x=1$, then the absolute value of $a-b$ is

Q. if $a\ne b\ne c$, are different, then the value of x satisfying
$\left|\begin{array}{ccc}0& x^2-a& x^3-b\\ x+a& 0& x-c\\ x+b& x+c& 0\end{array}\right|=0$ is

1. $0$
2. $b$
3. $c$
4. $a$

Q. The area (in square units) of the region bounded by the curves $x=y^2$ and $x=3-2y^2$ is

1. $\frac{3}{2}$
2. 2
3. 3
4. 4

Q. Let $h\left(x\right)=min\{x,x^2\}$, for every real number of $x$. Then-

1. $h$ is differential for all $x$
2. $h^{\prime }\left(x\right)=1$, for all $x>1$
3. $h$ is not differentiable at two values of $x$
4. $h$ is continuous for all $x$

Q. $f\left(x\right)=\frac{|x^3-3x^2+2x|}{x^3-3x^2+2x}.$Find the set of points a, where $\underset{x\to a}{lim}f\left(x\right)$ does not exist. Number of such points is?

Q. The function $f\left(x\right)=x-|x-x^2|$ is

1. continuous at $x=1$
2. discontinuous at $x=0$
3. not defined at $x=1$
4. not defined at $x=0$

Q. The set of all point where the function $f\left(x\right)=2x|x|$ is differentiable is -

1. $(-\infty ,0)\cup (0,\infty )$
2. [0, $\infty )$
3. $(0,\infty )$
4. $(-\infty ,\infty )$

Q. Which of the following is true about
$f\left(x\right)=\{\begin{array}{cc}\frac{(x-2)}{|x-2|}\left(\frac{x^2-1}{x^2+1}\right),& x\ne 2\\ \frac{3}{5},& x=2\end{array}$

1. $f\left(x\right)$ is continuous at $x=2$.
2. $f\left(x\right)$ has removable discontinuity at $x=2$.
3. discontinuity at $x=2$ can be removed by redefining the function at $x=2$.
4. $f\left(x\right)$ has non-removable discontinuity at $x=2$.

Q. If the area of the domain of the function $f(x,y)=\sqrt{16-x^2-y^2}-\sqrt{|x|-y}$ is $k\pi$ sq. units, then the value of $k$ is

Q. Let $f\left(x\right)=\{\begin{array}{ccc}x& if& xisrational\\ 2-x& if& xisirrational\end{array}$ Then $fof\left(x\right)$ is continuous

1. no where
2. at all rational $x$
3. everywhere
4. at all irrational $x$

Q. The function $f\left(x\right)$ is defined as $|\left[x\right]x|$ for $-1<x\le 2$.
The area under the curve $y=f\left(x\right)$ is

1. $1$
2. $2$
3. $4$
4. $3$

Q. The function $f\left(x\right)=\{\begin{array}{c}\begin{array}{cc}{(x+1)}^{2-(\frac{1}{\left|x\right|}+\frac{1}{x})},& x\ne 0\end{array}\\ \begin{array}{cc}0,& x=0\end{array}\end{array}$ is

1. Discontinuous at only one point
2. None of these
3. Discontinuous exactly at two points
4. Continuous everywhere

Q. Let $g:R\to R$ be a differentiable function with $g\left(0\right)=0,g^{\prime }\left(0\right)=0$ and $g^{\prime }\left(1\right)\ne 0$. Let $f\left(x\right)=\{\begin{array}{cc}\frac{x}{|x|}g\left(x\right),& x\ne 0\\ 0,& x=0\end{array}$ and $h\left(x\right)=e^{|x|}$ for all $x\in R$. Let $(f\cdot h)\left(x\right)$ denotes $f\left(h\right(x\left)\right)$ and $(h\cdot f)\left(x\right)$ denote $h\left(f\right(x\left)\right)$. Then which of the following is (are) true?

1. $f$ is differentiable at $x=0$
2. $f\cdot h$ is differentiable at $x=0$
3. $h$ is differentiable at $x=0$
4. $h\cdot f$ is differentiable at $x=0$

Q. $f\left(x\right)=\{\begin{array}{cc}{x}^2& forx\ge 1\\ ax+b& forx<1\end{array}$

1. $a=2,b=-1$
2. $a=-\frac{1}{2},b=3$
3. $a=-1,b=2$
4. None of these

Q. Let $f\left(x\right)=\{\begin{array}{cc}\left(\right[x]+\sqrt{\{x}\}),& x\ne 0\\ \lambda ,& x=0\end{array}$. If $f\left(x\right)$ is continuous at $x=0$, then the value of $\lambda$ is
(where $[.]$ represents the greatest integer function, {.} represents the fractional part function and $\lambda \in R$)

1. $-1$
2. $0$
3. $1$
4. $2$

Q. The function $f\left(x\right)$ is defined as $|\left[x\right]x|$ for $-1<x\le 2$.
For what values of $x$ is $f\left(x\right)$ discontinuous ?

1. $0$
2. $1$
3. $2$
4. $-1$