Continuous Function

Q. The function $f\left(x\right)=[x{]}^2-[x^2],$ where $\left[y\right]$ is the greatest integer less than or equal to $y$, is discontinuous at

1. all integers
2. all integers except $-1$
3. all integers except $0$
4. all integers except $1$

Q.

Let the functions $f:(-1,1)\to R$and $g:(-1,1)\to (-1,1)$ be defined by$f\left(x\right)=|2x-1|+|2x+1|$ and $g\left(x\right)=x-\left[x\right]$, where $\left[x\right]$ denotes the greatest integer less than or equal to $x$. Let $fog:(-1,1)\to R$ be the composite function defined by $(f\circ g)\left(x\right)=f\left(g\right(x\left)\right)$. Suppose $c$ is the number of points in the interval $(-1,1)$ at which $f\circ g\left(x\right)$ is NOT continuous, and suppose $d$ is the number of points in the interval $(-1,1)$ at which $f\circ g\left(x\right)$ is NOT differentiable. Then the value of $c+d$ is _____.

Q.

A function $f$ is defined on $[–3,3]$ as $f\left(x\right)=\left\{\begin{array}{l}min\left\{\left|x\right|,2-x^2\right\},-2\le x\le 2\\ \left[\left|x\right|\right],2<\left|x\right|\le 3\end{array}\right.$, where $\left[x\right]$ denotes the greatest integer $\le x$.

The number of points, where $f$ is not differentiable in $(–3,3)$ is __________ .

Q. Let $f:(a,b)\to R$ be twice differentiable function such that $f\left(x\right)={\int }_a^{x}g\left(t\right)dt$ for a differentiable function $g\left(x\right)$. If $f\left(x\right)=0$ has exactly five distinct roots in $(a,b)$, then $g\left(x\right)g^{\prime }\left(x\right)=0$ has at least

1. twelve roots in $(a,b)$
2. three roots in $(a,b)$
3. five roots in $(a,b)$
4. seven roots in $(a,b)$

Q. Which of the following is/are not true?

1. if $f\left(x\right)+g\left(x\right)$ is continuous at $x=a$, then $f\left(x\right)$ and $g\left(x\right)$ are both seperately continuous at $x=a$.
2. If $f\left(x\right).g\left(x\right)$ is continuous at $x=a$, then $f\left(x\right)$ and $g\left(x\right)$ are seperately continuous at $x=a$.
3. If $f\left(x\right).g\left(x\right)$ and $f\left(x\right)$ are discontinuous at $x=a$ then $g\left(x\right)$ is continuous at $x=a$.
4. If both $f\left(x\right)$ and $g\left(x\right)$ are discontinuous at $x=a$ then $f\left(x\right)+g\left(x\right)$ may not be discontinuous at $x=a$.

Q. Let $z_1,z_2$ be the roots of the equations $z^2+az+12=0$ and $z_1,z_2$ form an equilateral triangle with origin. Then, the value of $|a|$ is

Q. $f\left(x\right)=[x{]}^3-[x^3]$, (where [.] is greatest integer function) is discontinuous at all

1. integers $n$
2. integers $n$ except $n=0\text{and}1$
3. integers $n$ except $n=0\text{and}1$, since $f\left(n^{-}\right)\ne f\left(n\right)$
4. integers $n$ except $n=0\text{and}1$, since $f\left(n^{+}\right)\ne f\left(n\right)$

Q. If number of points of discontinuity of $f\left(x\right)=sgn\left(\cos 5x\right)$ is equal to the number of points of non- differentiability of g(x) = {n + m sin x} where $x\in (0,\pi ),n,m\in I$, then value of $m$ is
(where, {x} denotes fractional part of x and sgn(x) denotes signum function of x)

Q. If $f\left(x\right)=\{\begin{array}{cc}x& \text{if}x\text{is rational}\\ 1-x& \text{if}x\text{is irrational}\end{array}$, then the number of points for $xϵR$ where $f\left(f\right(x\left)\right)$ is/are discontinuous

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

Q. The number of Non-differentiable points of the function $f\left(x\right)=|5^{|x|}-4|$ is

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

Q. $f\left(x\right)=1/(1-e^{-1/x}),x\ne 0$ If f is continuous at $x=0$ then, Find f(0)

Q. Match $\text{List I}$ with the $\text{List II}$ and select the correct answer using the code given below the lists :

Let $\left[k\right]$ denote the greatest integer less than or equal to $k$ and $\text{sgn}$ denote the signum function.
$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{If}f\left(x\right)=\text{sgn}(x^2-ax+1)\text{has exactly one point of discontinuity,}& \text{(P)}& 1\\ & \text{then value(s) of}a\text{can be}& & \\ \text{(B)}& \text{If}f\left(x\right)=[2+3|n|\sin x]\text{has exactly}11\text{points of discontinuity in}& \text{(Q)}& 2\\ & x\in (0,\pi ),\text{then}n\text{cannot be}& & \\ \text{(C)}& \text{If}f\left(x\right)=|||x|-2|-p|\text{has exactly three points of non-differentiability,}& \text{(R)}& -1\\ & \text{then value(s) of}p\text{can be}& & \\ \text{(D)}& \text{If}\underset{x\to -\infty }{lim}\frac{\sqrt{4x^2+3}-x}{3x-2}=L,\text{then}L\text{equals}& \text{(S)}& -2\\ & & \text{(T)}& 3\end{array}$

Which of the following is a CORRECT combination?

1. $\left(C\right)\to \left(R\right),\left(S\right);\left(D\right)\to \left(S\right)$
2. $\left(C\right)\to \left(Q\right),\left(R\right),\left(S\right);\left(D\right)\to \left(R\right)$
3. $\left(C\right)\to \left(R\right),\left(S\right);\left(D\right)\to \left(R\right)$
4. $\left(C\right)\to \left(R\right),\left(T\right);\left(D\right)\to \left(S\right)$

Q. Find the numbers of integers satisfying the following Inequalities
$\left(i\right)x^4-5x^2+4\le 0$
$\left(ii\right)x^4-2x^2+63\le 0$

Q. If $f\left(x\right)=||x{|}^2-2|x|-3|$, then $f\left(x\right)$ is non differentiable at $x$ equal to

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

Q.

How do you find the domain of $f\left(x\right)=\sqrt{x^2-16}$?

Q. $f\left(x\right)$ is continuous function on [1, 3] and $f\left(1\right)=2,f\left(3\right)=-2,$ then which of the following not necessarily hold good?

1. $f\left(2\right)⩾0$
2. $x^{2}f\left(x\right)=0\text{has a root in}(1,3)$
3. $-2\le f\left(x\right)\le 2\forall x\in [1,3]$
4. $f\left(x\right)-x^2=0\text{has a root in}(1,3)$

Q. The figure shows a portion of the graph $y=2x–4x^3.$ The line $y=c$ is such that the areas of the regions marked
I and II are equal. If $a,b$ are the $x-$coordinates of $A,B$ respectively, then $a+b$ equals

1. $\frac{2}{\sqrt{7}}$
2. $\frac{3}{\sqrt{7}}$
3. $\frac{4}{\sqrt{7}}$
4. $\frac{5}{\sqrt{7}}$

Q. The function $\{\begin{array}{cc}(-1{)}^{\left[x\right]},& x<0\\ \underset{n\to \infty }{lim}\left(\frac{1}{1+x^n}\right),& x⩾0\end{array}$ then the number of integral values of $x$ in $[-3,5]$ where $f\left(x\right)$ is discontinuous is / are
([.] denotes greatest integer function)

Q. $Letf:[-\frac{1}{3},3]\to Randg:[-\frac{1}{3},3]\to Rdefinedbyf\left(x\right)=[x^2-4]andg\left(x\right)=|x-2|f\left(x\right)+|3x-5|f\left(x\right)$, where $\left[x\right]$ denotes the greatest integer less than or equal to $x$ for $x\in R,$ then

1. f is discontinuous exactly at eight points in $[-\frac{1}{3},3]$
2. f is discontinuous exactly at nine points in $[-\frac{1}{3},3]$
3. g is discontinuous exactly at ten points in $[-\frac{1}{3},3]$
4. g is discontinuous exactly at nine points in $[-\frac{1}{3},3]$

Q. $Letf:[-\frac{1}{3},3]\to Randg:[-\frac{1}{3},3]\to Rdefinedbyf\left(x\right)=[x^2-4]andg\left(x\right)=|x-2|f\left(x\right)+|3x-5|f\left(x\right)$, where $\left[x\right]$ denotes the greatest integer less than or equal to $x$ for $x\in R,$ then

1. f is discontinuous exactly at eight points in $[-\frac{1}{3},3]$
2. f is discontinuous exactly at nine points in $[-\frac{1}{3},3]$
3. g is discontinuous exactly at ten points in $[-\frac{1}{3},3]$
4. g is discontinuous exactly at nine points in $[-\frac{1}{3},3]$

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

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

Q.

Find the value of

$\sum {}_{k=1}^6$ $(sin\frac{2k\pi }{7}-icos\frac{2k\pi }{7})$.

__

Q. The range of $f\left(x\right)=x-\left[x\right]$ is

1. $f\left(x\right)=1,2,3\dots$
2. $f\left(x\right)\ge 0$
3. $f\left(x\right)<1$
4. $0\le f\left(x\right)<1$

Q. $f\left(x\right)=\frac{x+2}{|x+2|}$ find the point of discontinuity.

Q. If $D=4a^2-12b=4(a^2-3b)=0$ and let $x_1$ and $x_2$ be the roots of $f^{\prime }\left(x\right)=0$ such that $f\left(x_1\right)\ne 0$, then

1. $f\left(x\right)$ has repeated roots
2. $f\left(x\right)$ has one real and two non-real roots
3. $f\left(x\right)$ has all real roots
4. none of the above

Q. Match $\text{List I}$ with the $\text{List II}$ and select the correct answer using the code given below the lists :

Let $\left[k\right]$ denote the greatest integer less than or equal to $k$ and $\text{sgn}$ denote the signum function.
$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{If}f\left(x\right)=\text{sgn}(x^2-ax+1)\text{has exactly one point of discontinuity,}& \text{(P)}& 1\\ & \text{then value(s) of}a\text{can be}& & \\ \text{(B)}& \text{If}f\left(x\right)=[2+3|n|\sin x]\text{has exactly}11\text{points of discontinuity in}& \text{(Q)}& 2\\ & x\in (0,\pi ),\text{then}n\text{cannot be}& & \\ \text{(C)}& \text{If}f\left(x\right)=|||x|-2|-p|\text{has exactly three points of non-differentiability,}& \text{(R)}& -1\\ & \text{then value(s) of}p\text{can be}& & \\ \text{(D)}& \text{If}\underset{x\to -\infty }{lim}\frac{\sqrt{4x^2+3}-x}{3x-2}=L,\text{then}L\text{equals}& \text{(S)}& -2\\ & & \text{(T)}& 3\end{array}$

Which of the following is a CORRECT combination?

1. $\left(A\right)\to \left(P\right),\left(R\right);\left(B\right)\to \left(Q\right),\left(S\right)$
2. $\left(A\right)\to \left(Q\right),\left(S\right);\left(B\right)\to \left(Q\right),\left(R\right),\left(T\right)$
3. $\left(A\right)\to \left(P\right),\left(R\right);\left(B\right)\to \left(Q\right),\left(T\right)$
4. $\left(A\right)\to \left(Q\right),\left(S\right);\left(B\right)\to \left(P\right),\left(R\right),\left(T\right)$

Q. The function $f\left(x\right)=[x{]}^2-[x^2],$ where $\left[y\right]$ is the greatest integer less than or equal to $y$, is discontinuous at

1. all integers
2. all integers except $-1$
3. all integers except $0$
4. all integers except $1$