Continuity of a Function

Q.

The differential coefficient of $\log \tan x$ is

Q. Consider the real-valued function $f$ satisfying $2f\left(\sin x\right)+f\left(\cos x\right)=x$. Then,

1. Domain of $f$ is $R$
2. Domain of $f$ is $[-1,1]$
3. Range of $f$ is $[-\frac{2\pi }{3},\frac{\pi }{3}]$
4. Range of $f$ is $R$

Q. If $S_n=\frac{1}{3^2+1}+\frac{1}{4^2+2}+\frac{1}{5^2+3}+\cdots \cdots \text{upto}n\text{terms}$ and $S_{\infty }=\frac{a}{72},$ then $a$ equals

Q. If $\sum _{k=1}^{n}f\left(k\right)=\frac{n}{2(n+2)}$, then the value of $\sum _{k=1}^{10}\frac{1}{f\left(k\right)}$ is equal to

1. $480$
2. $560$
3. $550$
4. $570$

Q. The function $f\left(x\right)-p[x+1]+q[x-1]$, where is the greatest integer function is continuous at x =1, if

1. $p-q=0$
2. $p+q=0$
3. $p=0$
4. $q=0$

Q. Let $f\left(x\right)=[x^3-3],$ where $[.]$ denotes the greatest integer function. Then the number of points in the interval $(1,2)$ where the function is discontinuous, is

1. $2$
2. $4$
3. $6$
4. $3$

Q. Let $f:R\to R$ be a continuous function satisfying $f\left(0\right)=1$ and $f\left(2x\right)-f\left(x\right)=x$, for all $x\in R.$ Then $f\left(2020\right)$ equals

Q. If $f\left(x\right)=\{\begin{array}{cc}\frac{\sin (p+1)x+\sin x}{x},& x<0\\ q,& x=0\\ \frac{\sqrt{x+x^2}-\sqrt{x}}{x^{3/2}},& x>0\end{array}$ is continuous at $x=0$, then the ordered pair $(p,q)$ is equal to:

1. $(-\frac{3}{2},\frac{1}{2})$
2. $(\frac{5}{2},\frac{1}{2})$
3. $(-\frac{1}{2},\frac{3}{2})$
4. $(-\frac{3}{2},-\frac{1}{2})$

Q. Paragraph for below question

नीचे दिए गए प्रश्न के लिए अनुच्छेद

Consider the function $f\left(x\right)=\{\begin{array}{ccc}\frac{\tan x}{\sin x}& ,& x<0\\ a& ,& x=0\\ \frac{b{\log }_e(1+2x)}{x}& ,& x>0\end{array}$

फलन $f\left(x\right)=\{\begin{array}{ccc}\frac{\tan x}{\sin x}& ,& x<0\\ a& ,& x=0\\ \frac{b{\log }_e(1+2x)}{x}& ,& x>0\end{array}$ पर विचार कीजिए।


Q. If f(x) is continuous at x = 0, then the value of b is equal to

प्रश्न - यदि x = 0 पर f(x) सतत है, तब b का मान बराबर है

1. 1
2. 2
3. –1
4. $\frac{1}{2}$
   

Q. More than One Answer Type
एक से अधिक उत्तर प्रकार के प्रश्न

Which of the following functions is/are continuous for x ∈ R?

x ∈ R के लिए निम्नलिखित में से कौनसा/कौनसे फलन सतत है/हैं?

1. $f\left(x\right)={\tan }^{-1}x$
2. f(x) = {x}; {·} represents fractional part of x.
   
   f(x) = {x}; {·}, x के भिन्नात्मक भाग फलन को दर्शाता है
3. f(x) = |x + 4|
4. f(x) = sgn(e^x); sgn is signum function
   
   f(x) = sgn(e^x); sgn सिग्नम फलन है

Q. The value of $a$ for which the function $f\left(x\right)=a\sin x+\frac{1}{3}\sin 3x$ has an extremum at $x=\frac{\pi }{3}$ is

Q. If f(x) = $\{\begin{array}{cc}\frac{8^x-4^x-2^x+1}{x^2},& x>0\\ e^x\sin x+\pi x+\lambda ,& x⩽0\end{array}$ is continuous at x = 0, then the value of λ is

यदि f(x) = $\{\begin{array}{cc}\frac{8^x-4^x-2^x+1}{x^2},& x>0\\ e^x\sin x+\pi x+\lambda ,& x⩽0\end{array}$, x = 0 पर सतत है, तब λ का मान है

1. (log_e4)²
2. –log_e2
3. 2(log_e2)²
4. 4log_e2

Q. The function
f(x) = $\{\begin{array}{cc}\frac{1-\cos 3x}{x^2},& \text{if}x\ne 0\\ K,& \text{if}x=0\end{array}$
is continuous at x = 0 for

कौनसे विकल्प के लिए फलन

पर सतत है?

1. K = 9
2. K = $\frac{9}{2}$
3. K = 18
4. No value of K
   
   K का कोई मान नहीं

Q. The number of integral values of $k$ for which $h\left(x\right)=\text{sgn}(x^2-2kx+\text{sgn}(k+2\left)\right)$ is continuous for all $x\in R$ is
[Note : $\text{sgn}\left(y\right)$ denotes the signum function of $y$.]

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

Q. If the function $f\left(x\right)=\begin{array}{cc}(1+|sinx|{)}^{\frac{a}{|sinx|}},& -\frac{\pi }{6}<x<0\\ b,& x=0\\ e^{\frac{tan2x}{tan3x}},& 0<x<\frac{\pi }{6}\end{array}$, is continuous at x = 0, then

1. $a=lo{g}_{e}b,a=\frac{2}{3}$
2. $b=lo{g}_{e}a,a=\frac{2}{3}$
3. $a=lo{g}_{e}b,b=2$
4. $Noneofthese$

Q.

The function f(x) defined as f(x) = $\sqrt{(x-4{)}^2}$

1. continuous but not differentiable $\forall xϵR$

2. continuous and differentiable $\forall xϵR$

3. not continuous at x = 4

4. continuous only for $x\ge 4$

Q.

If f(x) be a continuous function defined for $1\le x\le 3.f\left(x\right)ϵQ\forall xϵ[1,3]andf\left(2\right)=10$ (Where Q is a set of all rational numbers). Then, f(1.8) is

1. 1

2. 5

3. 10

4. 20

Q. This section contains 1 Assertion-Reason type question, which has 4 choices (a), (b), (c) and (d) out of which ONLY ONE is correct.

इस खण्ड में 1 कथन-कारण प्रकार का प्रश्न है, जिसमें 4 विकल्प (a), (b), (c) तथा (d) दिये गये हैं, जिनमें से केवल एक सही है।

A : A function $f\left(x\right)=\{\begin{array}{cc}\frac{\sin x+\tan x}{2x};& x\ne 0\\ k;& x=0\end{array}$ is discontinuous at x = 0 if k = 1.

A : एक फलन $f\left(x\right)=\{\begin{array}{cc}\frac{\sin x+\tan x}{2x};& x\ne 0\\ k;& x=0\end{array}$, x = 0 पर असतत है, यदि k = 1 है।

R : If a function is continuous at x = a then $\underset{x\to a}{lim}f\left(x\right)=f\left(a\right)$.

R : यदि एक फलन, x = a पर सतत है, तब $\underset{x\to a}{lim}f\left(x\right)=f\left(a\right)$ है।

1. Both (A) and (R) are true and (R) is the correct explanation of (A)
   
   (A) तथा (R) दोनों सही हैं तथा (R), (A) का सही स्पष्टीकरण है
2. Both (A) and (R) are true but (R) is not the correct explanation of (A)
   
   (A) तथा (R) दोनों सही हैं लेकिन (R), (A) का सही स्पष्टीकरण नहीं है
3. (A) is true but (R) is false
   
   (A) सही है लेकिन (R) गलत है
4. (A) is false but (R) is true
   
   (A) गलत है लेकिन (R) सही है

Q. Let $S_1,S_2,S_3$ and $S_4$ be four sets defined as
$S_1=\{x:x\in Z\text{and}{\log }_2|4-3x|\le 2\}$
$S_2=\{x:x\in Z\text{and}|1-\frac{|x|}{1+|x|}|\ge \frac{1}{3}\}$
$S_3=\left\{x:x^2-3x+2\text{sgn}\left(x\right)=0\right\},$ where $\text{sgn}\left(x\right)$ represents the signum function.
$S_4=\left\{\right(x,y):x,y\in Z,x^2+y^2\le 4\}$.

$\text{List I}$ has four entries and $\text{List II}$ has five entries. Each entry of $\text{List I}$ is to be correctly matched with a unique entry of $\text{List II}.$

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& n\left(S_1\Delta S_2\right)& \text{(P)}& 9\\ \text{(B)}& n((S_1\times S_2)\cap (S_2\times S_1))& \text{(Q)}& 12\\ \text{(C)}& n(S_1\cap S_2\cap S_3^{\prime })& \text{(R)}& 36\\ \text{(D)}& n(S_4\times S_3)& \text{(S)}& 2\\ & & \text{(T)}& 0\end{array}$

Which of the following is the only CORRECT combination?

1. $\text{(C)}\to \text{(S)}$
2. $\text{(C)}\to \text{(Q)}$
3. $\text{(D)}\to \text{(R)}$
4. $\text{(D)}\to \text{(Q)}$

Q.

Form six expressions using $t$ and $4$. Use not more than one number operation and every expression must have $t$ in it.

Q. If the function $f\left(x\right)$ is continuous at $x=0,$ where $f\left(x\right)=\{\begin{array}{cc}\frac{\sin ((a+1)x)+\sin x}{x};& x<0\\ c;& x=0\\ \frac{(x+b{x}^2{)}^{1/2}-x^{1/2}}{b{x}^{3/2}};& x>0\end{array},$
then the possible values of $a,b,c$ can be

1. $a=-\frac{1}{2}$
2. $b=-\frac{3}{2}$
3. $b=1$
4. $c=\frac{1}{2}$

Q. The function $f\left(x\right)=\{\begin{array}{c}{x}^2/a,0\le x<1\\ a,1\le x<\sqrt{2}\\ \frac{2b^2-4b}{x^2},\sqrt{2}\le x<\infty \end{array}$
is continuous for $0\le x<\infty$, then the most suitable values of $a$ and $b$ are

1. $a=1,b=-1$
2. $a=-1,b=1+\sqrt{2}$
3. $a=-1,b=1$
4. None of the above

Q. If $f\left(x\right)=\{\begin{array}{cc}\frac{\sin (\alpha +2)x+\sin x}{x},& x<0\\ b,& x=0\\ \frac{(x+3x^2{)}^{\frac{1}{3}}-x^{\frac{1}{3}}}{x^{\frac{4}{3}}},& x>0\end{array}$ is continuous at $x=0$ then $a+2b$ is equal to :

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

Q. If the function $f\left(x\right)=\begin{array}{cc}(1+|sinx|{)}^{\frac{a}{|sinx|}},& -\frac{\pi }{6}<x<0\\ b,& x=0\\ e^{\frac{tan2x}{tan3x}},& 0<x<\frac{\pi }{6}\end{array}$, is continuous at x = 0, then

1. $a=lo{g}_{e}b,a=\frac{2}{3}$
2. $b=lo{g}_{e}a,a=\frac{2}{3}$
3. $a=lo{g}_{e}b,b=2$
4. $Noneofthese$

Q. Let $f\left(x\right)=\frac{{\cos }^{-1}(1-\{x{\}}^2\left){\sin }^{-1}\right(1-\left\{x\right\})}{\left\{x\right\}-\{x{\}}^3},x\ne 0$, where $\left\{x\right\}$ denotes fractional part of $x$. Then
(correct answer + 1, wrong answer - 0.25)

1. If $f\left(0\right)=\frac{\pi }{4},$ then $f\left(x\right)$ is continuous at $x=0$
2. If $f\left(0\right)=\frac{\pi }{\sqrt{2}},$ then $f\left(x\right)$ is continuous at $x=0$
3. If $f\left(0\right)=\frac{\pi }{2\sqrt{2}},$ then $f\left(x\right)$ is continuous at $x=0$
4. $f\left(x\right)$ is a discontinuous function.

Q. The function $f\left(x\right)-p[x+1]+q[x-1]$, where is the greatest integer function is continuous at x =1, if

1. $p-q=0$
2. $p+q=0$
3. $p=0$
4. $q=0$

Q. The value of p for which the function $f\left(x\right)=\frac{(4^x-1{)}^3}{sin\left(\frac{x}{p}\right)ln(1+\frac{x^2}{3})},x\ne 0=12(ln4{)}^3,x=0$ may be continuous at x = 0 is

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

Q. If $f\left(x\right)=\frac{x}{2}-1$, determine which of the following statement(s) is (are) true on the following interval $[0,\pi ].$

1. $\tan \left(f\right(x\left)\right)$ and $1/f\left(x\right)$ are both continuous
   
2. $\tan \left(f\right(x\left)\right)$ and $1/f\left(x\right)$ are both discontinuous
   
3. $\tan \left(f\right(x\left)\right)$ and $f^{-1}\left(x\right)$ are both continuous
   
4. $\tan \left(f\right(x\left)\right)$ is continuous but $1/f\left(x\right)$ is not.

Q. Let $f\left(x\right)=x\sin \pi x,x>0$. Then for all natural numbers $n,f^{\prime }\left(x\right)$ vanishes at

1. a unique point in the interval $(n,n+\frac{1}{2})$
2. a unique point in the interval $(n+\frac{1}{2},n+1)$
3. a unique point in the interval $(n,n+1)$
4. two points in the interval $(n,n+1)$

Q. If $f\left(x\right)=\{\begin{array}{cc}xsinx,& when0<x\le \frac{\pi }{2}\\ \frac{\pi }{2}sin(\pi +x),& when\frac{\pi }{2}<x<\pi \end{array},then$

1. f(x) is discontinous at $x=\frac{\pi }{2}$
2. f(x) is continuous at $x=\frac{\pi }{2}$
3. f(x) is continuous at x = 0
4. None of these