Differentiability

Q.

Let $\mathrm{g}\left(\mathrm{x}\right)=1+\mathrm{x}–\left[\mathrm{x}\right]$and $\mathrm{f}\left(\mathrm{x}\right)=\left\{\begin{array}{l}-1\mathrm{x}<0\\ 0\mathrm{x}=0\\ 1\mathrm{x}>0\end{array}\right.$Then for all$\mathrm{x}$, $\mathrm{f}\left(\mathrm{g}\right(\mathrm{x}\left)\right)$ is equal to

1. $\mathrm{x}$

2. $1$

3. $\mathrm{f}\left(\mathrm{x}\right)$

4. $\mathrm{g}\left(\mathrm{x}\right)$

Q.

Convert the following 75° C temperature into Fahrenheit (°F) scale?

Q.

The sum of the series sigma $\mathrm{n}=8\mathrm{to}17$, $\frac{1}{(n+2)(n+3)}$ is equal to

1. $\frac{1}{17}$

2. $\frac{1}{18}$

3. $\frac{1}{19}$

4. $\frac{1}{20}$

Q.

For all $x\in (0,1)$

1. $e^x<1+x$

2. ${\log }_e(1+x)<x$

3. $\sin x>x$

4. ${\log }_{e}x>x$

Q.

The L.C.M. of two numbers is $495$ and their H.C.F. is$5$. If the sum of the two numbers is $100$ then their difference is _____.

Q. If the function f(x) has LHD and RHD at $x_0$, which are finite and equal, then f(x) is differentiable at $x_0$, always.

1. True
2. False

Q. Let, $f\left(x\right)=\{\begin{array}{c}\left[x\right],-2\le x\le -1\\ |x|+1,-1<x\le 2\end{array}$ and $g\left(x\right)=\{\begin{array}{c}\left[x\right],-\pi \le x\le 0\\ \sin x,0<x\le \pi \end{array},$ then the number of integral points in the range of $g\left(f\right(x\left)\right)$ is

Q. The statement P(n) = $4^n-1\ge 3^n$ is true for which of the following options?

1. $n\in N-\left(1\right\}$
2. $n\in N$
3. $\mathrm{only}\mathrm{for}n\ge 7$
4. $\mathrm{no}\mathrm{natural}\mathrm{number}$

Q. The domain of the function f(x) given by $f\left(x\right)=\frac{1}{\sqrt{x+\left|x\right|}}$ is

1. R
2. $R^{+}$
3. $R^{-}$
4. $R-\{-1\}$

Q.

If $f\left(x\right)=\sqrt{2x+3}$, Then $f^{\prime }\left(x\right)=$___

1. $\sqrt[3]{3(2x+3{)}^2}$

2. $\frac{1}{2\sqrt{(2x+3)}}$

3. $\frac{1}{\sqrt{2x+3}}$

4. $\frac{2}{\sqrt{2x+3}}$

Q. The function $f\left(x\right)=|x^2-3x+2|+cos|x|$ is not differentiable at x is equal to -

1. - 1
2. \N
3. 1
4. 2

Q. The domain of $f\left(x\right)=\frac{{\sin }^{-1}x}{x^2}$ is

1. None of these
2. $(0,1]$
3. $(-1,1)-\left\{0\right\}$
4. $[-1,1]-\left\{0\right\}$

Q. If the domain of function f is set A and domain of function g is set B, then the domain of function f + g is

1. $A\cup B$
2. $A\cap B$
3. A+B
4. A-B

Q. The function $f\left(x\right)$ is given by $f\left(x\right)=(\begin{array}{c}{x}^2\sin \left(\frac{1}{x}\right),\mathrm{if}x\ne 0\\ 0,\mathrm{if}x=0\end{array}$
Which of the following statement(s) is/are true for $f\left(x\right)?$

1. $f\left(x\right)$ is continuous at x = 0
2. $f\left(x\right)$ is differentiable at x = 0
3. $f^{\prime }\left(0\right)=0$
4. $f\left(x\right)$ is not differentiable at x = 0

Q.

$\mathrm{If}11\times 11=121,\mathrm{then}11\times 0.11=\mathrm{\_\_}$

1. $1.21$

2. $12.1$

Q. Let $f\left(x\right)=x^2-6x+5$ and $m$ is the number of points of non-derivability of $y=|f\left(|x|\right)|.$ If $|f\left(|x|\right)|=k,k\in R$ has at least $m$ distinct solution(s), then the number of integral values of $k$ is

1. $2$
2. $3$
3. $4$
4. $5$

Q. If $f\left(x\right)=\{\begin{array}{cc}{x}^k\sin \left(\frac{1}{x}\right),& x\ne 0\\ 0,& x=0\end{array}$ is differentiable at $x=0,$ then (where $k$ is an integer)

1. $k>0$
2. $k>1$
3. $k\ge 1$
4. $k\ge 0$

Q. Let $g\left(x\right)=\{\begin{array}{cc}2(x+1),& -\infty <x\le -1\\ \sqrt{1-x^2},& -1<x<1\\ |||x|-1|-1|,& 1\le x<\infty \end{array}.$ Then

1. $g\left(x\right)$ is non-differentiable at exactly three points
2. $g\left(x\right)$ is continuous in $(-\infty ,1]$
3. $g\left(x\right)$ is differentiable in $(-\infty ,-1)$
4. $g\left(x\right)$ has finite type of discontinuity at $x=1$, but continuous at $x=-1$

Q. $\text{List I}$ has four entries and $\text{List II}$ has four entries. Each entry of $\text{List I}$ has a unique match in $\text{List II}.$

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{Let}f\text{be a real-valued differentiable}& \text{(P)}& 18\\ & \text{function on}R\text{such that}{f}^{\prime }\left(1\right)=6& & \\ & \text{and}{f}^{\prime }\left(2\right)=2.\text{Then}& & \\ & \underset{h\to 0}{lim}\frac{f(3\cos h+4\sin h-2)-f\left(1\right)}{f(3e^h-5\sec h+4)-f\left(2\right)}& & \\ & \text{is equal to}& & \\ \text{(B)}& \text{If}(0.5{)}^{{\log }_3{\log }_{1/5}(x^2-\frac{4}{5})}>1,\text{then the least}& \text{(Q)}& 4\\ & \text{positive integral value of}x\text{is}& & \\ \text{(C)}& \text{The value of}\underset{x\to 1}{lim}\frac{{\sin }^2(x^3+x^2+x-3)}{1-\cos (x^2-4x+3)}\text{is}& \text{(R)}& 5\\ \text{(D)}& \text{Number of points of non-differentiability of}& \text{(S)}& 2\\ & f\left(x\right)=min(|\sin x|,|\cos x|,\frac{1}{4})\text{in}(0,\pi )\text{is}& & \end{array}$

Which of the following is the only CORRECT combination?

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

Q. If f(a) = 2, f’(a) = 1, g(a) = –1, g’(a) = 2, then the value of $\underset{x\to a}{lim}\frac{g\left(x\right)f\left(a\right)-g\left(a\right)f\left(x\right)}{x-a}$ is

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

Q. Let $f:R\to R$ be differentiable at $c\in R$ and $f\left(c\right)=0$. If $g\left(x\right)=\left|f\right(x\left)\right|$, then at $x=c,g$ is :

1. differentiable if $f^{\prime }\left(c\right)\ne 0$
2. differentiable if $f^{\prime }\left(c\right)=0$
3. not differentiable if $f^{\prime }\left(c\right)=0$
4. not differentiable

Q. Let $f\left(x\right)=|\sin x|$. Then which of the following statement(s) is/are true?

1. f is everywhere differentiable
2. f is not differentiable at $x=n\pi ,n\in Z$.
3. f is everywhere continuous but not differentiable at $x=(2n+1)\frac{\pi }{2},n\in Z$
4. f is continuous everywhere

Q.

Let f(x+y) = f(x).f(y) for all x and y. Given that f(3) = 3 and f'(0)= 11. Then the value of f'(3) is ___ .

Q. If $f\left(x\right)=\{\begin{array}{c}1,x<0\\ 1+\sin x,0\le x<\frac{\pi }{2}\end{array}$, then at $x=0$ the derivative $f^{\prime }\left(x\right)$ is

1. $1$
2. $0$
3. infinite
4. not defined

Q. If $f\left(x\right)=\{\begin{array}{cc}{x}^k\sin \left(\frac{1}{x}\right),& x\ne 0\\ 0,& x=0\end{array}$ is differentiable at $x=0,$ then (where $k$ is an integer)

1. $k>0$
2. $k>1$
3. $k\ge 1$
4. $k\ge 0$

Q. Let $f\left(x\right)=\{\begin{array}{cc}x|x|;& -\infty \le x\le 1\\ 2x^3-1;& x>1\end{array}$

Choose the correct answer.

माना $f\left(x\right)=\{\begin{array}{cc}x|x|;& -\infty \le x\le 1\\ 2x^3-1;& x>1\end{array}$

सही उत्तर का चयन कीजिए।


f(x) is non-differentiable at
f(x) अन-अवकलनीय है

1. x = 1
   
   x = 1 पर
2. x = –1
   
   x = –1 पर
3. x = 0
   
   x = 0 पर
4. No value of x
   
   x के किसी भी मान पर नहीं

Q. If the function f(x) has LHD and RHD at $x_0$, which are finite and equal, then f(x) is differentiable at $x_0$, always.

1. True
2. False

Q.

$Rs15:Rs75=\_\_:200$