Differentiability

Q. The function $f\left(x\right)=(\begin{array}{c}{x}^2+3x+a,\mathrm{if}x\le 1\\ \mathrm{bx}+2,\mathrm{if}x>1\end{array}$
is differentiable at each

$x\in R$. Then, the value of a is and b is .

Q.

If $\frac{\log \left(x\right)}{b-c}=\frac{\log \left(y\right)}{c-a}=\frac{\log \left(z\right)}{a-b}$, then which of the following is true?

1. $xyz=1$

2. $x^a{y}^b{z}^c=1$

3. $xyz=x^a{y}^b{z}^c$

4. None of these

Q.

If $y=2^{\frac{1}{{\log }_x\left(4\right)}}$, then $x$ is equal to

1. $y$

2. $y^2$

3. $y^3$

4. None of these

Q. Let $A_1,A_2,....A_n$ be the vertices of an $n-$sided regular polygon such that $\frac{1}{A_1{A}_2}=\frac{1}{A_1{A}_3}+\frac{1}{A_1{A}_4}$. Then the value of $n$ is:

1. $7$
2. $8$
3. $9$
4. $10$

Q.

The set of all points, where the function $f\left(x\right)=\frac{x}{1+|x|}$ is differentiable, is

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

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. Find the value of cot(7.5)°

Q. Let $f\left(x\right)=\frac{\sqrt{x^2+kx+1}}{x^2-k}$ is continuous for every $x\in R$ then set of values of $k$ is

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

Q.

Let $T_n$ denote the number of triangles which can be formed by using the vertices of a regular polygon of $n$ sides. If $T_{n+1}-T_n=21$, then $n$ equals

1. $5$

2. $7$

3. $6$

4. $4$

Q. For every twice differentiable function $f:R\to [-2,2]$ with $\left(f\right(0){)}^2+(f^{\prime }\left(0\right){)}^2=85$, which of the following statement(s) is (are) TRUE?

1. There exist $r,s\in R$, where $r<s$, such that $f$ is one-one on the open interval $(r,s)$
2. There exists $x_0\in (-4,0)$ such that $|f^{\prime }\left(x_0\right)|\le 1$
3. $\underset{x\to \infty }{lim}f\left(x\right)=1$
4. There exists $\alpha \in (-4,4)$ such that $f\left(\alpha \right)+f^{\prime \prime }\left(\alpha \right)=0$ and $f^{\prime }\left(\alpha \right)\ne 0$

Q.

Find the value of $\log \left(1\right)+\log \left(2\right)+\log \left(3\right)$

Q.

If ${\log }_5\left(a\right).{\log }_a\left(x\right)=2$, then $x$ is equal to

1. $5$

2. $125$

3. $25$

4. None of these

Q.

Show that $f\left(x\right)=|x-5|$ is continuous but not differentiable at x=5

Q. If $f\left(x\right)=\text{sgn}({\sin }^{2}x-\sin x-1)$ has exactly four points of discontinuity for $x\in (0,n\pi ),n\in N,$ then

1. Minimum value of $n$ is $5.$
2. Maximum value of $n$ is $6.$
3. There are exactly two possible values of $n$
4. Minimum value of $n$ is $7$

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

Q. Let $f:R\to R$ and $g:R\to R$ be respectively given by $f\left(x\right)=|x|+1$ and $g\left(x\right)=x^2+1$. Define $h:R\to R$ by
$h\left(x\right)=\{\begin{array}{c}max\left\{f\right(x),g(x\left)\right\},x\le 0\\ min\left\{f\right(x),g(x\left)\right\},x>0.\end{array}$
The number of points at which $h\left(x\right)$ is not differentiable is

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. Prove that the function f(x) = log (2x-1) notderivable at x=0.

Q. $\left|\begin{array}{ccc}1& 1& 1\\ cos\left(nx\right)& cos(n+1)x& cos(n+2)x\\ sin\left(nx\right)& sin(n+1)x& sin(n+2)x\end{array}\right|$ does not depend

1. Either on x or on n
2. On n
3. On x
4. Both on x and n

Q. Let $I=\int \frac{{\sin }^{2}x+\sin x}{1+\sin x+\cos x}dx$, $J=\int \frac{{\cos }^{2}x+\cos x}{1+\sin x+\cos x}dx$ and $c$ is the constant of integration.

$$\begin{array}{cc}\text{Function}& \text{Integral}\\ \begin{array}{c}\text{(a) I}\end{array}& \text{(p)}\frac{1}{2}(x-\sin x-\cos x)+c\\ \begin{array}{c}\text{(b) J}\end{array}& \text{(q)}\frac{1}{2}(x+\sin x+\cos x)+c\\ \begin{array}{c}\text{(c) I + J}\end{array}& \text{(r)}x+c\\ \begin{array}{c}\text{(d) I - J}\end{array}& \text{(s)}c-\cos x-\sin x\\ \begin{array}{}\end{array}& \text{(t)}c+\cos x+\sin x\\ \begin{array}{}\end{array}& \text{(u)}-\frac{1}{2}(x+\sin x+\cos x+c)\end{array}$$

Then the value of $\frac{d(I+J)}{dx}$ at $x=\sqrt{2}$ is

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

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. If f(x)= \vert x - 2020\vert, then derivative of f(x) at x = 2019

Q. if log2 (x2 - 3x +2)= log2|x-1)|+ log2(|x-2)| find x ( detailed explained steps)

Q.

The domain of the function log2log3log4(x) is

Q. Number of points where $f\left(x\right)=\left|x\text{sgn}(1-x^2)\right|$ is non-differentiable is

Q.

Area (in sq units) of the triangle formed by the lines $y=2x,y=3x$ and $y=5$ is

1. $\frac{25}{6}$

2. $\frac{25}{12}$

3. $\frac{5}{6}$

4. $\frac{17}{12}$

5. $6$

Q. Let $A$ and $B$ be two $3\times 3$ invertible matrices such that $A$ is an idempotent matrix and
$det\left(adjB\right)=det(A{)}^{12}+det(A-A^T{)}^{13}+det\left(adj\right(A)-I{)}^{14}+L,$
where $L=\underset{x\to 0}{lim}\frac{\underset{0}{\overset{{\sin }^{2}x}{\int }}\ln (1+t^3)dt}{\underset{0}{\overset{x}{\int }}(e^t-1-t)(\tan t-t)(1-\cos t)dt}.$
The absolute value of $det\left(B\right)$ is

Q. If $f\left(x\right)=x^3+3x^2+4x+b\sin x+c\cos x\forall x\in R$ is a one-one function, then the maximum value of $(b^2+c^2)$ is

Q.

If $f\left(x\right)=\frac{x^n-a^n}{x-a}$, then $f^{\prime }\left(a\right)$ is

1. 0

2. does not exist

3. 1

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

Q. Sin20°sin40° sin60°. Sin80° =3/10