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If the function $f:(1,\infty )\to (1,\infty )$ is defined by $f\left(x\right)=2^{x(x-1)}$ , then $f^{-1}\left(x\right)$ is

On which of the following intervals is the function f (x) = x¹⁰⁰ + sin x – 1 not increasing ?

Add vectors $\overrightarrow{A},\overrightarrow{B}$ and $\overrightarrow{C}$ each having magnitude of 100 unit and inclined to the X-axis at angles ${45}^0,{135}^0$ and ${315}^0$ respectively.

A die is tossed thrice. Find the probability of getting an odd number atleast once.

Calculate
the wavelength of an electron moving with a velocity of 2.05 × 10⁷
ms^–1.

Let $f\left(x\right)$ and $g\left(x\right)$ be two differentiable functions in $R$ and $f\left(2\right)=8,g\left(2\right)=0,f\left(4\right)=10$ and $g\left(4\right)=8$ for at least one $x\in (2,4),$ $g^{\prime }\left(x\right)=$

If the distance between the foci of an ellipse is 6 and the length of the minor axis is 8, then the eccentricity Is

If $\underset{x\to 0}{lim}[1+x\ln (1+b^2){]}^{\frac{1}{x}}=2b{\sin }^2\theta ,b>0$ and $\theta \in (-\pi ,\pi ),$ then which of the following options(s) is/are correct:

If $y={\log }_{\cos x}\left(\tan x\right)$, then $\frac{dy}{dx}{|}_{x=\frac{\pi }{4}}$ is equal to

$\int \frac{x^2+x+1}{(x+1{)}^2(x+2)}dx.$

$\text{If sum of the roots of the equation}(a+1)x^2-(2a+3)x+(3a+4)=0is-2,\text{then the product of the roots is}$

What is the orthogonal trajectory of $x^2-y^2=c$?

Let $△=\left|\begin{array}{ccc}\sin n\pi & \cos n\pi & 2\\ \cos 2x& \cos x& \sin x\\ \tan n\pi & -1& 2\cos \left(2n\pi \right)\end{array}\right|$ and $△=0\forall x\in R.$ Then the value of $n$ can be

Find the value of $tan\frac{1}{2}[si{n}^{-1}\frac{2x}{1+x^2}+co{s}^{-1}\frac{1-y^2}{1+y^2}]$

$|x|<1,y>0andxy<1$

The equation(s) of lines(s) belonging to the family of lines $(\lambda +1)x+(\lambda –2)y+3=0$, (where $\lambda$ is a parameter) and making an angle ${45}^{\circ }$ with the line $3x–4y–5=0$ is/are

Construct
a 3 ×
4 matrix, whose elements are given by

(i) (ii)

If ∝, β are the roots of the equation $x^2$ - ax + b = 0, then ${\alpha }^4$+${\alpha }^3$β+${\alpha }^2$${\beta }^2$+α${\beta }^3$+${\beta }^4$

If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are unit vectors such that $\overrightarrow{a}.\overrightarrow{b}=0$ and $(\overrightarrow{a}-\overrightarrow{c}).(\overrightarrow{b}+\overrightarrow{c})=0$ and $\overrightarrow{c}=\lambda \overrightarrow{a}+\mu \overrightarrow{b}+\omega (\overrightarrow{a}\times \overrightarrow{b})$ for some scalars $\lambda ,\mu ,\omega$, then

Let $I=\int \frac{e^x}{e^{4x}+e^{2x}+1}dx,J=\int \frac{e^{-x}}{e^{-4x}+e^{-2x}+1}dx.$ Then the value for $J-I$ equals to

( where $C$ is integration constant)

Let f(x) = x + sin x. The area bounded by $y=f^{-1}\left(x\right),y=x,xϵ[0,\pi ]$ is

Let the system of linear equations $\left[\begin{array}{ccc}1& \lambda & {\lambda }^2+5\\ 1& \gamma & {\gamma }^2+5\\ 1& 5& 30\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}2\\ 4\\ -1\end{array}\right]$ has a unique solution. Then which of the following is TRUE?

Let $△=\left|\begin{array}{ccc}f& -g& -h\\ -a& b& -c\\ -x& y& -z\end{array}\right|,$ then which of the following statement(s) is(are) correct?

(where $M_{ij}$ and $C_{ij}$ are minor and co-factor of element $a_{ij}$)

The value of $\left(\begin{array}{c}30\\ 0\end{array}\right)$$\left(\begin{array}{c}30\\ 10\end{array}\right)$ - $\left(\begin{array}{c}30\\ 1\end{array}\right)$$\left(\begin{array}{c}30\\ 11\end{array}\right)$

+ $\left(\begin{array}{c}30\\ 2\end{array}\right)$ $\left(\begin{array}{c}30\\ 12\end{array}\right)$ + ..............+ $\left(\begin{array}{c}30\\ 20\end{array}\right)$ $\left(\begin{array}{c}30\\ 30\end{array}\right)$

The value of ${[\tan \{\frac{\pi }{4}+\frac{1}{2}{\sin }^{-1}\left(\frac{a}{b}\right)\}+\tan \{\frac{\pi }{4}-\frac{1}{2}{\sin }^{-1}\left(\frac{a}{b}\right)\}]}^{-1},$ where $0<a<b,$ is

The vertex of an equilateral triangle is $(2,-1)$ and the equation of its base is $x+2y=1.$ The length of its side is

If $x_0$ is the solution of the equation $2^{1+{\left({\log }_{2}x\right)}^2}+(x^{{\log }_{2}x}{)}^2=3,$ then the value of ${\sin }^{-1}\left(x_0\right)+{\tan }^{-1}\left(\frac{2x_0}{2-(x_0{)}^2}\right)+{\cot }^{-1}\left(2\right)$ equals

If $f\left(x\right)$ is a real valued function defined by $f\left(x\right)=x^2+x^2\underset{-1}{\overset{1}{\int }}tf\left(t\right)dt+x^3\underset{-1}{\overset{1}{\int }}f\left(t\right)dt$ then which of the following hold(s) good