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Why can the variable be raisesd only to a whole number? Why can't it be raised to any real no.s?

If $\int \frac{1}{\sqrt{{\left(\log \frac{1}{2}\right)}^2-x^2}}dx=f\left(x\right)+c,$ then $f\left(x\right)$ equals to

A data consists of $n$ observations:

$x_1,x_2,....,x_n$. If $\sum _{i=1}^n(x_i+1{)}^2=9n$ and $\sum _{i=1}^n(x_i-1{)}^2=5n$, then the standard deviation of this data is :

If $\int \frac{dx}{{(1+\sqrt{x})}^{2010}}=\frac{2}{\alpha {(1+\sqrt{x})}^{\alpha }}-\frac{2}{\beta {(1+\sqrt{x})}^{\beta }}+C,$ where $C$ is arbitrary constant of integration and $\alpha ,\beta >0,$ then

If the order and degree of the differential equation satisfying $\sqrt{1-x^2}+\sqrt{1-y^2}=b(x-y),$ where $b$ is a parameter, is $\lambda$ and $\mu$ respectively, then $\lambda +\mu$ is

How many numbers can be formed with the digits, 1, 2, 3, 4, 3, 2, 1 so that the odd digits always occupy the odd places ?

If the radius of the circle $x^2+y^2+ax+(1-a)y+5=0$
does not exceed 5, write the number of integral values a.

The equation of curve satisfying differential equation $\frac{dy}{dx}=x\ln x$ and passing through the point $(e,1)$ is

Given $f\left(x\right)=\tan x$. Then $f^{\prime }\left(x\right)$ is equal to

If $y={\tan }^{-1}(\sec x-\tan x),x\in (0,\frac{\pi }{2}),$ then $\frac{dy}{dx}$ is equal to

If $f\left(x\right)$ is a quadratic polynomial with leading coefficient $1$ such that $\underset{x\to 0}{lim}f\left(x\right)=3=\underset{x\to 0}{lim}{f}^{\prime }\left(x\right)$, then $\underset{x\to 2}{lim}f\left(x\right)=$

$\left(f^{\prime }\right(x)=\frac{df\left(x\right)}{dx})$

If $x$ and $y$ co-ordinates of a point $P$ in the $xy$ plane are given by $x=\left(u\cos \alpha \right)t,y=\left(u\sin \alpha \right)t-\frac{1}{2}g{t}^2$ where $t$ is a parameter and $g,u$ and $\alpha$ are given constants. Then the locus of the point $P$ is a parabola whose vertex is

What is the value of |2i|??

Give reasons.

If $\alpha ,\beta$ and $\gamma$ are three consecutive terms of a non-constant G.P. such that the equations $\alpha x^2+2\beta x+\gamma =0$ and $x^2+x–1=0$ have a common root, then $\alpha (\beta +\gamma )$ is equal to :

Let $\alpha ,\beta$ are two real and different roots of a quadratic equation and the determinant $\Delta =\left|\begin{array}{ccc}4-\alpha & 3& 4-\beta \\ \alpha & 1& \beta \\ {\alpha }^3& 1& {\beta }^3\end{array}\right|$ is zero,$(\alpha ,\beta \ne 1).$ Then the quadratic equation can be

Find the coordinates of points on the parabola $y^2=8x$ whose focal distance is 4.

Let $A=\left[a_{ij}\right]$ and $B=\left[b_{ij}\right]$ be two $3\times 3$ real matrices such that $b_{ij}=(3{)}^{(i+j-2)}{a}_{ji}$, where $i,j=1,2,3$. If the determinant of $B$ is $81$, then the determinant of $A$ is :

The least value of $|z|$ where $z$ is complex number which satisfies the inequality $\exp \left(\frac{(|z|+3)(|z|-1)}{||z|+1|}{\log }_{e}2\right)\ge {\log }_{\sqrt{2}}|5\sqrt{7}+9i|,$ $i=\sqrt{-1},$ is equal to :

A paper contains $10$ questions with total marks of $40$ and solving each question is mandatory. In the paper, there are $3$ types of questions i.e., $2$ marks, $4$ marks and $6$ marks questions. If someone attempts the paper and got $38$ marks, then the probability that question paper has only $3$ question of $6$ marks is (there is no negative marking)

If $A=\left[\begin{array}{cc}1& \tan x\\ -\tan x& 1\end{array}\right]$, then $A^T{A}^{-1}$ is

If secθ = 2, such that θ lies in the fourth quadrant, then the value of sin 2θ is

A person wants to buy one fountain pen, one ball pen and one pencil from a stationery shop. If there are 10 fountain pen varieties, 12 ball pen varieties and 5 pencil varieties, in how many ways can he select these articles?

If $cotA+2tanA=tanB$ then $cotA$ is

find x if ${\log }_2(x-5)>3$

A triangle $ABC$ consists of vertex points $A(0,0),B(1,0)$ and $C(0,1)$. The value of the integral $\int \int 2xdxdy$ over the triangle is

If ${}^n{C}_{10}{=}^n{C}_{12}$, find ${}^{23}{C}_n$

The value of $1+\frac{4}{7}+\frac{9}{7^2}+\frac{16}{7^3}+\frac{25}{7^4}+\cdots \text{upto}\infty$ is

For non-zero distinct real numbers $a_1$ and $a_2$, let $f\left(x\right)=a_1{x}^2+b_{1}x+c_1,$$g\left(x\right)=a_2{x}^2+b_{2}x+c_2$ and $p\left(x\right)=f\left(x\right)-g\left(x\right)$. If $p\left(x\right)=0$ only at $x=-1$ and $p(-2)=2$, then the value of $p\left(2\right)$ is