If $f(x)$ is a polynomial of degree $n$ with rational coefficients and $1+2i, 2-\sqrt{3},$ and $5$ are three roots of $f(x)=0$,then the least value of $n$ is

The solution set of the equation $x^{\log_x(1-x)^2} = 9$ is

Arrange the following compounds in the increasing order of their dipole moment:
$(I)$ Toluene
$(II)$ $m$-Dichlorobenzene
$(III)$ $o$-Dichlorobenzene
$(IV)$ $p$-Dichlorobenzene

An electron makes a transition from orbit $n = 4$ to the orbit $n = 2$ of the hydrogen atom. The wave number of the emitted radiation ($R =$ Rydberg's constant) will be

The values of a function $f(x)$ at different values of $x$ are given in the table below:

$x$	$0$	$1$	$2$	$3$	$4$	$5$
$f(x)$	$2$	$3$	$6$	$11$	$18$	$27$

Then,the approximate area (in square units) bounded by the curve $y=f(x)$ and the $x$-axis between $x=0$ and $x=5$,using the Trapezoidal rule,is

$\int (1+x-x^{-1}) e^{x+x^{-1}} dx$ is equal to :