The number of real roots of the equation $\log_{e} x + ex = 0$ is

Let $a_1, a_2, \ldots, a_{100}$ be non-zero real numbers such that $a_1+a_2+\ldots+a_{100}=0$. Then,

Using the fact that $\sin (A+B)=\sin A \cos B+\cos A \sin B$ and the differentiation,obtain the sum formula for cosines.

The function defined by $f(x) = \begin{cases} |x - 3|, & x \ge 1 \\ \frac{1}{4}x^2 - \frac{3}{2}x + \frac{13}{4}, & x < 1 \end{cases}$ is

If $y = \operatorname{Tan}^{-1}\left(\frac{3x - x^3}{1 - 3x^2}\right) + \operatorname{Tan}^{-1}\left(\frac{7x}{1 - 12x^2}\right)$,then at $x = 0$,$\frac{dy}{dx} = $

Let $C$ be the curve $y = x^3$ (where $x$ takes all real values). The tangent at $A(t, t^3)$ meets the curve again at $B(T, T^3)$. If the gradient at $B$ is $K$ times the gradient at $A$,then $K$ is equal to