The graph of an exponential function $y = 2 + ae^{-x}$ is shown in the figure. What is the value of $a$?

If $\angle P - \angle Q = 50^{\circ}$,then find the values of $\angle P$ and $\angle Q$.

The value of $\frac{d}{dx}(\log_e x)$ is:

In the given figure,each box represents a function machine. $A$ function machine illustrates what it does with the input. Which of the following statements is correct?

$\int \frac{dx}{{(2ax - x^2)}^{1/2}} = a^n \sin^{-1} \left( \frac{x}{a} - 1 \right)$. In this formula,$n =$ . . . . . . .

$\frac{d}{dx}(\sin 30^\circ)$ is equal to