If $\frac{dy}{dx} = (x - 1)^3 (x - 2)^4$,then $y$ has:

For $a > 0$,if the function $f(x) = 2x^3 - 9ax^2 + 12a^2x + 1$ attains its maximum value at $p$ and minimum value at $q$ such that $p^2 = q$,then $a =$

The lateral edge of a regular rectangular pyramid is $a \text{ cm}$ long. The lateral edge makes an angle $\alpha$ with the plane of the base. The value of $\alpha$ for which the volume of the pyramid is greatest,is

The number of points in $(-\infty, \infty)$ for which $x^2 - x \sin x - \cos x = 0$ is

For every twice differentiable function $f : R \rightarrow [-2, 2]$ with $(f(0))^2 + (f'(0))^2 = 85$,which of the following statement$(s)$ is (are) $TRUE$?
$(A)$ There exist $r, s \in R$,where $r < s$,such that $f$ is one-one on the open interval $(r, s)$.
$(B)$ There exists $x_0 \in (-4, 0)$ such that $|f'(x_0)| \leq 1$.
$(C)$ $\lim_{x \rightarrow \infty} f(x) = 1$.
$(D)$ There exists $a \in (-4, 4)$ such that $f(a) + f''(a) = 0$ and $f'(a) \neq 0$.

If the function $f(x) = x^4 - 62x^2 + ax + 9$ is maximum at $x = 1$,then the value of $a$ is