If the tangent of the curve $4 y^3=3 a x^2+x^3$ drawn at the point $(a, a)$ forms a triangle of area $\frac{25}{24}$ sq. units with the coordinates axes, then $a=$

If the function $f(x)=\sin x-\cos ^2 x$ is defined on the interval $[-\pi, \pi]$, then $f$ is strictly increasing in the interval

If the Lagrange' mean value theorem is applied to the function $f(x)=e^x$ defined on the interval $[1,2]$ and the value of $c \in(1,2)$ is $k$, then $e^{k-1}=$

Consider the quadratic equation $a x^2+b x+c=0$, where $2 a+3 b+6 c=0$ and let $g(x)=\frac{a x^3}{3}+\frac{b x^2}{2}+c x$

Statement I The given quadratic equation $a x^2+b x+c=0$ has atleast one root in $(0,1)$.

Statement II Rolle's theorem is applicable to $g(x){\text {on }}$ [0, 1].

Then