Higher-order derivatives

Let $\mathbb R$ denote the set of all real numbers. Consider the polynomial function $f:\mathbb R\to\mathbb R$ defined by
$$f(x)=\frac{d^{10}}{dx^{10}}\left(\left(x^2-1\right)^{10}\right),\qquad\text{for all }x\in\mathbb R.$$
Here $\dfrac{d^{10}}{dx^{10}}\left(\left(x^2-1\right)^{10}\right)$ is the $10^{\text{th}}$ order derivative of the function $\left(x^2-1\right)^{10}$.

Then which of the following statements is (are) TRUE ?

Let $f$ be a real polynomial of degree $n$ such that $f(x)=f'(x)f''(x)$, for all $x\in\mathbb R$. If $f(0)=0$, then $36\left(f'(2)+f''(2)+\displaystyle\int_0^2f(x)\,dx\right)$ is equal to:

Let $f(x)$ and $g(x)$ be twice differentiable functions satisfying $f''(x)=g''(x)$ for all $x\in\mathbf R$, $f'(1)=2g'(1)=4$ and $g(2)=3f(2)=9$. Then $f(25)-g(25)$ is equal to :

If $y=\tan^{-1}\left(\sec x^3-\tan x^3\right)$, $\dfrac\pi2<x^3<\dfrac{3\pi}2$, then

If $\cos^{-1}\left(\dfrac y2\right)=\log_e\left(\dfrac x5\right)^5$, $|y|<2$, then :

If $\displaystyle\int\frac{\left(x^2+1\right)e^x}{(x+1)^2}dx=f(x)e^x+C$, where C is a constant, then $\dfrac{d^3f}{dx^3}$ at $x=1$ is equal to :

If $y(x)=\left(x^x\right)^x$, $x>0$, then $\dfrac{d^2x}{dy^2}+20$ at $x=1$ is equal to ______.

For the curve $C:\left(x^2+y^2-3\right)+\left(x^2-y^2-1\right)^5=0$, the value of $3y'-y^3y''$, at the point $(\alpha,\alpha)$, $\alpha>0$, on C, is equal to ______.

Let $x(t)=2\sqrt2\cos t\sqrt{\sin2t}$ and $y(t)=2\sqrt2\sin t\sqrt{\sin2t}$, $t\in\left(0,\dfrac\pi2\right)$. Then $\dfrac{1+\left(\frac{dy}{dx}\right)^2}{\frac{d^2y}{dx^2}}$ at $t=\dfrac\pi4$ is equal to :

If $y(x)=\tan x-\displaystyle\int_0^{2x}(x^2-xt+1)\sin t\,dt$, then at $x=\dfrac\pi3$, the value of $2y'''-y''$ is equal to

If $y=\left[x+\sqrt{x^2-1}\right]^{15}+\left[x-\sqrt{x^2-1}\right]^{15}$, then $(x^2-1)\dfrac{d^2y}{dx^2}+x\dfrac{dy}{dx}$ is equal to :

If $2x=y^{1/5}+y^{-1/5}$ and $(x^2-1)\dfrac{d^2y}{dx^2}+\lambda x\dfrac{dy}{dx}+ky=0$, then λ + k is equal to :

Match List I with List II and select the correct answer using the code given below the lists.

If $y=e^{nx}$, then $\left(\dfrac{d^2y}{dx^2}\right)\left(\dfrac{d^2x}{dy^2}\right)$ is equal to :

Let for $i=1,2,3$, $p_i(x)$ be a polynomial of degree 2 in $x$, $p_i'(x)$ and $p_i''(x)$ be the first and second order derivatives of $p_i(x)$ respectively. Let,
$$A(x)=\begin{bmatrix}p_1(x)&p_1'(x)&p_1''(x)\\p_2(x)&p_2'(x)&p_2''(x)\\p_3(x)&p_3'(x)&p_3''(x)\end{bmatrix}$$
and $B(x)=[A(x)]^TA(x)$. Then determinant of $B(x)$ :

Which of the following is true for $0<x<1$?

Let $f$ and $g$ be real valued functions defined on interval $(-1,1)$ such that $g''(x)$ is continuous, $g(0)\ne0$, $g'(0)=0$, $g''(0)\ne0$, and $f(x)=g(x)\sin x$.

STATEMENT-1: $\displaystyle\lim_{x\to0}\left[g(x)\cot x-g(0)\operatorname{cosec}x\right]=f''(0)$.

and

STATEMENT-2: $f'(0)=g(0)$.

If $f(-10\sqrt2)=2\sqrt2$, then $f''(-10\sqrt2)=$

Let $g(x)=\log f(x)$ where $f(x)$ is a twice differentiable positive function on $(0,\infty)$ such that $f(x+1)=xf(x)$. Then, for $N=1,2,3,\ldots$,
$$g''\left(N+\frac12\right)-g''\left(\frac12\right)=$$

Which of the following is true?