If $y=e^{-x}\cos xan{dy}_n+k_{n}y=0,where{y}_n=\frac{d^{n}y}{{dx}^n}and{k}_n$ are constants $\forall n\in N,$ then (a)$k_4=4$ (b) $k_8=-16$ (c)$k_{12}=20$ (d) $k_{16}=-24$

Let $g\left(x\right)=f\left(x\right)\sin x,wheref\left(x\right)$ is a twice differentiable function on $\left(-\infty ,\infty \right)$ such that $f\left(-\pi \right)=1.$ The value of $\left|g^{-\pi }\right|$ equals __________

If $y=\frac{x^2}{2}+\frac{1}{2}x\sqrt{x^2+1}+{\left(\log \right)}_e\sqrt{x+\sqrt{x^2+1}}$ , prove that $2y=x{y}^{prime}+{\left(\log \right)}_e{y}^{prime},where{y}^{\prime }$ denotes the derivative w.r.t $x\dot{}$

If $y=x^{{\left(\log x\right)}^{\text{log}\left(\log \dot{x}\right)}},then\frac{dy}{dx}is$ (a)$\frac{y}{x}\left(1n{x}^{\infty x-1}\right)+21nx1n\left(1nx\right))$ (b)$\frac{y}{x}{\left(\log x\right)}^{\text{log}\left(\log x\right)}\left(2\log \left(\log x\right)+1\right)$ (c)$\frac{y}{x1nx}\left[{\left(1nx\right)}^2+21n\left(1nx\right)\right]$ (d)$\frac{y}{x}\frac{\log y}{\log x}\left[2\log \left(\log x\right)+1\right]$

A nonzero polynomial with real coefficient has the property that $f\left(x\right)=f^{prime}\left(x\right){\dot{f}}^{primeprime}\left(x\right)\dot{}$ If $a$ is the leading coefficient of $f\left(x\right),$ then the value of $\frac{1}{2a}$ is____

Let $f\left(x+y\right)=f\left(x\right)+f\left(y\right)+2xy-1$ for all real $xandy$ and $f\left(x\right)$ be a differentiable function. If $f^{prime}\left(0\right)=\cos \alpha ,$ the prove that $f\left(x\right)>0\forall x\in R\dot{}$

${\left(\text{lim}\right)}_{x\vec{0}}\frac{x^a{\sin }^{b}x}{\sin \left(x^c\right)}$, where a , b , c in R ~{0},exists and has non-zero value. Then, $a+c$ = (a) b (b) $-1$ $0$ (d) none of these

If $f\left(x\right)=\{\begin{array}{cc}\sin x& x\ne n\pi \text{ and }n\in I_2\\ 2& x=n\pi \end{array}$ and $g\left(x\right)=\{\begin{array}{cc}{x}^2+1& x\ne 0\\ 4& x=0\\ 5& x=2\end{array}$ then $\underset{x\to 0}{\lim }g\left\{f\left(x\right)\right\}$ is

If $x=t^2,y=t^3,then\frac{d^{2}y}{{dx}^2}=$ (a)$\frac{3}{2}$ (b) $\frac{3}{\left(4t\right)}$ (c) $\frac{3}{2\left(t\right)}$ (d) $\frac{3t}{2}$

Let $f\left(x\right)=\{\underset{n\to \infty }{\lim }\frac{ax\left(x-1\right){\left(\frac{\cot \left(\pi x\right)}{4}\right)}^n+\left(p{x}^2+2\right)}{{\left(\frac{\cot \left(\pi x\right)}{4}\right)}^n+1},x\in \left(0,1\right)\cup \left(1,2\right)\text{and}0,x=1$ If f(x) is differentiable for all $x\in \left(0,2\right)$ then $\left(a^2+p^2\right)$ equals