Let $$f(x)=2x+{\tan }^{-1}x$$ and $$g(x)={\log }_e(\sqrt{1+x^2}+x),x\in [0,3]$$. Then

Let $$y=f(x)={\sin }^3\left(\frac{\pi }{3}\left(\cos \left(\frac{\pi }{3\sqrt{2}}{\left(-4x^3+5x^2+1\right)}^{\frac{3}{2}}\right)\right)\right)$$. Then, at x = 1,

Let $$f$$ and $$g$$ be the twice differentiable functions on $$\mathbb{R}$$ such that $$f^{\prime \prime }(x)=g^{\prime \prime }(x)+6x$$ $$f^{\prime }(1)=4g^{\prime }(1)-3=9$$ $$f(2)=3g(2)=12$$. Then which of the following is NOT true?

Let $$y(x)=(1+x)(1+x^2)(1+x^4)(1+x^8)(1+x^{16})$$. Then $$y^{\prime }-y^{\prime \prime }$$ at $$x=-1$$ is equal to

If $$f(x)=x^3-x^2{f}^{\prime }(1)+x{f}^{\prime \prime }(2)-f^{\prime \prime \prime }(3),x\in \mathbb{R}$$, then

For the differentiable function $$f:\mathbb{R}-\{0\}\to \mathbb{R}$$, let $$3f(x)+2f\left(\frac{1}{x}\right)=\frac{1}{x}-10$$, then $$\left|f(3)+f^{\prime }\left(\frac{1}{4}\right)\right|$$ is equal to

Let $$f(x)=\frac{\sin x+\cos x-\sqrt{2}}{\sin x-\cos x},x\in [0,\pi ]-\left\{\frac{\pi }{4}\right\}$$. Then $$f\left(\frac{7\pi }{12}\right)f^{\prime \prime }\left(\frac{7\pi }{12}\right)$$ is equal to

If $$2x^y+3y^x=20$$, then $$\frac{dy}{dx}$$ at $$(2,2)$$ is equal to :

Let $$\alpha x=\exp \left(x^{\beta }{y}^{\gamma }\right)$$ be the solution of the differential equation $$2x^{2}y\mathrm{d}y-\left(1-x{y}^2\right)\mathrm{d}x=0,x>0,y(2)=\sqrt{{\log }_{e}2}$$. Then $$\alpha +\beta -\gamma$$ equals :

Let $y=y(x)$ be the solution of the differential equation $\left(3y^2-5x^2\right)y\mathrm{d}x+2x\left(x^2-y^2\right)\mathrm{d}y=0$ such that $y(1)=1$. Then $\left|(y(2){)}^3-12y(2)\right|$ is equal to :

The area enclosed by the closed curve $$\mathrm{C}$$ given by the differential equation $$\frac{dy}{dx}+\frac{x+a}{y-2}=0,y(1)=0$$ is $$4\pi$$. Let $$P$$ and $$Q$$ be the points of intersection of the curve $$\mathrm{C}$$ and the $$y$$-axis. If normals at $$P$$ and $$Q$$ on the curve $$\mathrm{C}$$ intersect $$x$$-axis at points $$R$$ and $$S$$ respectively, then the length of the line segment $$RS$$ is :

If $$y=y(x)$$ is the solution curve of the differential equation $$\frac{dy}{dx}+y\tan x=x\sec x,0\le x\le \frac{\pi }{3},y(0)=1$$, then $$y\left(\frac{\pi }{6}\right)$$ is equal to

Let a differentiable function $$f$$ satisfy f(x)+\int_\limits{3}^{x} \frac{f(t)}{t} d t=\sqrt{x+1}, x \geq 3. Then $$12f(8)$$ is equal to :

The solution of the differential equation $\frac{dy}{dx}=-\left(\frac{x^2+3y^2}{3x^2+y^2}\right),y(1)=0$ is :

Let the solution curve $$y=y(x)$$ of the differential equation $$\frac{\mathrm{d}y}{\mathrm{d}x}-\frac{3x^5{\tan }^{-1}\left(x^3\right)}{{\left(1+x^6\right)}^{3/2}}y=2x\exp \left\{\frac{x^3-{\tan }^{-1}{x}^3}{\sqrt{\left(1+x^6\right)}}\right\}\text{ pass through the origin. Then }y(1)\text{ is equal to : }$$

Let $$y=y(x)$$ be the solution of the differential equation x{\log _e}x{{dy} \over {dx}} + y = {x^2}{\log _e}x,(x > 1). If $$y(2)=2$$, then $$y(e)$$ is equal to

Let $$y=f(x)$$ be the solution of the differential equation $$y(x+1)dx-x^{2}dy=0,y(1)=e$$. Then $$\underset{x\to 0^{+}}{\lim }f(x)$$ is equal to

Let $$y=y(t)$$ be a solution of the differential equation {{dy} \over {dt}} + \alpha y = \gamma {e^{ - \beta t}} where, $$\alpha >0,\beta >0$$ and $$\gamma >0$$. Then $$\underset{t\to \infty }{\lim }y(t)$$

Let $$y=y(x)$$ be the solution curve of the differential equation {{dy} \over {dx}} = {y \over x}\left( {1 + x{y^2}(1 + {{\log }_e}x)} \right),x > 0,y(1) = 3. Then {{{y^2}(x)} \over 9} is equal to :

Let $$y=y(x)$$ be the solution of the differential equation $$(x^2-3y^2)dx+3xydy=0,y(1)=1$$. Then $$6y^2(e)$$ is equal to