Application of ...

If $$m$$ be the slope of a tangent to the curve $${ e }^{ 2y }=1+4{ x }^{ 2 }$$, then 

The slope of the tangent to the curve $$y=x^{2}-x$$ at the point where the line $$y=2$$ cuts the curve in the first quadrant is

The slope of the tangent to the locus $$y=\cos^{-1}\left ( \cos x \right )$$ at $$x=\displaystyle \frac{\pi }{4}$$ is

If at each point of the curve $$y=x^{3}-ax^{2}+x+1$$ the tangent is inclined at an acute angle with the positive direction of the x-axis then

The slope of the tangent to the curve $$y=\sqrt{4-x^{2}}$$ at the point where the ordinate and the abscissa are equal, is

The equation of the curve is given by $$x=e^{t}\sin t$$, $$y=e^{t}\cos t$$. The inclination of the tangent to the curve at the point $$t=\displaystyle \frac{\pi }{4}$$ is

The function $$\: f\left( x \right) =x^{ 3 }+\lambda x^{ 2 }+5x+\sin  2x$$ will be an invertible function if $$\: \lambda $$ belongs to

The curve given by $$x+y=e^{xy}$$ has a tangent as the $$y$$-axis at the point

Let $$\displaystyle f(x)=e\:^{x}\sin x$$ be the equation of a curve. If at $$\displaystyle x=a,0\leq a\leq 2\pi$$, the slope of the tangent is the maximum then the value of $$a$$ is 

If $$m$$ be the slope of tangent to the curve $$e^{y}=1+x^{2}$$ then