Application of ...

A point on the curve $$y = 2{x^3} + 13{x^2} + 5x + 9$$, the tangent at which passes through the origin is 

The value of n for which the length of the sub-normal at any point of the curve $$y^3= a^{1-n}x^{2n}$$ must be constant, is

The value of 'a' for which the function $$f\left( x \right) = \left( {a + 2} \right){x^3} - 3a{x^2} + 9ax - 1$$ decreases for all real values of x is

If the tangent at any point on the curve $$x^{4} +y^{4}=a^{4}$$ cuts off intercepts $$p$$ and $$q$$ on the coordinate axes the value of $$p^{-4/3}+q^{-4/3}$$ is

The slope of the tangent to the curve at a point $$(x,y) $$ on it is proportional to $$(x-2).$$ If the slope of the tangent to the curve at $$(10,-9)$$  on it is $$-3$$. The equation of the curves is .

At any two points of the curve represented parametrically by $$x = a\left( {2\cos t - \cos 2t} \right);y = a\left( {2\sin t - \sin 2t} \right)$$ the tangent are parallel to the axis of $$x$$ corresponding to the values of the parameter $$1$$ differing from each other by

Given $$P(x)=x^4+ax^3+bx^2+cx+d$$ such that $$x=0$$ is the only real root of $$P(x)=0$$. If $$P(-1) < P(1)$$, then in the interval $$[-1, 1]$$.

Find the slope of tangent of the curve$$x = a\,{\sin ^3}t,y = b\,\,{\cos ^3}t$$ at $$t = \frac{\pi }{2}$$

The set of all values of $$a$$ for which $$ f\left( x \right) =\left( { a }^{ 2 }-3a+2 \right) \left( \cos ^{ 2 }{ \dfrac { x }{ 4 } -\sin ^{ 2 }{ \dfrac { x }{ 4 }  }  }  \right) +\left( a-1 \right) x+\sin { 1 }$$ does not possess critical points is

Let $$f(x) = \underset{x}{\overset{x + \dfrac{\pi}{3}}{\int}} |\sin \, \theta | \, d \theta \, \, (x \in [0, \pi])$$