Tangents and it...

If the tangent at $$(1, 1)$$ on  $$\displaystyle y^{2}= x\left ( 2-x^{2} \right )$$ meets the curve again at $$P$$, then $$P$$ is

Tangent is drawn to ellipse $$\displaystyle \frac{x^{2}}{27}+y^{2}=1$$ at $$\left ( 3\sqrt{3}\cos \theta ,\sin \theta  \right )$$ (where $$\theta \in \left ( 0,\dfrac{\pi}2 \right )).$$ Then the value of $$\theta$$ such that sum of intercepts on axes made by this tangent is least is

The tangent to the curve $$\displaystyle 3xy^{2}-2x^{2}y=1$$  at $$(1, 1)$$ meets the curve again at the point

The angle at which the curve $$y=ke^{kx}$$ intersects the $$y$$ -axis is

Let $$\displaystyle f\left ( x \right )=x^{3}+ax+b$$ with $$\displaystyle a\neq b$$ and suppose the tangent lines to the graph of $$f$$ at $$x = a$$ and $$x = b$$ have the same gradient Then the value of $$f (1)$$ is equal to

If a variable tangent to the curve $$\displaystyle x^{2}y=c^{3}$$ makes intercepts $$a, b$$ on $$x$$ and $$y$$ axis respectively then the value of $$\displaystyle x^{2}$$ is

A curve with equation of the form $$\displaystyle y=ax^{4}+bx^{3}+cx+d$$ has zero gradient at the point (0, 1) and also touches the x-axis at the point (-1, 0) then the values of x for which the curve has a negative gradient are

The slope of the tangent to the curve represented by $$x= t^{2}+3t-8$$ and $$y= 2t^{2}-2t-5$$ at the point $$M\left ( 2,-1 \right )$$ is

The curve $$y= ax^{3}+bx^{2}+cx+8$$  touches $$x$$-axis at $$P\left ( -2,0 \right )$$ and cuts the $$y$$-axis at a point $$Q(0,8)$$ where its gradient is 3. The values of $$a$$, $$b$$, $$c$$ are respectively

The number of values of c such that the straight line $$3x + 4y = c$$ touches the curve $$\displaystyle \frac{x^{4}}{2}=x+y$$ is