Application of ...

For the curve $$y=3  \sin \theta  \cos  \theta,  x= e^{\theta} \sin \theta,  0  \leq \theta  \leq  \pi$$, the tangent is parallel to x-axis when $$\theta$$ is :

The tangent at the point $$(2, -2)$$ to the curve, $$x^2y^2-2x=4(1-y)$$ does not pass through the point.

If the tangent at $$(1, 7)$$ to the curve $$x^{2} = y - 6$$ touches the circle $$x^{2} + y^{2} + 16x + 12y + c = 0$$ then the value of $$c$$ is

If the tangent to the conic, $$y - 6 = x^2$$ at (2, 10) touches the circle, $$x^2 + y^2 + 8x - 2y = k$$ (for some fixed k) at a point $$(\alpha, \beta)$$; then $$(\alpha, \beta)$$ is;

The set of all values of a for which the function $$f(x) = (a^{2} =- 3a + 2)(\cos^{2}x / 4 - \sin^{2}x / 4) + (a - 1)x + \sin 1$$ does not process critical points is

What is the $$x$$-coordinate of the point on the curve $$f(x) = \sqrt {x}(7x - 6)$$, where the tangent is parallel to $$x$$-axis?

Consider the following statements in respect of the function $$f(x) = x^{3} - 1, \quad x\epsilon [-1, 1]$$
I. $$f(x)$$ is increasing in $$[-1, 1]$$
II. $$f'(x)$$ has no root in $$(-1, 1)$$.
Which of the statements given above is/ are correct?

If $$\dfrac{x^2}{f(4a)}=\dfrac{y^2}{f(a^2-5)}$$ respresents and ellipse with major axis as y-axis and $$f$$ is a decreasing function, then 

The values of $$\mathrm{x}$$ at which $$\mathrm{f}(\mathrm{x})=\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{x}$$ is stationary are given by

The number of stationary points of $$\mathrm{f}(\mathrm{x})=\mathrm{s}\mathrm{i}\mathrm{n} \mathrm{x}$$ in $$[0, 2{\pi}]$$ are