Tangents and it...

lf the chord joining the points where $$x= p,\ x =q$$ on the curve $$y=ax^{2}+bx+c$$ is parallel to the tangent drawn to the curve at $$(\alpha, \beta)$$ then $$\alpha=$$

The arrangement of the following curves in the ascending order of slopes of their tangents at the given points.
$$A) \displaystyle y=\frac{1}{1+x^{2}}$$ at $$x=0$$

$$B) y=2e^{\frac{-x}{4}},$$ where it cuts the y-axis
$$C) y= cos(x)$$ at $$\displaystyle x=\frac{-\pi}{4}$$
$$D) y=4x^{2}$$ at $$x=-1$$

Observe the following lists for the curve $$y=6+x-x^{2}$$ with the slopes of tangents at the given points; I, II, III, IV

lf the tangent to the curve $$2y^{3}=ax^{2}+x^{3}$$ at the point $$(a,\ a)$$ cuts off intercepts $$\alpha$$ and $$\beta$$ on the coordinate axes such that $$\alpha^{2}+\beta^{2}=61$$, then $$a$$ is equal to

Match List-I with List-II and select the correct answer using the code given below. A B C D

Area of the triangle formed by the tangent, normal at $$(1, 1)$$ on the curve $$\sqrt{x}+\sqrt{y}=2$$ and the y axis is (in sq. units)

Assertion (A): The points on the curve $$y=x^{3}-3x$$ at which the tangent is parallel to $$x$$-axis are $$(1, -2)$$ and $$(-1, 2).$$
Reason (R): The tangent at $$(x_{1}, y_{1})$$ on the curve $$y=f(x)$$ is vertical then $$\displaystyle \frac{dy}{dx}$$ at $$(x_{1}, y_{1})$$ is not defined.

lf the tangent to the curve $$f(x)=x^{2}$$ at any point $$(c, f(c))$$ is parallel to the line joining points $$(a, f(a))$$ and $$(b,f(b))$$ on the curvel then $$a,\ c,\ b$$ are in

I. lf the curve $$y=x^{2}+ bx +c$$ touches the straight line $$y=x$$ at the point $$(1, 1)$$ then $$b$$ and $$c$$ are given by $$1, 1.$$
II. lf the line $$Px+ my +n=0$$ is a normal to the curve $$xy=1$$, then $$P> 0,\ m <0$$.
Which of the above statements is correct

lf the parametric equation of a curve given by $$x=e^{t}\cos t,\ y=e^{t}\sin t$$, then the tangent to the curve at the point $$t=\dfrac{\pi}{4}$$ makes with axis of $$x$$ the angle.