If a tangent to the curve $y = 6x - x^2$ is parallel to the line $4x - 2y - 1 = 0$,then the point of tangency on the curve is

Find the angle of intersection of the curves $y^{2}=x$ and $x^{2}=y$.

At the point $P(a, a^n)$ on the graph of $y = x^n$ $(n \in N)$ in the first quadrant,a normal is drawn. The normal intersects the $y-$ axis at the point $(0, b)$. If $\mathop {Lim}\limits_{a \to 0} b = \frac{1}{2}$,then $n$ equals

An angle between the curves $x^2 y = 1$ and $y(x^2 + 1) = 2$ is

Let $f: R \rightarrow R$ be a bijection. $A$ curve represented by $y=f(x)$ is such that $f^{\prime}(x)>0$ for all $x \in R$. The tangent and normal drawn at $P(\alpha, 1)$ on the curve cut the $X$-axis at $A$ and $B$ respectively, and $C$ is the foot of the perpendicular from $P$ onto the $X$-axis. If $P(\alpha, 1)$ is such a point that $AC+CB$ is minimum, then the tangent at $P$ is parallel to the line

The curve $y=ax^3+bx^2+cx+5$ touches the $X$-axis at $P(-2,0)$ and cuts the $Y$-axis at a point $Q$,where its gradient is $3$. Then: