If the slope of the tangent to the curve $y=ax^3+bx+4$ at the point $(2, 14)$ is $21$, then the values of $a$ and $b$ are respectively:

If the tangent drawn at $A(2,1)$ to the curve $x=1+\frac{1}{y^2}$ meets the curve again at $B$,then

Let $n \in (0, \infty)$. If all the curves $y = x^n \log x$ for distinct values of $n$ always have $y = x - 1$ as the tangent drawn at a fixed point $(\alpha, \beta)$,then $\alpha + \beta =$

The curve $y=x^3-2x^2+3x-4$ intersects the horizontal line $y=-2$ at the point $P(h, k)$. If the tangent drawn to this curve at $P$ meets the $X$-axis at $(x_1, y_1)$, then $x_1=$

Let $\ell$ be a line which is normal to the curve $y=2x^2+x+2$ at a point $P$ on the curve. If the point $Q(6,4)$ lies on the line $\ell$ and $O$ is the origin,then the area of the triangle $OPQ$ is equal to.......

If the lengths of the tangent,subtangent,normal and subnormal for the curve $y=x^2+x-1$ at the point $(1,1)$ are $a, b, c$ and $d$ respectively,then their increasing order is