If 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,where $\alpha^2 + \beta^2 = 61$,then the value of $|a|$ is

The curve $y=ax^3+bx^2+cx+5$ touches the $x$-axis at $(-2,0)$ and cuts the $y$-axis at a point $Q$ where its gradient is $3$. Then the value of $a+b+c$ is

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:

The abscissa of the point on the curve $\sqrt{xy} = a + x$ where the tangent cuts off equal intercepts from the coordinate axes is $(a > 0)$.

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.......

$V$ is the set of points on the curve $y^3 - 3xy + 2 = 0$ where the tangent is vertical. Then $V = $