The tangent to the curve $y=x^3+ax-b$ at the point $(1,-5)$ is perpendicular to the line $y-x+4=0$. Which one of the following points lies on the curve?

The equation of the tangent to the curve $y = x + \frac{4}{x^2}$ which is parallel to the $X$-axis is:

An angle between the curves $x^2-y^2=4$ and $x^2+y^2=4 \sqrt{2}$ is

The angle $\theta$,at which the curves $y=3^x$ and $y=7^x$ intersect,is given by

Let the quadratic curve passing through the point $(-1, 0)$ and touching the line $y = x$ at $(1, 1)$ be $y = f(x)$. Then the $x$-intercept of the normal to the curve at the point $(\alpha, \alpha + 1)$ in the first quadrant is $..........$.

If $T$ is the length of the subtangent drawn at any point on the curve $3 y^2 = 4 x^3$ and $N$ is the length of the subnormal at the same point,then $(\beta T)^2 =$