The slope of the normal to the curve $y=2x^{2}+3\sin x$ at $x=0$ is

The equation of the tangent to the curve $y = 2 \cos x$ at $x = \frac{\pi}{4}$ is

The length of the subtangent at any point of the curve $x^my^n = a^{m+n}$ is proportional to

The tangent to the curve $y = e^x$ at the point $(c, e^c)$ intersects the line joining the points $(c - 1, e^{c-1})$ and $(c + 1, e^{c+1})$ at a point whose $x$-coordinate is:

If the tangent and the normal drawn to the curve $xy^2 + x^2y = 12$ at the point $(1, 3)$ meet the $X$-axis in $T$ and $N$ respectively, then $TN =$

If the tangent drawn at a point $P$ on the curve $y=3x^2-5x+7$ is parallel to its chord joining the points $(1, y_1)$ and $(2, y_2)$ on it, then the $x$-coordinate of the point $P$ is