If the normal to the curve $y = f(x)$ at the point $(3, 4)$ makes a positive angle of $3\pi /4$ with the $x$-axis,then $f'(3) = .....$

The sum of intercepts on coordinate axes made by the tangent to the curve $\sqrt{x}+\sqrt{y}=\sqrt{a}$ is

If the angle between the curves $y=e^{2(1+x)-4}$ and $x^2 y=1$ at the point $(1,1)$ is $\theta$,then $|\sin \theta|+|\cos \theta|=$

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 to the curve $2y^3 = ax^2 + x^3$ at the point $(a, a)$ cuts the coordinate axes at $p$ and $q$ such that $p^2 + q^2 = 61$,then what is the value of $a$?

If the tangent to the curve $y = f(x) = x \log_{e} x$ $(x > 0)$ at a point $(c, f(c))$ is parallel to the line segment joining the points $(1, 0)$ and $(e, e)$,then $c$ is equal to: