The equation of the plane passing through the line $\frac{x - 1}{5} = \frac{y + 2}{6} = \frac{z - 3}{4}$ and the point $(4, 3, 7)$ is

The term independent of $x$ in the expansion of $\left(\sqrt{x}-\frac{2}{\sqrt{x}}\right)^{18}$ is

The potential energy of a particle of mass $1\, kg$ in motion along the $x-$axis is given by $U = 4(1 - \cos 2x)$,where $x$ is in $meters$. The period of small oscillation (in $seconds$) is

As shown in the figure,two particles each of mass $m$ tied at the ends of a light string of length $2 a$ are kept on a frictionless horizontal surface. When the mid-point $(P)$ of the string is pulled vertically upwards with a small but constant force $F$,the particles move towards each other on the surface. The magnitude of acceleration of each particle,when the separation between them becomes $2 x$,is:

How is the rate of an enzymatic reaction increased?

Three rods of the same dimensions have thermal conductivities $3 K, 2 K$,and $K$. They are arranged as shown in the figure below. The temperature of the junction in the steady state is: