The normal at a point on the parabola $y^2=4x$ passes through $(5,0)$. If two more normals to this parabola also pass through $(5,0)$,then the centroid of the triangle formed by the feet of these three normals is

Three immiscible liquids of densities $d_1 > d_2 > d_3$ and refractive indices $\mu_1 > \mu_2 > \mu_3$ are put in a beaker. The height of each liquid column is $h/3$. $A$ dot is made at the bottom of the beaker. For near normal vision,find the apparent depth of the dot.

$A$ tuning fork of frequency $392 \ Hz$ resonates with $50 \ cm$ length of a string under tension $T$. If the length of the string is decreased by $2\%$,keeping the tension constant,the number of beats heard when the string and the tuning fork are made to vibrate simultaneously is:

If $\lambda_0$ is the de-Broglie wavelength for a proton accelerated through a potential difference of $100 \ V$,the de-Broglie wavelength for an $\alpha$-particle accelerated through the same potential difference is

If $g$ is the acceleration due to gravity on the earth's surface,the gain in the potential energy of an object of mass $m$ raised from the surface of the earth to a height equal to the radius $R$ of the earth is

$\lim _{x \rightarrow \infty}\left(\frac{x+6}{x+1}\right)^{x+4}$ is equal to