If $f$ is a differentiable function such that $f(2x+1) = f(1-2x)$ for all $x \in R$,then find the minimum number of roots of the equation $f'(x) = 0$ in the interval $x \in (-5, 10)$,given that $f(2) = f(5) = f(10)$.

$_6 C^{12}$ and $_1 T^3$ are formed in nature due to the nuclear reaction of a neutron with:

If $\Delta$ is the area of the triangle formed by the positive $x$-axis and the normal and tangent to the circle $x^2+y^2=4$ at $(1, \sqrt{3})$,then $\Delta$ is equal to

The bob of a simple pendulum is displaced from its equilibrium position $O$ to a position $Q$ which is at height $h$ above $O$ and the bob is then released. Assuming the mass of the bob to be $m$ and time period of oscillations to be $2.0\ s$,the tension in the string when the bob passes through $O$ is

The plates of a parallel plate capacitor are charged up to $100\, V$. $A$ $2\, mm$ thick plate is inserted between the plates. To maintain the same potential difference,the distance between the capacitor plates is increased by $1.6\, mm$. The dielectric constant of the plate is:

$A$ charge $Q$ is enclosed by a Gaussian spherical surface of radius $R$. If the radius is doubled,then the outward electric flux will