An archaeologist analyses the wood in a prehistoric structure and finds that the ratio of ${C^{14}}$ (half-life $= 5700 \, years$) to ${C^{12}}$ is only one-fourth of that found in the cells of living plants. The age of the wood is about ........ $years$.

After absorbing a slowly moving neutron of mass $m_N$ (momentum $\sim 0$),a nucleus of mass $M$ breaks into two nuclei of masses $m_1$ and $5m_1$ $(6m_1 = M + m_N)$. If the de-Broglie wavelength of the nucleus with mass $m_1$ is $\lambda$,then the de-Broglie wavelength of the other nucleus will be:

If $\alpha, \beta, \gamma$ are the roots of $2x^3 - 2x - 1 = 0$,then $(\Sigma \alpha \beta)^2$ is equal to

Let $PQ$ and $RS$ be tangents at the extremities of the diameter $PR$ of a circle of radius $r$. If $PS$ and $RQ$ intersect at a point $X$ on the circumference of the circle,then $2r$ equals

The internal and external diameters of a hollow cylinder measured with vernier calipers are $(5.73 \pm 0.01) \text{ cm}$ and $(6.01 \pm 0.01) \text{ cm}$ respectively. Then the thickness of the cylinder wall is

Two rigid bodies $A$ and $B$ rotate with rotational kinetic energies $E_A$ and $E_B$ respectively. The moments of inertia of $A$ and $B$ about the axis of rotation are $I_A$ and $I_B$ respectively. If $I_A = \frac{I_B}{4}$ and $E_A = 100E_B$,then the ratio of the angular momentum of $A$ $(L_A)$ to the angular momentum of $B$ $(L_B)$ is: