If $\alpha = \int\limits_{0}^{2\sqrt{3}} \log_{2}(x^{2} + 4) \, dx + \int\limits_{2}^{4} \sqrt{2x - 4} \, dx$, then $\alpha^{2}$ is equal to ________.

The dimensional formula of $\frac{1}{2} \epsilon_0 E^2$ ($\epsilon_0$ = permittivity of vacuum and $E$ = electric field) is $M^a L^b T^c$.

The value of $2a - b + c =$ ________.

The diameter of a wire measured by a screw gauge of least count 0.001 cm is 0.08 cm. The length measured by a scale of least count 0.1 cm is 150 cm. When a weight of 100 N is applied to the wire, the extension in length is 0.5 cm, measured by a micrometer of least count 0.001 cm. The error in the measured Young’s modulus is $\alpha \times 10^9 \ \mathrm{N/m}^2$. The value of $\alpha$ is ________.

(Ignore the contribution of the load to Young’s modulus error calculation)

The velocity of a particle is given as $\vec{v} = -x \hat{i} + 2y \hat{j} - z \hat{k}$ m/s. The magnitude of acceleration at point (1, 2, 4) is ________ m/s².