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.

The position of an object having mass 0.1 kg as a function of time $t$ is given as

$\vec{r} = \left( 10 t^2 \hat{i} + 5 t^3 \hat{j} \right)$ m. At $t = 1$ s, which of the following statements are correct?

A. The linear momentum $\vec{p} = \left( 2 \hat{i} + 1.5 \hat{j} \right)$ kg·m/s.

B. The force acting on the object $\vec{F} = \left( 2 \hat{i} + 3 \hat{j} \right)$ N.

C. The angular momentum of the object about its origin $\vec{L} = 15 \hat{k}$ J·s.

D. The torque acting on the object about its origin $\vec{\tau} = 20 \hat{k}$ N·m.

Choose the correct answer from the options given below: