A travelling microscope has 20 divisions per $$\mathrm{cm}$$ on the main scale while its vernier scale has total 50 divisions and 25 vernier scale divisions are equal to 24 main scale divisions, what is the least count of the travelling microscope?

In an experiment to find out the diameter of wire using screw gauge, the following observations were noted :

(A) Screw moves $$0.5 \mathrm{~mm}$$ on main scale in one complete rotation

(B) Total divisions on circular scale $$=50$$

(C) Main scale reading is $$2.5 \mathrm{~mm}$$

(D) $$45^{\text {th }}$$ division of circular scale is in the pitch line

(E) Instrument has 0.03 mm negative error

Then the diameter of wire is :

Consider the efficiency of carnot's engine is given by $$\eta=\frac{\alpha \beta}{\sin \theta} \log_e \frac{\beta x}{k T}$$, where $$\alpha$$ and $$\beta$$ are constants. If T is temperature, k is Boltzmann constant, $$\theta$$ is angular displacement and x has the dimensions of length. Then, choose the incorrect option :

The dimensions of $$\left(\frac{\mathrm{B}^{2}}{\mu_{0}}\right)$$ will be :

(if $$\mu_{0}$$ : permeability of free space and $$B$$ : magnetic field)