A system consists of a cylinder surrounded by a cylindrical shell. A cylinder has a radius \(R\) and is made of material of thermal conductivity \(K,\) whereas a cylindrical shell has an inner radius \(R\) and outer radius \(2R\) and is made of material of thermal conductivity twice as that of a cylinder. Assuming the system is in a steady state and has negligible heat loss across the cylindrical surface, what is the effective thermal conductivity of the system if the two ends of the combined system are maintained at two different temperatures?
1. \(3K\)
2. \(\frac23K\)
3. \(\frac{7K}{4}\)
4. \(\frac{5K}{4}\)

An isotropic point source emits light with wavelength 500 nm. The radiation power of the source is P = 10 W. The number of photons passing through a unit area per second at a distance of 3 m from the source is:
1. 5.92 × 10¹⁷ m/s²
2. 2.23 × 10¹⁷ m/s²
3. 2.23 × 10¹⁸ m/s²
4. 5.92 × 10¹⁸ m/s²

A bat emitting an ultrasonic wave of frequency 4.5 × 10⁴ Hz at speed of 6 m/s between two parallel walls. The two frequencies heard by the bat will be:
1. 4.67 × 10⁴ Hz, 4.34 × 10⁴ Hz
2. 4.34 × 10⁴ Hz, 4.67 × 10⁴ Hz
3. 4.5 × 10⁴ Hz, 5.4 × 10⁴ Hz
4. 4.67 × 10³ Hz, 4.34 × 10⁴ Hz

A rod made up of metal is 1.2 m long and 0.8 cm in diameter. Its resistance is 3.5 × 10^-3 \(\Omega.\) Another disc made of the same metal is 2.0 cm in diameter and 1.25 mm thick. What is the resistance between the round faces of the disc?
1. 1.35 × 10^-8 \(\Omega\)
2. 2.70 × 10^-7 \(\Omega\)
3. 5.82 × 10^-7 \(\Omega\)
4. 8.10 × 10^-5 \(\Omega\)

A circular current-carrying coil has a radius \(R.\) The distance from the centre of the coil, on the axis, where \(B\) will be \(\frac18\) of its value at the centre of the coil is:
1. \(\frac{R}{\sqrt3}\)
2. \(\sqrt3R\)
3. \(2\sqrt3R\)
4. \(\frac{2R}{\sqrt3}\)