(A) The graph between magnetic susceptibility and magnetising field is as shown in (III).
(B) For \( x < a \), the magnetic field due to a current-carrying wire is given by: \[ B = \frac{\mu_0 I x}{2 \pi a^2}, \] matching graph (IV).
(C) For \( x > a \), the magnetic field due to a current-carrying wire is given by: \[ B = \frac{\mu_0 I}{2 \pi x}, \] matching graph (I).
(D) The magnetic field inside a solenoid varies with distance as shown in (II).
In the given circuit the sliding contact is pulled outwards such that the electric current in the circuit changes at the rate of 8 A/s. At an instant when R is 12 Ω, the value of the current in the circuit will be A.
Let $ P_n = \alpha^n + \beta^n $, $ n \in \mathbb{N} $. If $ P_{10} = 123,\ P_9 = 76,\ P_8 = 47 $ and $ P_1 = 1 $, then the quadratic equation having roots $ \alpha $ and $ \frac{1}{\beta} $ is:
For $ \alpha, \beta, \gamma \in \mathbb{R} $, if $$ \lim_{x \to 0} \frac{x^2 \sin \alpha x + (\gamma - 1)e^{x^2} - 3}{\sin 2x - \beta x} = 3, $$ then $ \beta + \gamma - \alpha $ is equal to: