Consider the operator S·n̂ with eigenkets |±⟩n and eigenvalues ±ℏ/2 where n̂ is a unit vector and S is the spin operator. A partially polarized beam of spin-1/2 particles contains a 25-75 mixture of two pure ensembles, one with |+⟩z and the other with |+⟩x respectively. What is the ensemble average of S·x̂/ℏ?

Which of the following territories was annexed by Lord Dalhousie under the pretext of misgovernance, and not directly through the Doctrine of Lapse?

What was the primary stated reason for the Partition of Bengal in 1905 by Lord Curzon?

Which of the following events is considered the immediate trigger for the launch of the Non-Cooperation Movement by Mahatma Gandhi?