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Number of orbitals = \(2\ell + 1 = 2(3) + 1 = 7\)
Max electrons = \(7 \times 2 = 14\)
This is the 4f subshell. Note that \(n = 4\) is valid since \(\ell\) can be 0, 1, 2, or 3 for \(n = 4\). The 4f subshell holds exactly 14 electrons, which accounts for the 14 lanthanoids.
Evaluate: $\displaystyle\lim_{x\to0}\dfrac{\sin^2 x}{\sqrt{2}-\sqrt{1+\cos x}}$
Rationalise the denominator by multiplying by $\dfrac{\sqrt{2}+\sqrt{1+\cos x}}{\sqrt{2}+\sqrt{1+\cos x}}$:
Denominator becomes $2-(1+\cos x)=1-\cos x$.
$\lim_{x\to0}\dfrac{\sin^2x(\sqrt{2}+\sqrt{1+\cos x})}{1-\cos x}$
Use $\dfrac{\sin^2x}{1-\cos x}=\dfrac{1-\cos^2x}{1-\cos x}=1+\cos x$:
$=\lim_{x\to0}(1+\cos x)(\sqrt{2}+\sqrt{1+\cos x})=(1+1)(\sqrt{2}+\sqrt{2})=2\cdot2\sqrt{2}=\mathbf{4\sqrt{2}}$
Wait: $2\times2\sqrt{2}=4\sqrt{2}$... but option A is 4. Let me recheck: $(2)(\sqrt{2}+\sqrt{2})=2\cdot2\sqrt{2}=4\sqrt{2}$. Answer: $4\sqrt{2}$ is not listed as option A (which is 4). The correct answer is $\mathbf{4\sqrt{2}}$ (index 1).
It is a commonly held belief that history represents a story of moral progress — that human societies, however haltingly, have grown more just, more humane, and more inclusive over time. The abolition of slavery, the extension of suffrage, the dismantling of colonial empires, and the codification of universal human rights are often cited as evidence for this view. Yet to accept this narrative uncritically is to commit what might be called the "retrospective fallacy" — the tendency to evaluate the past by the standards of the present while assuming that those standards are themselves the product of inevitable forward momentum.
What this comfortable narrative obscures is the profound contingency of moral change. The abolition of chattel slavery in the United States was not the result of a gradual awakening of collective conscience but of a catastrophic, politically destabilizing war that killed over 600,000 people and whose outcome was uncertain until nearly the end. The suffragette movement succeeded not only because of the moral persuasiveness of its arguments but because of the instrumental needs of governments that required women in wartime economies. Progress, in other words, has typically required crisis, and often produces new forms of injustice in the process of resolving old ones.
Furthermore, the metrics by which we measure moral progress are themselves contested. When philosophers such as Peter Singer argue that the extension of moral consideration to animals represents the next frontier of moral progress, they implicitly concede that earlier generations failed by the standards of a future ethics not yet fully articulated. This raises a disquieting possibility: that many of our own most confident moral commitments will appear to future generations as indefensible as the endorsement of slavery appears to us. If moral progress is real, its scope may be far larger than we currently imagine — and we may already be on the wrong side of it.
None of this implies that moral progress is illusory. It does suggest, however, that we should hold our sense of moral advancement with a certain epistemic humility. Progress is neither linear nor automatic. It requires not only argument but structural change, political will, and often, terrible cost. The smug confidence with which contemporary societies congratulate themselves on their enlightenment may itself be a symptom of the very complacency that has always impeded genuine moral advance.
Sub-Questions:
The author's primary purpose in describing the abolition of slavery and the suffragette movement is to:
Question 1. Rationale: C is correct. The author uses both examples to challenge the view that moral progress results from a 'gradual awakening of collective conscience,' instead pointing to war, crisis, and instrumental political needs. Option A is too absolute — the author does not claim moral arguments play no role. Option B overstates the case; the author does not prescribe violence. Option D is too narrow — 'wartime economies' is only one example cited.
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