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Stress is defined as $\frac{\text{Force}}{\text{Area}}$.
- Force (Weight) $\propto \text{Volume} \propto L^3$
- Area $\propto L^2$
- $\text{Stress} \propto \frac{L^3}{L^2} \propto L$
- Since $L$ increases by factor $9$, stress increases by factor $9$.
Reaction of an alcohol with sodium metal produces:
Alcohols react with Na to form sodium alkoxide and hydrogen gas: \(2\mathrm{ROH} + 2\mathrm{Na} \rightarrow 2\mathrm{RO}^-\mathrm{Na}^+ + \mathrm{H}_2\).
Find all $x$ satisfying $(\cot^{-1}x)^2 - 7(\cot^{-1}x)+10>0$.
Let $u=\cot^{-1}x$. Solve $u^2-7u+10>0$, i.e., $(u-2)(u-5)>0$.
This gives $u<2$ or $u>5$.
Since $\cot^{-1}x$ is a decreasing function with range $(0,\pi)$:
- $u<2 \Rightarrow \cot^{-1}x<2 \Rightarrow x>\cot2$ (decreasing function reverses inequality)
- $u>5 \Rightarrow \cot^{-1}x>5 \Rightarrow x<\cot5$
Solution: $x\in(-\infty,\cot5)\cup(\cot2,\infty)$
Bilateral symmetry (also called axial or mirror symmetry) means one half of the composition is the mirror image of the other about a central axis.
Examples in architecture: Taj Mahal (bilateral symmetry on all four sides), most classical Greek temples, and many government buildings.
Other design principles: Rhythm = repetition of elements creating movement. Radial balance = elements arranged around a central point. Contrast = juxtaposition of opposing elements (light/dark, old/new).
The first All Pakistan Education Conference was held in Karachi in November 1947, shortly after independence. It laid the foundational vision for Pakistan’s national education system. The correct answer is D — None of these (Karachi), as none of the first three options is correct.
Correct Answer: D — None of these (Karachi).
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