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Cardiac Tamponade: Beck's Triad
- Hypotension โ reduced cardiac output due to compression
- Muffled (distant) heart sounds โ fluid surrounding the heart dampens sounds
- Jugular venous distension (JVD) โ elevated venous pressure as blood cannot enter compressed heart
- Pulsus paradoxus \(> 10 \text{ mmHg}\): Exaggerated fall in systolic BP during inspiration
- Tachycardia (compensatory)
- Electrical alternans on ECG (alternating QRS amplitude)
- Enlarged cardiac silhouette on CXR ('water bottle' heart)
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Suppose $x, y, z$ are in arithmetic progression (A.P.) and $\tan^{-1}x,\ \tan^{-1}y,\ \tan^{-1}z$ are also in A.P. Then:
If $\tan^{-1}x, \tan^{-1}y, \tan^{-1}z$ are in A.P., then $2\tan^{-1}y=\tan^{-1}x+\tan^{-1}z$.
Also $x,y,z$ in A.P. means $2y=x+z$.
From $2\tan^{-1}y=\tan^{-1}x+\tan^{-1}z$: using the addition formula, $\tan(2\tan^{-1}y)=\tan(\tan^{-1}x+\tan^{-1}z)$, i.e., $\dfrac{2y}{1-y^2}=\dfrac{x+z}{1-xz}$.
Substituting $x+z=2y$: $\dfrac{2y}{1-y^2}=\dfrac{2y}{1-xz}$. So $1-y^2=1-xz$, giving $y^2=xz$.
Combined with $x+z=2y$ (A.P.) and $y^2=xz$ (G.P.) $\Rightarrow x=y=z$.
Which sub-shell does not exist?
For n=1, l can be 0 only (s orbital). 1p would require l=1, not possible. 1s exists; 5d and 6f exist for higher n.
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