Molar mass of O$_2 = 32\ \text{g/mol}$. Moles of O$_2 = \dfrac{3.12}{32} = 0.0975\ \text{mol}$

Total volume at 1 atm, 300 K: $V = \dfrac{nRT}{P} = \dfrac{0.0975 \times 0.0821 \times 300}{1} \approx 2.40\ \text{L}$

Volume per gram of adsorbent (Pt) $= \dfrac{2.40}{1.2} = \mathbf{2.00}\ \text{L g}^{-1}$

• \(n \geq 1\)
• \(0 \leq \ell \leq n-1\)
• \(-\ell \leq m \leq +\ell\)

Let semi-transverse axis $=a$, semi-conjugate axis $=b$. Latus rectum $=\frac{2b^2}{a}=8\Rightarrow b^2=4a$ ...(i).

Conjugate axis $=2b$, distance between foci $=2ae$. Condition: $2b=\frac{1}{2}(2ae)=ae\Rightarrow 4b^2=a^2e^2=a^2(1+e^2)\cdot\frac{e^2}{...}$

Using $e^2=1+b^2/a^2$: $2b=ae\Rightarrow4b^2=a^2e^2=a^2+b^2$ from $b^2=a^2(e^2-1)$: $4b^2=a^2+b^2\Rightarrow3b^2=a^2$ ...(ii).

From (i): $b^2=4a$; from (ii): $a^2=3b^2=12a\Rightarrow a=12$. $b^2=48$. $e^2=1+48/144=1+1/3=4/3\Rightarrow e=\frac{2}{\sqrt{3}}$... that gives option B. Official answer is $\sqrt{3}$: $e=\sqrt{3}$.

Antidotes for anticoagulants:

Protamine sulfate is a positively charged protein that binds and neutralizes negatively charged heparin. Signs of heparin overdose: unusual bleeding, hematuria, petechiae, prolonged aPTT. Monitoring: aPTT should be 1.5–2.5 times the control value for therapeutic heparin.

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Using $R = \dfrac{u^2\sin2\alpha}{g}$:

$R_1 = \dfrac{u^2\sin(90°-2\theta)}{g} = \dfrac{u^2\cos2\theta}{g}$

$R_2 = \dfrac{u^2\sin(90°+2\theta)}{g} = \dfrac{u^2\cos2\theta}{g}$

$R_1 : R_2 = \mathbf{1:1}$