The osmotic pressure (\( \pi \)) for a solution is given by the formula:
\[ \pi = \frac{nRT}{V} \] where:
\( n \) is the number of moles of solute,
\( R \) is the gas constant,
\( T \) is the temperature,
\( V \) is the volume of the solution.
For non-electrolyte (A) and glucose, the osmotic pressures are the same, so we can equate the osmotic pressures: \[ \frac{12 / M_A}{1} = 0.05 \times 1 \] where \( M_A \) is the molar mass of A, and 12 g is the mass of A. Simplifying the equation: \[ \frac{12}{M_A} = 0.05 \quad \Rightarrow \quad M_A = \frac{12}{0.05} = 240 \, \text{g/mol}. \] Thus, the molecular mass of A is 240 g/mol.
At STP \(x\) g of a metal hydrogen carbonate (MHCO$_3$) (molar mass \(84 \, {g/mol}\)) on heating gives CO$_2$, which can completely react with \(0.02 \, {moles}\) of MOH (molar mass \(40 \, {g/mol}\)) to give MHCO$_3$. The value of \(x\) is:
Let one focus of the hyperbola $ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $ be at $ (\sqrt{10}, 0) $, and the corresponding directrix be $ x = \frac{\sqrt{10}}{2} $. If $ e $ and $ l $ are the eccentricity and the latus rectum respectively, then $ 9(e^2 + l) $ is equal to:
The largest $ n \in \mathbb{N} $ such that $ 3^n $ divides 50! is: