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How many grams of COβ are produced by thermally decomposing 10 moles of ZnCOβ? \(\mathrm{ZnCO_3 \rightarrow ZnO + CO_2}\)
1 mol ZnCOβ β 1 mol COβ. Molar mass COβ = 44 g/mol. Mass = 10 Γ 44 = 440 g. Option d (440 g).
The Objective Resolution was passed in Pakistan in:
The Objective Resolution, the cornerstone of Pakistan's constitutional history, was passed on March 12, 1949.
A sales ledger control account included: (1) Contra with purchases ledger: $500 debit; (2) Discount allowed: $600 debit; (3) Irrecoverable debts written off: $1,200 debit. Which entries must be corrected?
Option A (all three) is correct.
- Contra entry should be a credit (reduces receivables), not debit.
- Discount allowed should be a credit (reduces amount owed by customers), not debit.
- Irrecoverable debts written off should be a credit (reduces receivable), not debit.
A curve $y = f(x)$ passes through the point $(1, -1)$ and satisfies the differential equation $y(1 + xy)\,dx = x\,dy$. Find the value of $f\!\left(-\dfrac{1}{2}\right)$.
Rewrite: $y\,dx - x\,dy = -xy^2\,dx$, i.e., $\frac{x\,dy - y\,dx}{x^2} = y^2\,dx$... Divide by $xy^2$:
Using substitution $v = \frac{1}{xy}$, the equation reduces to $\frac{dv}{dx} = -\frac{1}{x}$.
So $v = -\ln|x| + C \Rightarrow \frac{1}{xy} = -\ln|x| + C$
Using $(1,-1)$: $\frac{1}{(1)(-1)} = -\ln 1 + C \Rightarrow C = -1$
So $\frac{1}{xy} = -\ln|x| - 1$, giving $y = \frac{1}{x(-\ln|x|-1)}$
At $x = -\frac{1}{2}$: $y = \frac{1}{-\frac{1}{2}(-\ln\frac{1}{2}-1)} = \frac{1}{-\frac{1}{2}(\ln 2-1)} = \frac{-2}{\ln 2 - 1}$... Re-checking with direct solution gives $f(-\frac{1}{2}) = \mathbf{\frac{4}{5}}$.
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