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- $2$ moles of resin ($2 \times 206\text{ g}$) react with $1$ mole of $Ca^{2+}$.
- Total mass of resin = $412\text{ g}$.
- Uptake per gram = $\frac{1\text{ mole } Ca^{2+}}{412\text{ g resin}}$.
A software firm plans to migrate its infrastructure to a new VPS. The project requires an initial outlay of $10,000 and is expected to generate net cash flows of $4,000, $5,000, and $6,000 over three years. If the cost of capital is 10%, what is the Net Present Value (NPV)?
Using the formula $NPV = -I_0 + \sum \frac{CF_t}{(1+i)^t}$, we get: $-10,000 + 3,636 + 4,132 + 4,508 = 2,276$. (Calculated precisely: $NPV = 2104$).
For \(n = 4\): \(4^2 = 16\) orbitals
Breakdown:
- 4s: 1 orbital
- 4p: 3 orbitals
- 4d: 5 orbitals
- 4f: 7 orbitals
- Total = 1+3+5+7 = 16
The dimensional formula of torque is:
Torque $\tau = r \times F$ (moment of force)
$[\tau] = [r][F] = L \times MLT^{-2} = ML^2T^{-2}$
Note: Torque has the same dimensional formula as energy ($ML^2T^{-2}$), but they are physically different quantities. Torque is a vector while energy is a scalar.
What is the most appropriate basis for apportioning factory rent to each department?
Option A (Floor space) is correct.
Rent is a cost of occupying space โ floor space is the logical basis. Number of employees suits canteen costs; production output suits variable costs; value of machinery suits depreciation/insurance.
0.5 moles of gas A and $x$ moles of gas B together exert a pressure of 200 Pa in a 10 m$^3$ container at 1000 K. Given that $R$ is the gas constant in J K$^{-1}$ mol$^{-1}$, the value of $x$ is:
Apply the ideal gas law: $PV = n_{\text{total}}RT$
$n_{\text{total}} = 0.5 + x$
$200 \times 10 = (0.5 + x) \times R \times 1000$
$2000 = 1000R(0.5 + x)$
$2 = R(0.5 + x) = 0.5R + Rx$
$Rx = 2 - 0.5R$
$x = \dfrac{2 - 0.5R}{R} = \dfrac{4 - R}{2R}$
Find a value of $\theta$ for which $\dfrac{2+3i\sin\theta}{1-2i\sin\theta}$ is purely imaginary.
Multiply numerator and denominator by the conjugate of the denominator $(1+2i\sin\theta)$:
Numerator real part: $(2+3i\sin\theta)(1+2i\sin\theta)$ real part $= 2-6\sin^2\theta$
For purely imaginary, the real part must be zero: $2-6\sin^2\theta=0\Rightarrow\sin^2\theta=\frac{1}{3}\Rightarrow\sin\theta=\frac{1}{\sqrt{3}}$
Therefore $\theta=\sin^{-1}\!\left(\dfrac{1}{\sqrt{3}}\right)$
Formula for Minute Ventilation:
\[\dot{V}_E = V_T \times RR\]
\[\dot{V}_E = 700 \text{ mL} \times 14 = 9800 \text{ mL/min} = 9.8 \text{ L/min}\]
Clinical significance:
- Normal \(\dot{V}_E\) = \(5 - 8 \text{ L/min}\)
- This patient's \(\dot{V}_E\) of \(9.8 \text{ L/min}\) is elevated, risking hyperventilation and respiratory alkalosis
- Lung-protective ventilation targets \(V_T = 6 \text{ mL/kg}\) ideal body weight (IBW) to prevent ventilator-induced lung injury (VILI)
- If IBW = 70 kg, then target \(V_T = 6 \times 70 = 420 \text{ mL}\)
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