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For each $t\in\mathbb{R}$, let $[t]$ denote the greatest integer $\leq t$. Evaluate: $\displaystyle\lim_{x\to1^+}\dfrac{(1-|x|+\sin|1-x|)\sin\!\left(\dfrac{\pi}{2}[1-x]\right)}{|1-x|\cdot[1-x]}$
As $x\to1^+$: let $h=x-1>0$, so $h\to0^+$. Then $|1-x|=h$, $[1-x]=[-h]=-1$.
$\sin\!\left(\dfrac{\pi}{2}[-h]\right)=\sin\!\left(\dfrac{\pi}{2}\cdot(-1)\right)=\sin(-\pi/2)=-1$
Numerator: $(1-|x|+\sin h)\cdot(-1)$. As $x\to1^+$: $|x|=x=1+h$, so $1-|x|=-h$.
Numerator $= (-h+\sin h)(-1) = h-\sin h$
Denominator: $h\cdot(-1)=-h$
Limit $= \dfrac{h-\sin h}{-h} = -(1-\dfrac{\sin h}{h})\to-(1-1)=\mathbf{0}$
Resistance of the bulb: $R_b = \frac{V^2}{P} = \frac{(120)^2}{60} = 240\,\Omega$
Before heater is switched on: Current $= \frac{120}{240+6} = \frac{120}{246}\,\text{A}$. Voltage across bulb $= \frac{120 \times 240}{246} \approx 116.9\,\text{V}$.
Resistance of heater: $R_h = \frac{(120)^2}{240} = 60\,\Omega$. Parallel combination: $R_p = \frac{240 \times 60}{300} = 48\,\Omega$.
After heater is switched on: Voltage across combination $= \frac{120 \times 48}{48+6} = \frac{5760}{54} \approx 106.67\,\text{V}$.
Decrease $= 116.9 - 106.67 \approx 10.04\,\text{V}$. Answer: D.
Before Maria changed jobs, her salary was 24 percent more than Julio's salary. After Maria changed jobs, her new salary was 24 percent less than her old salary.
Compare:
Column A: Julio's salary
Column B: Maria's new salary
Let Julio's salary be $J$ and Maria's old salary be $M_{old}$.
Given: $M_{old} = 1.24J$
Maria's new salary: $M_{new} = M_{old} - 0.24M_{old} = 0.76M_{old} = 0.76(1.24J) = 0.9424J$
Comparing:
- Column A: $J = 1.0J$
- Column B: $0.9424J$
Since $1.0 > 0.9424$, Column A (Julio's salary) is greater than Column B (Maria's new salary).
Therefore, Column A is greater.
Financial data: Profit from operations $125,000; Profit for year $116,000; Shareholders' equity $423,000; Long-term loan $80,000; Current liabilities $45,000. What is the ROCE?
Option D (24.85%) is correct.
ROCE = Profit from operations ÷ Capital Employed × 100
Capital Employed = Shareholders' equity + Non-current liabilities = $423,000 + $80,000 = $503,000
(Current liabilities are excluded from capital employed.)
ROCE = $125,000 ÷ $503,000 × 100 = 24.85%
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