Daily Practice Questions

Explanation: The voltage across an inductor is governed by $V_L = L \frac{dI}{dt}$, meaning the voltage magnitude is proportional to the slope of the current-time graph. If the current $I(t)$ is a sine wave, its derivative is a cosine wave. Consequently, the voltage leads the current by $90^\circ$ (or $\pi/2$ radians). A common error is selecting the graph that looks like the current itself; however, since the slope of a sine wave is zero at its peaks and maximum at its zero-crossings, the voltage must be at its maximum when the current is zero. Therefore, the correct graph is the one shifted by a quarter period relative to the current, corresponding to choice (D).

Explanation: Each of the alcohols listed has one hydroxyl (O-H) group. This is the part of the alcohol that can form hydrogen bonds with water molecules. Because there is only one hydroxyl group per molecule, each alcohol listed can form the same number of hydrogen bonds. However, the portion of the molecule that is nonpolar increases as you go down the list. The six-carbon alcohol at the bottom is mostly hydrocarbon and only “a little” alcohol, resulting in mostly nonpolar behavior.

Explanation: “Which” is used to set aside a nonessential description, while “that” is used at the beginning of an essential description. In this case, the clarification about the constancy of interest rates could be removed from the sentence and the sentence would still be logical. Therefore, surrounding the description of the fixed interest rates with commas make sense. It is not choice (A) because this is a nonessential description, and commas typically do not precede the word “that”. It is not (C) because there is no pause before “which.” It is not (D) because there is not a complete sentence after the semicolon.

Explanation: To find the volume, we use the disk method integrating with respect to $y$ because the rotation is around the $y$-axis. The radius of each disk is the horizontal distance from the axis to the curve $x = \ln(y)$, bounded between $y=1$ and $y=2$. The integral $V = \pi \int_{1}^{2} (\ln y)^2 \, dy$ evaluates to approximately $0.592$, making (A) correct. Choices like (B), (C), and (D) likely result from incorrect integration limits or failing to square the radius function in the disk method formula.

Explanation: Option E is correct because it uses the past perfect tense ('had struggled') appropriately to show that the struggling occurred before the current evaluation. It also maintains proper parallelism and clarity in the sentence structure. The other options either use incorrect verb tenses or create grammatical inconsistencies.

Explanation: To find the 80th percentile score, compute 80% of 40 students: 0.8 × 40 = 32. So the 32nd score (when ordered from lowest to highest) is the minimum for an A. Cumulative counts: <70 (2), 70–74 (2+7=9), 75–79 (9+15=24), 80–84 (24+8=32). The 32nd student falls in the 80–84 range, but since percentile cutoffs typically require scoring *at or above* the value where cumulative count reaches 80%, and the next range starts at 85, we check: 24 students scored below 80, so scores of 80–84 cover positions 25 to 32. Thus, the minimum score in that range is 80 — however, standard percentile definitions often use interpolation or require the *lowest* score in the range containing the 80th percentile. But many tests define percentile rank as the percentage of scores *less than or equal to* a given score. To be in the 80th percentile or above, you need to be among the top 20% (8 students). The top 8 scores are in 85–89 (3) and 90–94 (2) and >94 (3), totaling 8. So the 33rd score is the first in the top 20%, i.e., the lowest score among the top 8 is 90 (since 3+2+3=8). Therefore, the minimum score for an A is 90.