Angles

The minute hand moves at a constant rate, completing one full circle in 60 minutes. A full circle measures 360 degrees, or 2\(\pi\) radians. Therefore, the angle turned depends on the fraction of an hour represented by the elapsed minutes. Only the time difference matters; the starting clock hour does not change the calculation.

From 4:12 to 4:47, the elapsed time is \(47-12=35\) minutes. The fraction of a complete revolution is \(35/60\), so the angle is \((35/60)\times2\pi=70\pi/60=7\pi/6\) radians. Hence option C is correct. Option A represents only 25 minutes, while option B represents 30 minutes, so neither matches the stated interval.

The governing relation for arc length is s = rθ, where θ must be expressed in radians. Convert the central angle first: 225° = 225π/180 = 5π/4 radians. Substituting the given arc length s = 15π cm gives 15π = r(5π/4). Solving, r = 15π × 4/(5π) = 12 cm. Thus option B is correct. The π factors cancel, leaving a radius in centimetres, as required. A common error is to put 225 directly into s = rθ, even though that formula requires radians, or to use the wrong conversion factor. Those errors explain why the other numerical options are not valid.

The governing concept is the clock-angle method. The minute hand moves 6° per minute, while the hour hand moves 30° per hour, or 0.5° per minute. At 54 minutes, the minute hand is 54 × 6 = 324° from 12 o’clock. At 11:54, the hour hand is 11 × 30 + 54 × 0.5 = 330 + 27 = 357° from 12. Their difference is |357 − 324| = 33°. Since 33° is already less than 180°, it is the smaller angle. Therefore option C is correct. Treating the hour hand as fixed at 11 would give 30° and explains a tempting distractor, but the hour hand continues moving between hour marks.

The governing concept is the radian formula for arc length, s = rθ, where θ is measured in radians. For the two concentric circles, the arc lengths corresponding to the same central angle θ are 5θ and 13θ. Their difference is therefore 13θ − 5θ = 8θ. The question states that this difference equals 6π, so 8θ = 6π. Dividing by 8 gives θ = 6π/8 = 3π/4 radians. Hence option A is correct. The other choices would produce different arc-length differences: 2π/3 gives 16π/3, 5π/6 gives 20π/3, and π gives 8π. The common angle must be the same for both circles.

The governing conversion is π radians = 180°. Therefore, to convert a radian measure to degrees, multiply by 180°/π. Thus (7π/12) × (180°/π) = 7 × 15° = 105°. The π cancels, so option C is correct. Another equivalent check is that π/12 radians equals 15°, and seven such portions equal 7 × 15° = 105°. Option A is 10° too small, option B does not result from the conversion factor, and option D is 5° too large. The cancellation of π and multiplication by 180/12 are the essential steps; treating the numerical coefficient 7π/12 as degrees without conversion would be conceptually incorrect.

Coterminal angles have the same terminal side and differ by an integer multiple of 360°. To obtain the least positive representative, subtract 360° repeatedly from 845°. We get 845° − 360° = 485°, and 485° − 360° = 125°. Since 125° is positive and less than 360°, it is the least positive coterminal angle, so option A is correct. Equivalently, 845° − 2(360°) = 125°. The other options are not congruent to 845° modulo 360°: adding 360° to each does not recover 845° as an integer multiple relationship. The key is to use 360° for a full revolution, not 180°.

The governing concept is conversion between degrees and radians using the fundamental equivalence 180° = π radians. Convert 130° into radians: 130° × (π⁄180°) = 130π⁄180 = 13π⁄18 radians, after dividing numerator and denominator by 10. The given expression is kπ⁄18 radians, so comparison with 13π⁄18 gives k = 13. Therefore option C is correct. Alternatively, start with kπ⁄18 = 130° and multiply by 180°⁄π; this gives 10k = 130 and hence k = 13. If k were 10, 11, or 15, the corresponding angles would be 100°, 110°, or 150°, respectively, so those distractors do not satisfy the equality.