CSAT Solved Papers/ 2022/Q68

2022 CSAT — Q68

Quant Statement validity 2.5 marks Hard

Consider the following statements:

  1. Between 3:163{:}16 p.m. and 3:173{:}17 p.m., both hour hand and minute hand coincide.

  2. Between 4:584{:}58 p.m. and 4:594{:}59 p.m., both minute hand and second hand coincide.

Which of the above statements is/are correct?

  1. A 1 only
  2. B 2 only
  3. C Both 1 and 2 Answer
  4. D Neither 1 nor 2

Worked rationale

Statement 1 — after 3:003{:}00 the hour and minute hands coincide once. The minute hand gains on the hour hand at 112∘\tfrac{11}{2}^{\circ} per minute, and at 3:003{:}00 the hour hand leads by 90∘90^{\circ}, so the catch-up time is

9011/2=18011≈16.36 min=3:16:22,\frac{90}{11/2} = \frac{180}{11} \approx 16.36 \text{ min} = 3{:}16{:}22,

which lies between 3:163{:}16 and 3:173{:}17. Correct.

Statement 2 — measure time as 4:58+t4{:}58 + t min (0≤t<10 \le t < 1). Minute-hand angle =(58+t)⋅6=348+6t= (58+t)\cdot 6 = 348 + 6t; second-hand angle =360t(mod360)= 360t \pmod{360}. Coincidence:

360t≡348+6t(mod360)  ⇒  354t=348  ⇒  t=348354≈0.983 min=4:58:59,360t \equiv 348 + 6t \pmod{360} \;\Rightarrow\; 354t = 348 \;\Rightarrow\; t = \frac{348}{354} \approx 0.983 \text{ min} = 4{:}58{:}59,

which lies between 4:584{:}58 and 4:594{:}59. Correct.

Answer: (c) Both 1 and 2.

Why the other options miss

  • A
    missed the second check: verifies the hour–minute coincidence but assumes the minute–second one falls outside the minute, skipping the 354t=348354t = 348 check.
  • B
    mis-times the hour–minute coincidence (e.g. 3:153{:}15), wrongly excluding Statement 1.
  • D
    uses additive (not relative) angular speeds and rejects both.

Specialist insight

Both reduce to a relative-angular-speed catch-up. Hour–minute: lead 90∘90^{\circ} at 112∘/min\tfrac{11}{2}^{\circ}/ \text{min} gives 18011≈16.36\tfrac{180}{11} \approx 16.36 min. Minute–second: solving 354t=348354t = 348 lands the coincidence at ≈58.98\approx 58.98 s into the minute — just inside the interval, which is exactly why the examiner chose 4:584{:}58–4:594{:}59. Compute, don’t eyeball, near interval boundaries.

The trap, in one line

Hour–minute meet at 3:16:223{:}16{:}22; minute–second meet at 4:58:594{:}58{:}59 — both inside their intervals ⇒\Rightarrow (c).

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