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Time taken by Drone 2 to travel from Camp to Spot = 5:05 PM − 5:02 PM = 3 minutes.

Since speed is constant, Drone 1 also took 3 minutes; thus, Sound was received at 5:02 PM − 3 min = 4:59 PM.

Distance = Speed × Time = 30 m/s × 180 s = 5400 m.

Time for Sound to reach camp = 5400 m ÷ 300 m/s = 18 seconds.

Explosion Time = Sound Arrival Time − Sound Travel Time = 4:59:00 PM − 18 s = 4:58:42 PM.

∴ The correct answer is 4:58:42 PM.

Alternate Method

Given:

Speed of Drone (v_d) = 30 m/s

Speed of Sound (v_s) = 300 m/s

Arrival of Picture 1 = 5:02 PM

Arrival of Picture 2 = 5:05 PM

Formula Used:

Distance = Speed × Time | Time = Distance ÷ Speed

⇒ Let the Explosion occur at time T₀ and distance D from the camp.

⇒ Time when the Sensor receives sound (T_s) = T₀ + D/300

⇒ Drone 1 starts at T_s and reaches the spot at 5:02 PM.

⇒ T_s + D/30 = 5:02 PM — (1)

⇒ Drone 2 starts at 5:02 PM and reaches the spot at 5:05 PM.

⇒ 5:02 PM + D/30 = 5:05 PM

⇒ D/30 = 3 minutes = 180 seconds

⇒ D = 180 × 30 = 5400 m

⇒ From equation (1): T_s = 5:02 PM − 3 min = 4:59:00 PM

⇒ Time of explosion T₀ = T_s − D/300

⇒ T₀ = 4:59:00 PM − (5400 / 300) s

⇒ T₀ = 4:59:00 PM − 18 seconds = 4:58:42 PM

∴ The correct answer is 4:58:42 PM.

Additional Information

Speed-Distance Relationship

For a fixed distance, the time taken is inversely proportional to the speed (t ∝ 1/v). Here, the ratio of speeds of sound and drone is 10:1, so the ratio of times taken is 1:10.

Sound Propagation

Sound travels at a constant speed in a uniform medium. The speed of sound in air is approximately 330-344 m/s at standard temperature, though 300 m/s is used here for simplification.

Time Conversion

Always convert minutes to seconds (1 min = 60 s) when dealing with speeds in m/s to maintain unit consistency in calculations.

Shortcut Trick

The unit digit of 6 raised to any positive integer power is always 6.

The cyclicity of 7 is 4 (7, 9, 3, 1).

Divide the power 307 by 4: 307 ÷ 4 → Remainder = 3.

Unit digit of 7³⁰⁷ is the same as 7³, which is 3 (since 343 ends in 3).

Required unit digit = (6 × 3) = 18 → 8.

∴ The correct answer is 8.

Alternate Method

Given Expression: 6¹²⁹ × 7³⁰⁷

Step 1: Finding unit digit of 6¹²⁹

⇒ Powers of 6: 6¹=6, 6²=36, 6³=216...

⇒ Unit digit of 6¹²⁹ = 6.

Step 2: Finding unit digit of 7³⁰⁷

⇒ Cyclicity of 7: 7¹=7, 7²=49, 7³=343, 7⁴=2401.

⇒ Divide the exponent 307 by 4.

⇒ 307 = (4 × 76) + 3.

⇒ Unit digit of 7³⁰⁷ = Unit digit of 7³ = 3.

Step 3: Calculating final unit digit

⇒ Unit digit of (6¹²⁹ × 7³⁰⁷) = (Unit digit of 6¹²⁹) × (Unit digit of 7³⁰⁷)

⇒ Unit digit = 6 × 3 = 18.

⇒ Unit digit = 8.

∴ The correct answer is 8.

Additional Information

Cyclicity of Digits

Digits 0, 1, 5, and 6 have a cyclicity of 1. Their unit digit remains unchanged regardless of the power.

Cyclicity of 4 and 9

These digits have a cyclicity of 2. For 4: 4^odd ends in 4, 4^even ends in 6. For 9: 9^odd ends in 9, 9^even ends in 1.

Cyclicity of 2, 3, 7, and 8

These digits have a cyclicity of 4. To find the unit digit, divide the power by 4 and use the remainder as the new power. If remainder is 0, use 4 as the power.

Shortcut Trick

Let initial Salary = 100, then Saving = 10 and Expenditure = 90.

New Salary = 112 (12% increase) and New Expenditure = 90 + 9 = 99 (10% increase).

New Saving = 112 − 99 = 13.

Percentage change in Saving = [(13 − 10) ÷ 10] × 100 = 30%.

∴ The correct answer is 30% increase.

Alternate Method

Given: Initial savings = 10%, Salary increase = 12%, Expenditure increase = 10%.

Formula Used: Salary = Expenditure + Savings; % Change = [(New − Old) ÷ Old] × 100.

⇒ Let Initial Salary = 100

⇒ Initial Savings = 10% of 100 = 10

⇒ Initial Expenditure = 100 − 10 = 90

⇒ New Salary = 100 + 12% of 100 = 112

⇒ New Expenditure = 90 + 10% of 90 = 90 + 9 = 99

⇒ New Savings = 112 − 99 = 13

⇒ Percentage Change in Savings = [(13 − 10) ÷ 10] × 100

⇒ Percentage Change = (3 ÷ 10) × 100 = 30%

∴ The correct answer is 30% increase.

Additional Information

Income-Expenditure Relationship

The basic identity used in these problems is Income = Expenditure + Savings. Any percentage change in two components allows for the calculation of the third.

Percentage Change Formula

Percentage change is calculated as [(Final Value − Initial Value) ÷ Initial Value] × 100. A positive result indicates an increase, while a negative result indicates a decrease.

Base Value Selection

In percentage problems involving comparisons, choosing 100 as the Base Value for the original quantity (like Salary) simplifies calculation of subsequent changes.

Given:

1. P has one son and one daughter.
2. S is the mother of T.
3. S is R's spouse.
4. Q and R are children of P.

Since R is the spouse of S and S is female, R must be male.

Therefore:

R = son of P

Q = daughter of P

So, Q is the sister of R.

And R is the husband of S.

Therefore, Q is the sister of the husband of S.

Hence, the correct answer is "Option 4".

Shortcut Trick

List the powers of 2 starting from the first three-digit number:

2⁶ = 64 (Two-digit)

2⁷ = 128 (First three-digit)

2⁸ = 256 (Three-digit)

2⁹ = 512 (Three-digit)

2¹⁰ = 1024 (First four-digit)

∴ The correct answer is 3.

Alternate Method

Given: Three-digit numbers range from 100 to 999.

Expression: The numbers must be of the form 2ⁿ where n is an integer.

⇒ For a number to be a three-digit number, 100 ≤ 2ⁿ < 1000.

⇒ If n = 6, 2⁶ = 64 (Not in range).

⇒ If n = 7, 2⁷ = 128 (In range).

⇒ If n = 8, 2⁸ = 256 (In range).

⇒ If n = 9, 2⁹ = 512 (In range).

⇒ If n = 10, 2¹⁰ = 1024 (Not in range).

⇒ The numbers are 128, 256, and 512.

∴ The correct answer is 3.

Additional Information

Powers of 2 and Digit Boundaries

2¹⁰ is 1024, which is the smallest power of 2 that exceeds the three-digit limit (999). It is often used as a benchmark (1 KB = 1024 bytes).

Logarithmic Relation

To find the number of digits in 2ⁿ, use the formula: Number of digits = floor(n × log₁₀2) + 1. For 3 digits, the result must be 3.

Range of n-digit Numbers

An n-digit number N satisfies 10ⁿ⁻¹ ≤ N < 10ⁿ. For n = 3, the range is 100 to 999.

The logic followed here is based on the evaluation of the provided statements in the context of environmental policy and economic intervention as described in the passage.

For Statement 1:

The passage discusses the transition from dirty fuels (like coal) to cleaner alternatives (like natural gas) as a strategy for air quality improvement. This implies that the energy source used is the fundamental factor in the battle for cleaner air. Thus, this statement is a valid inference.

For Statement 2:

The passage focuses on the economic impact, industrial competitiveness, and pricing mechanisms. It does not mention the public will or societal conviction regarding the bans. As there is no supporting evidence in the text for these social aspects, this statement cannot be inferred.

For Statement 3:

The text highlights that there is judicial approval for a policy that intervenes fiscally (by managing price differences) to ensure that cleaner fuel remains cheaper and more accessible than dirty fuel. This facilitate benevolent pricing for eco-friendly energy. Thus, this statement is a valid inference.

Based on the analysis, only Statement 1 and Statement 3 are correct inferences from the text.

Therefore, the correct answer is 1 and 3.

The correct answer is 2 only.

Key Points

• Statement 1 refers to the cessation of operations at power plants within the capital. While facilities like the Badarpur Thermal Power Station and Rajghat Power House were indeed shut down to mitigate air pollution, these closures were phased based on environmental standards and age rather than a summary, instantaneous shutdown of all units. Hence, statement 1 is incorrect.
• Statement 2 pertains to the logistical requirements of the Supreme Court mandate in the M.C. Mehta case, which ordered the conversion of public transport to Compressed Natural Gas (CNG). To ensure the viability of this ban on diesel, the government had to guarantee a robust clean fuel supply infrastructure to maintain urban mobility. Hence, statement 2 is correct.
• Statement 3 discusses the prohibition of solid fuels. While the Commission for Air Quality Management (CAQM) recently moved to restrict coal in industrial applications, historically, the legal focus was on banning specific high-sulfur 'dirty' fuels like Pet Coke and Furnace Oil, rather than an absolute blanket ban on all coal across every sector in the National Capital Region (NCR) during that period. Hence, statement 3 is incorrect.

 Additional Information

• Clean Fuel Policy and Economics:
• The transition to Natural Gas is sensitive to pricing; if Natural Gas remains significantly more expensive than coal or diesel due to taxation, industries lose their competitive edge.
• CNG is primarily composed of Methane (CH₄) and is favored in the NCR because it emits significantly lower levels of Particulate Matter (PM 2.5) and Nitrogen Oxides (NOx) compared to traditional fuels.
• The Supreme Court has consistently used Article 21 of the Indian Constitution to intervene in environmental policy, advocating for fiscal measures that make cleaner energy sources more accessible to the public.

Solution

Given

Statements:

Every red is blue.

Every blue is green.

Every green is yellow.

The least possible Venn diagram for the given statement is:

Conclusions:

P. Every blue is yellow - True (Since all blue are green and all green are yellow, the entire blue circle lies within the yellow circle.)

Q. Every red is green - True (Since all red are blue and all blue are green, the entire red circle lies within the green circle.)

R. Every red is yellow - True (Since red is inside blue, blue is inside green, and green is inside yellow, it follows that all red are yellow.)

So, P, Q and R follow.

∴ The correct answer is P, Q and R.

We need to count how many times digit 5 appears in all two-digit positive integers (10 to 99).

Case 1: 5 in the tens place

Numbers are:

50, 51, 52, 53, 54, 55, 56, 57, 58, 59

Here, 5 appears 10 times.

Case 2: 5 in the units place

Numbers are:

15, 25, 35, 45, 55, 65, 75, 85, 95

Here, 5 appears 9 times.

Note: In 55, the digit 5 appears twice, and both occurrences are counted separately.

Total occurrences of digit 5:

10 + 9 = 19

Hence, the correct answer is "Option 2".

Shortcut Trick

Ignore the middle segment (2 km at 10 kmph) as it is identical for both X and Y.

Total time for X (remaining parts) = (1/5 + 3/4) × 60 = 12 + 45 = 57 minutes.

Total time for Y (remaining parts) = (1/4 + 3/5) × 60 = 15 + 36 = 51 minutes.

Time difference = 57 − 51 = 6 minutes.

∴ The correct answer is 6 minutes.

Alternate Method

Given: Total journey = 6 km. X and Y travel in three segments of 1 km, 2 km, and 3 km.

Formula: Time = Distance ÷ Speed | 1 hour = 60 minutes.

Total Time taken by X (T_X):

⇒ T_X = (1/5 + 2/10 + 3/4) hours

⇒ T_X = (0.2 + 0.2 + 0.75) hours = 1.15 hours

⇒ T_X in minutes = 1.15 × 60 = 69 minutes

Total Time taken by Y (T_Y):

⇒ T_Y = (1/4 + 2/10 + 3/5) hours

⇒ T_Y = (0.25 + 0.2 + 0.6) hours = 1.05 hours

⇒ T_Y in minutes = 1.05 × 60 = 63 minutes

Difference in arrival time:

⇒ 69 minutes − 63 minutes = 6 minutes

∴ The correct answer is 6 minutes.

Additional Information

Average Speed Formula

Average Speed = Total Distance ÷ Total Time. For multiple segments, it is ∑d_i ÷ ∑(d_i/s_i).

Unit Conversion

To convert speed from kmph to m/s, multiply by 5/18. To convert time from hours to minutes, multiply by 60.

Time, Speed, and Distance Relationship

When distance is constant, time is inversely proportional to speed (T ∝ 1/S). When time is constant, distance is directly proportional to speed (D ∝ S).