Decay Rate

Step 1: Concept
Decay follows the rule: , where is the number of half-lives.
Step 2: Analysis
75% decay means 25% (or ) remains.
.
Total time .
Step 3: Conclusion
The required time is 20 days.
Final Answer: (A)

Step 1: Concept
Fraction remaining , where is the number of half-lives.
Step 2: Analysis
3 hours = 180 min. Number of half-lives .
Fraction remaining .
Step 3: Conclusion
Fraction decayed .
Final Answer: (B)

Step 1: Chemical Reaction
Methanol reacts with salicylic acid in the presence of concentrated (catalyst).
Step 2: Esterification
The reaction is an esterification process that yields methyl salicylate.
Step 3: Result
Methyl salicylate has a characteristic smell known as "oil of winter green."
Final Answer: (d)

Step 1: Geometry
A pyramidal structure ( hybridization) typically involves four electron pairs.
Step 2: Bonding
In , three 'B' atoms are attached to the central 'A' atom, forming 3 bond pairs.
Step 3: Lone Pair
To achieve the pyramidal shape (like ), there must be 1 lone pair on the central atom that distorts the tetrahedral arrangement.
Step 4: Conclusion
There is 1 lone pair and 3 bond pairs.
Final Answer: (d)

Step 1: Concept
The activity reduces to of the original after half-lives.
Step 2: Analysis
To reach the safe level, intensity must reduce to of its current value. $ t = n \times T = 6 \times 2 \text{ h} = 12 \text{ h}$.
Final Answer: (A)

Step 1: Understand the decay process.
The half-life of radon ( ) is 4 days. The number of atoms of radon after time can be calculated using the formula: where is the initial number of atoms, is the number of atoms after time , and is the half-life.

Step 2: Calculate the remaining atoms after 12 days.
Substitute the values into the formula:

Step 1: Understand the relationship between decay constants.
The decay constant of a substance determines how quickly it decays. For radium bromide, the decay constant is the same as that of radium, since the chemical bonding does not alter the nuclear properties of radium.

Step 2: Decay constant of radium bromide.
Since the decay constant is a characteristic of the isotope and does not depend on the chemical form, the decay constant of radium bromide remains .

Step 1: Understanding the Concept:
Water drops fly off tangentially with tangential speed . Time to fall . Horizontal range = . Locus radius = .
Step 2: Detailed Explanation:
Radius of umbrella rim .
Angular velocity .
Tangential speed .
Time to fall: .
Horizontal range = .
Locus radius = .
Step 3: Final Answer:
Thus, radius = .

Step 1: Understanding the Concept:
Rotational KE = . Using with .
Step 2: Detailed Explanation:
Given KE = , .
.
Using : .
Step 3: Final Answer:
Thus, time = .

Step 1: Remaining after 6 days

Step 2: After 10 days

Step 3: Decayed fraction