Logarithms

Step 1: Use the formula for an arithmetic progression (A.P.).
If the terms , , and are in A.P., then the middle term is the average of the first and third terms:
Step 2: Simplify the trigonometric expressions.
Using trigonometric identities, we simplify the expressions and find that:
Step 3: Conclusion.
Thus, the correct answer is .

Step 1: Apply the doubling formula.
The time taken to double a population growing at a constant rate is given by the formula: Given (5%), we substitute the values:
Step 2: Compute .
Using a calculator or logarithmic tables, we find:
Step 3: Calculate the time.
Now, calculate :
Step 4: Conclusion.
Thus, the time taken for the population to double is approximately 13.240 years, which corresponds to option (B).

Step 1: Express 99 suitably.
Step 2: Use logarithmic approximation.
For small , Thus, Step 3: Substitute values.
Step 4: Final calculation.

Step 1: Write the growth formula.
For population growth, Here, and the population becomes double, so Step 2: Simplify the equation.
Taking logarithm on both sides, Step 3: Substitute given values.
Using and , Step 4: Calculate the time.

Step 1: Recall the logarithmic identity
We are asked to find the value of . Recall that the logarithmic identity means that . In this case, means that .
Step 2: Express 32 as a power of 2
We know that: Thus, the equation becomes:
Step 3: Solve for
Since the bases are the same, we can equate the exponents:
Answer:
Therefore, . So, the correct answer is option (1).

Concept:
In circular permutations: Also, for restriction “no two girls together”, we use the gap method. Step 1: Arrange the boys first. Since it is a circular table:
Step 2: Identify gaps created by boys. 6 boys create 6 gaps:
Step 3: Place the girls. We must place 5 girls in these 6 gaps so that no two are together.

• Choose 5 gaps out of 6:
• Arrange 5 girls in those gaps:

Step 4: Total arrangements.
Step 5: Correct circular consideration. Since arrangement of boys was already circularly fixed, no further division is needed. However, arranging girls in selected gaps gives: Thus: Now considering all placements distinctly in circular seating:
Step 6: Conclusion.