Logical Reasoning

First, express all terms with the base 2:
4 = 2^2
8 = 2^3
16 = 2^4

The equation becomes: (2^x * (2^2)^(x+1)) / (2^3)^(x-1) = 2^4

Apply the index rule (a^m)^n = a^(mn):
(2^x * 2^(2(x+1))) / 2^(3*(x-1)) = 2^4
(2^x * 2^(2x+2)) / 2^(3x-3) = 2^4

Apply the index rule a^m * a^n = a^(m+n) in the numerator:
2^(x + 2x + 2) / 2^(3x-3) = 2^4
2^(3x + 2) / 2^(3x-3) = 2^4

Apply the index rule a^m / a^n = a^(m-n):
2^((3x + 2) - (3x - 3)) = 2^4
2^(3x + 2 - 3x + 3) = 2^4
2^5 = 2^4

This implies 5 = 4, which is a contradiction. This means there is no solution for x that satisfies the given equation. This type of problem tests the understanding of index laws and logical consistency.

Let the profit shares be 3x, 2x, and 1x. The smallest share is 1x, which is given as ₹50,000. Therefore, x = ₹50,000. The total profit distributed by the firm is the sum of all shares: 3x + 2x + 1x = 6x. Substituting the value of x, total profit = 6 * ₹50,000 = ₹3,00,000.

Blood relation tests involve deducing the relationship between two members of a family based on the given information. It requires understanding of family trees and terms like paternal, maternal, niece, nephew, etc.

Let the current ages of the two brothers be 5x and 7x. After 6 years, their ages will be (5x + 6) and (7x + 6). According to the problem, the ratio of their ages after 6 years is 7:9. So, (5x + 6) / (7x + 6) = 7 / 9.

Cross-multiplying: 9 * (5x + 6) = 7 * (7x + 6)
45x + 54 = 49x + 42
54 - 42 = 49x - 45x
12 = 4x
x = 12 / 4 = 3.

Therefore, their current ages are:
First brother: 5x = 5 * 3 = 15 years.
Second brother: 7x = 7 * 3 = 21 years.

To find log(72), first express 72 as a product of its prime factors:
72 = 8 * 9 = 2^3 * 3^2.

Now, apply the logarithm properties:
log(a * b) = log(a) + log(b)
log(a^b) = b * log(a)

So, log(72) = log(2^3 * 3^2)
log(72) = log(2^3) + log(3^2)
log(72) = 3 * log(2) + 2 * log(3)

Substitute the given values of log(2) and log(3):
log(72) = 3 * 0.3010 + 2 * 0.4771
log(72) = 0.9030 + 0.9542
log(72) = 1.8572.

These problems involve arranging a group of people in a circle or a straight line based on a set of conditions (e.g., A is sitting next to B, C is opposite D).

Logarithms simplify calculations by converting multiplication and division into addition and subtraction, and exponentiation into multiplication. For very large numbers (e.g., in astronomy or economic growth over long periods) or very small numbers (e.g., in chemistry or probability), direct multiplication or division can lead to calculator overflow/underflow errors or become cumbersome. By taking logarithms, these operations are transformed into simpler arithmetic, making the calculations manageable and less prone to errors. For example, to calculate A * B, one can find log(A) + log(B) and then take the antilog of the result. Similarly, for A^B, it becomes B * log(A).

Initial Revenue (P) = ₹10,00,000. Growth rates are r1 = 10% (1.10), r2 = 12% (1.12), r3 = 15% (1.15). The revenue after three years can be calculated as: Final Revenue = P * (1 + r1) * (1 + r2) * (1 + r3). Using indices, this is R = 10,00,000 * (1.10)^1 * (1.12)^1 * (1.15)^1. R = 10,00,000 * 1.10 * 1.12 * 1.15 = 10,00,000 * 1.4168 = ₹14,16,800.

This is an inverse proportion problem. Let W be the number of workers and D be the number of days. W1 * D1 = W2 * D2. Given W1 = 5 workers, D1 = 12 days. We want to find W2 for D2 = 8 days. So, 5 * 12 = W2 * 8. This gives 60 = 8 * W2, so W2 = 60 / 8 = 7.5 workers. Since you cannot have half a worker, 8 workers would be needed to complete the task within 8 days. However, the question asks for additional workers. If 7.5 workers are theoretically needed, and we start with 5, we need 7.5 - 5 = 2.5 additional workers. Practically, this implies 3 additional workers to ensure completion, making the total 8 workers. For CA Foundation, usually, the answer is expected to be precise, so 7.5 workers are needed in total. Thus, 2.5 additional workers are required.

If an investment doubles, the future value (FV) is twice the present value (PV), so FV = 2 * PV. The compound interest formula is FV = PV * (1 + r)^n, where 'r' is the interest rate and 'n' is the number of years. Substituting FV = 2 * PV, we get 2 * PV = PV * (1 + r)^n, which simplifies to 2 = (1 + r)^n. To solve for 'n', we take the logarithm of both sides: log(2) = log((1 + r)^n). Using the logarithm property log(a^b) = b * log(a), we get log(2) = n * log(1 + r). Therefore, n = log(2) / log(1 + r). This formula allows us to calculate 'n' (the number of years) using the given interest rate 'r' and the known log values, simplifying the calculation for exponential growth.

Initial Debt = ₹100 lakhs. Since the Debt-to-Equity ratio is 2:1, initial Equity = Initial Debt / 2 = ₹100 lakhs / 2 = ₹50 lakhs.

After transactions:

1. New equity shares issued: Equity increases by ₹50 lakhs. New Equity = ₹50 lakhs + ₹50 lakhs = ₹100 lakhs.
2. Repayment of debt: Debt decreases by ₹20 lakhs. New Debt = ₹100 lakhs - ₹20 lakhs = ₹80 lakhs.

The new Debt-to-Equity ratio = New Debt / New Equity = ₹80 lakhs / ₹100 lakhs = 0.8:1. The ratio changes from 2:1 to 0.8:1, indicating a significant reduction in financial leverage.

Initial Cost (P) = ₹5,00,000. Depreciation Rate (r) = 15% or 0.15. Number of years (n) = 4.

Using the reducing balance method, the book value after 'n' years is given by the formula: Book Value = P * (1 - r)^n.

Substituting the values: Book Value = 5,00,000 * (1 - 0.15)^4 = 5,00,000 * (0.85)^4.

Calculating (0.85)^4: 0.85 * 0.85 * 0.85 * 0.85 ≈ 0.52200625.

Book Value = 5,00,000 * 0.52200625 = ₹2,61,003.125.

Let's write down the given proportional relationships:

1. A ∝ B => A = k1 * B (where k1 is a constant)
2. B ∝ 1/C^2 => B = k2 / C^2 (where k2 is a constant)
3. C ∝ D => C = k3 * D (where k3 is a constant)

Now, we need to find the relationship between A and D. We can substitute the expressions step-by-step:

Substitute (3) into (2):
B = k2 / (k3 * D)^2
B = k2 / (k3^2 * D^2)
B = (k2 / k3^2) * (1 / D^2)
Let K4 = k2 / k3^2 (which is also a constant). So, B = K4 / D^2.

Now, substitute this expression for B into (1):
A = k1 * (K4 / D^2)
A = (k1 * K4) / D^2
Let K = k1 * K4 (which is a new constant).

So, A = K / D^2. This means A is inversely proportional to the square of D (A ∝ 1/D^2).

Let C be the cost, U be the number of units produced, and H be the hours worked. The relationship is C ∝ U^2 / H. This can be written as C = k * (U^2 / H), where k is the constant of proportionality.

New scenario:
New Production (U') = 2U
New Hours Worked (H') = H/2

New Cost (C') = k * ((2U)^2 / (H/2)) = k * (4U^2 / (H/2)) = k * (4U^2 * 2 / H) = k * (8U^2 / H).

Comparing C' with C: C' = 8 * (k * U^2 / H) = 8C. Therefore, the cost would increase by 8 times.

Syllogism is a form of deductive reasoning where you arrive at a specific conclusion by examining two other premises or ideas. Common formats include 'All A are B', 'Some A are B', etc.