Question 1

Find the logarithm of 144 to base 23, that is, evaluate log₂₃ 144 and choose the closest correct option.

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Introduction / Context: This problem is about estimating the value of a logarithm with an unusual base. Instead of asking for a detailed decimal approximation, the question simply wants to know which listed option could be equal to log₂₃ 144. You can solve this by comparing powers of 23 with 144 without using a calculator. Given Data / Assumptions: We need log₂₃ 144. Base is 23, argument is 144. Available options are 2, 4, 8, and None of these. All numbers are positive, and the logarithm is real. Concept / Approach: The logarithm log₂₃ 144 answers the question, "To what power must 23 be raised to obtain 144?" Therefore, we should compare 23¹, 23², and nearby powers to see the range in which 144 falls. If 144 were exactly a simple power of 23, its logarithm would be an integer, but otherwise it will be some non-integer value between two integers. This reasoning alone is enough to choose the correct option from the list. Step-by-Step Solution: Start with 23¹ = 23. Next, compute 23² = 23 × 23 = 529. Now compare 144 with these powers. We see that 23 < 144 < 529. Therefore, 23¹ < 144 < 23². This implies 1 < log₂₃ 144 < 2, because logarithms are increasing functions for bases greater than 1. Hence log₂₃ 144 is not equal to 2, 4, or 8, all of which are integers outside the interval (1, 2). Verification / Alternative check: If log₂₃ 144 were exactly 2, then 23² would equal 144, but 23² is 529. If it were 4, then 23⁴ would need to equal 144, but 23⁴ is huge, far above 144. Thus, any integer value from the options is clearly impossible. Why Other Options Are Wrong: 2: Would require 144 = 23² = 529, which is false. 4: Would require 144 = 23⁴, which is vastly larger than 144. 8: Would require 144 = 23⁸, an even more enormous number, so this is clearly incorrect. Common Pitfalls: Sometimes students try to overcomplicate such questions by attempting long logarithm calculations without real need. Here, conceptual understanding of logarithm behaviour and simple power comparisons are enough. Another mistake is to assume that because the number 144 is a perfect square (12²), it must somehow match a simple integer logarithm base 23, which is not true. Final Answer: Therefore, none of the integer values listed match log₂₃ 144. The correct choice is None of these.

Question 2

If log 64 (to base 10) is 1.8061, then what is the approximate value of log 16 (to base 10)?

Accepted Answer

Introduction / Context: This is a straightforward logarithm problem that uses the laws of indices and logarithms. You are given the common logarithm of 64 and asked to deduce the logarithm of 16. Since both numbers are powers of 2, the connection between them can be exploited to find log 16 without using tables or a calculator directly. Given Data / Assumptions: log 64 = 1.8061 (base 10). We need log 16 to the same base. 64 = 2⁶ and 16 = 2⁴. Concept / Approach: We make use of the identity log(a^k) = k log a. Since 64 and 16 are powers of the same base, we can express both in terms of log 2 and relate log 16 to log 64. Express log 64 as 6 log 2, then find log 2 and use it to compute log 16 as 4 log 2. Step-by-Step Solution: Write 64 as 2⁶, so log 64 = log(2⁶) = 6 log 2. We are given 6 log 2 = 1.8061. Therefore, log 2 = 1.8061 / 6. Divide: 1.8061 / 6 ≈ 0.3010167. Next, write 16 as 2⁴. So log 16 = log(2⁴) = 4 log 2. Substitute log 2: log 16 = 4 × 0.3010167 ≈ 1.2040668. Rounded to four decimal places, log 16 ≈ 1.2040. Verification / Alternative check: We know that 10¹ = 10 and 10² = 100. Since 16 lies between 10 and 100, log 16 should lie between 1 and 2. A value of about 1.2 is therefore reasonable. This cross-check with the position of 16 on the number line confirms that 1.2040 is a sensible approximation. Why Other Options Are Wrong: 1.9048 and 1.4521: These are too large for log 16, because they correspond to numbers much closer to 80 or 90 or beyond, not 16. 0.9840: This is less than 1, which would correspond to a number less than 10, whereas 16 is greater than 10. Common Pitfalls: A common mistake is to take a simple ratio of 16 and 64 and attempt to apply it directly to the logarithms without using the power rule. Another pitfall is rounding too early; dividing 1.8061 coarsely may introduce more error than necessary, although in this problem the answer options are far enough apart that small rounding differences do not affect the choice. Final Answer: The approximate value of log 16 is 1.2040.

Question 3

A fast moving superfast express train crosses a slower passenger train in 20 seconds. The speed of the faster train is 72 km/h and the speed of the slower train is 27 km/h. The length of the faster train is 100 metres. If both trains are moving in t…

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Introduction / Context: This question is a standard relative speed and trains problem. When one train overtakes another, the time taken depends on their relative speed and the total distance that must be covered, which is the sum of their lengths. We are given the speeds, the time of overtaking, and the length of the faster train, and must determine the length of the slower train. Given Data / Assumptions: Speed of the faster train = 72 km/h. Speed of the slower train = 27 km/h. Length of the faster train = 100 metres. Time taken to completely cross the slower train = 20 seconds. Both trains move in the same direction on parallel tracks. Concept / Approach: When two bodies move in the same direction, their relative speed is the difference of their speeds. During overtaking, the faster train has to cover a distance equal to the sum of the lengths of both trains with respect to the slower train. We convert the speeds from km/h to m/s, compute the relative speed, multiply by time to get the total distance covered, and then subtract the known length of the faster train to obtain the length of the slower train. Step-by-Step Solution: Convert speeds to m/s using 1 km/h = 5/18 m/s. Speed of faster train = 72 × 5/18 = 20 m/s. Speed of slower train = 27 × 5/18 = 7.5 m/s. Relative speed when moving in the same direction = 20 - 7.5 = 12.5 m/s. Time taken to overtake = 20 seconds. Total distance covered relative to the slower train = relative speed × time = 12.5 × 20 = 250 metres. This total distance equals the sum of the lengths of both trains. Let L be the length of the slower train. Then 100 + L = 250. So L = 250 - 100 = 150 metres. Verification / Alternative check: We can verify by reversing the reasoning. If the slower train is 150 metres long, then combined length = 100 + 150 = 250 metres. At a relative speed of 12.5 m/s, time to cross is distance / speed = 250 / 12.5 = 20 seconds, matching the given value. So the answer is consistent. Why Other Options Are Wrong: 100 m: Would imply combined length 200 metres, giving time = 200 / 12.5 = 16 seconds, not 20 seconds. 125 m and 175 m: These lengths would produce combined distances of 225 or 275 metres, which lead to overtaking times of 18 seconds or 22 seconds, neither matching the given 20 seconds. Common Pitfalls: Common mistakes include adding speeds instead of subtracting them for same-direction motion, forgetting to convert km/h to m/s, or mistakenly treating the distance to be covered as only one train length instead of the sum of both lengths. Always identify whether trains move in the same or opposite directions and convert units before applying formulas. Final Answer: The length of the slower passenger train is 150 metres.

Question 4

For a natural number x > 1, let p = logₓ (x + 1) and q = log₍ₓ₊₁₎ (x + 2). Which one of the following is correct about p and q?

Accepted Answer

Introduction / Context: This problem is a conceptual logarithm inequality question. Instead of asking for a numeric value, it compares two expressions involving logarithms with slightly different bases and arguments. Such questions test your understanding of how logarithmic functions behave as the base and argument change, and they are common in higher level aptitude and entrance exams. Given Data / Assumptions: x is a natural number with x > 1. p = logₓ (x + 1). q = log₍ₓ₊₁₎ (x + 2). We must decide whether p < q, p = q, p > q, or if the relation cannot be determined. Concept / Approach: For bases greater than 1, logarithmic functions are increasing in their argument and decreasing in their base. In p, the base is x and the argument is x + 1. In q, the base has increased to x + 1 and the argument to x + 2. We can compare p and q by writing both in terms of natural logarithms and carefully analysing how the ratios behave. Another powerful tool is to test several values of x to observe a pattern and then reason more generally. Step-by-Step Solution: Write p and q using natural logs: p = ln(x + 1) / ln x and q = ln(x + 2) / ln(x + 1). Evaluate numerically for x = 2. Then p = ln 3 / ln 2 ≈ 1.585 and q = ln 4 / ln 3 ≈ 1.262. So p > q for x = 2. For x = 3, p = ln 4 / ln 3 ≈ 1.262 while q = ln 5 / ln 4 ≈ 1.113, again p > q. For x = 4, p = ln 5 / ln 4 ≈ 1.161 while q = ln 6 / ln 5 ≈ 1.113, still p > q. These consistent results suggest that p is greater than q for all natural x > 1. To see why, notice that ln(x + 1) / ln x decreases slowly with x, but ln(x + 2) / ln(x + 1) is evaluated at a larger base and is pulled down more, making q smaller. Verification / Alternative check: We can reason qualitatively: for large x, (x + 1) is only slightly larger than x, so p is close to 1 from above. Meanwhile, x + 2 is only slightly larger than x + 1, so q is also close to 1 but with a slightly larger base in the denominator, which pushes q a bit lower than p. The numerical checks at several integer values confirm this consistent inequality p > q. Why Other Options Are Wrong: p < q: Contradicted by direct substitution for x = 2, 3, 4, all giving p > q. p = q: If they were equal for all x, numeric testing would show equality, but it does not. can't be determined: The pattern is clear and supported both numerically and conceptually, so the relationship is determinable as p > q. Common Pitfalls: A frequent mistake is to treat the increase in argument (from x + 1 to x + 2) as dominant and ignore the effect of increasing the base. Another error is assuming that logs with nearby arguments and bases automatically become equal. It is essential to remember that for base greater than 1, increasing the base while also increasing the argument does not guarantee any specific order without proper analysis. Final Answer: For all natural numbers x > 1, we have p > q.

Question 5

In common (base 10) logarithms, what is the characteristic of log 0.0000134?

Accepted Answer

Introduction / Context: This question focuses on the concept of the characteristic of a common (base 10) logarithm. For numbers less than 1, the characteristic is negative and tells us how far the number is from 1 in terms of powers of 10. Understanding this is essential for reading and using logarithm tables correctly. Given Data / Assumptions: We are working with common logarithms (base 10). The number is 0.0000134. We are asked only for the characteristic, not the mantissa. Concept / Approach: For a positive number N, the common logarithm log₁₀ N can be written as C + M, where C is the characteristic (integer part) and M is the mantissa (fractional part). For N between 0 and 1, the characteristic is negative. A useful rule is: if the first non-zero digit after the decimal occurs in the (k + 1)th place, then the characteristic is -(k + 1). Alternatively, rewrite the number in scientific notation N = a × 10^n, where 1 ≤ a < 10, and then log N = n + log a; here n will be the characteristic if a is between 1 and 10. Step-by-Step Solution: Write 0.0000134 in scientific notation. We can move the decimal point 5 places to the right: 0.0000134 = 1.34 × 10^-5. Then log(0.0000134) = log(1.34 × 10^-5). Using log(ab) = log a + log b, this becomes log 1.34 + log 10^-5. log 10^-5 = -5, and log 1.34 is a positive fraction between 0 and 1. Therefore log(0.0000134) = -5 + (a fractional mantissa). The characteristic is the integer part, which is -5. Verification / Alternative check: We can also count the zeros after the decimal point before the first non-zero digit. In 0.0000134, after the decimal point there are four zeros before the 1. The first significant digit 1 is in the fifth place after the decimal. By the rule, characteristic = -(number of digits to the first significant digit) = -5. This matches the previous method. Why Other Options Are Wrong: 5: This would correspond to a number around 10^5, which is huge compared to 0.0000134, which is very small. 6 and -6: These would indicate the number is closer to 10^6 or 10^-6. But 0.0000134 is equal to 1.34 × 10^-5, so the power of 10 involved is -5, not -6 or 6. Common Pitfalls: Students often miscount the zeros or confuse the sign for numbers less than 1. Another mistake is to think that the characteristic is simply the number of zeros, not the number of digits to the first non-zero digit. Always check by converting to scientific notation and identifying the power of 10 explicitly. Final Answer: The characteristic of log 0.0000134 is -5.

Question 6

Find the value of the sum log (tan 10°) + log (tan 20°) + log (tan 30°) + … + log (tan 80°), where all logarithms are to base 10.

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Introduction / Context: This trigonometric logarithm problem asks you to evaluate a sum of logarithms of tangent values at specific angles. The angles come in complementary pairs, and the key insight is to use the identity involving tan θ and tan(90° − θ). Understanding this symmetry helps dramatically simplify the expression without computing any actual tangent values. Given Data / Assumptions: The sum is S = log(tan 10°) + log(tan 20°) + log(tan 30°) + log(tan 40°) + log(tan 50°) + log(tan 60°) + log(tan 70°) + log(tan 80°). All logarithms are base 10. Angles are in degrees, and 0° < angle < 90°. Concept / Approach: We make use of the identity tan(90° − θ) = cot θ. The product tan θ × tan(90° − θ) = 1, since tan θ × cot θ = 1. Taking logarithms, log(tan θ) + log(tan(90° − θ)) = log 1 = 0. The given angles can be grouped into complementary pairs: (10°, 80°), (20°, 70°), (30°, 60°), and (40°, 50°). Each pair contributes zero to the sum of logs, so the entire sum collapses to zero. Step-by-Step Solution: Recall that tan(90° − θ) = cot θ and tan θ × cot θ = 1. Therefore tan θ × tan(90° − θ) = 1 for 0° < θ < 90°. Taking logarithms, log(tan θ) + log(tan(90° − θ)) = log 1 = 0. Now group the terms in S into complementary pairs: Pair 1: log(tan 10°) + log(tan 80°) = 0. Pair 2: log(tan 20°) + log(tan 70°) = 0. Pair 3: log(tan 30°) + log(tan 60°) = 0. Pair 4: log(tan 40°) + log(tan 50°) = 0. Add all pairs: S = 0 + 0 + 0 + 0 = 0. Verification / Alternative check: If desired, you can approximate a couple of pairs numerically. For example, tan 30° = 1/√3 and tan 60° = √3, so tan 30° × tan 60° = 1. Therefore log(tan 30°) + log(tan 60°) = log 1 = 0. Similar behaviour holds for (10°, 80°) and other pairs, confirming that every complementary pair contributes zero to the sum. Why Other Options Are Wrong: -1, 1/2, and 1: All these would imply that the product of all the tangent values is something other than 1, but the pairing argument shows that each pair multiplies to 1, and hence the overall product is 1 and the sum of logs must be 0. Common Pitfalls: A typical mistake is attempting to compute or approximate each tangent value directly, which is unnecessary and time consuming. Another pitfall is forgetting the identity tan(90° − θ) = cot θ, which is central to the simplification. Also, some students mis-pair angles or assume an incorrect period for tangent, leading to sign mistakes. Final Answer: The value of the sum of logarithms is 0.

Question 7

Evaluate the logarithmic expression: Find the numerical value of log base 2 of (log base 5 of 625), that is, compute log base 2 (log base 5 625).

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Introduction: This problem checks your understanding of nested logarithms, where one logarithm expression appears inside another. Correctly evaluating such expressions requires applying basic logarithm rules in the right order and simplifying step by step. Given Data / Assumptions: We must find the value of log base 2 (log base 5 625). All logarithms are assumed to have positive arguments and valid bases (not equal to 1). 625 is a power of 5, which will simplify the inner logarithm. Concept / Approach: To solve nested logarithms, first evaluate the inner logarithm completely, then treat its result as a regular number when evaluating the outer logarithm. Here, the inner term log base 5 of 625 can be simplified by expressing 625 as a power of 5. Once that is done, the outer log base 2 becomes straightforward because we are left with the logarithm of an integer. Step-by-Step Solution: Step 1: Simplify the inner logarithm log base 5 625. Observe that 625 = 5^4. Therefore, log base 5 625 = log base 5 (5^4) = 4. Step 2: Substitute this value into the outer logarithm. The original expression becomes log base 2 (4). Note that 4 = 2^2. Therefore, log base 2 (4) = log base 2 (2^2) = 2. Thus, the value of log base 2 (log base 5 625) is 2. Verification / Alternative check: We can verify by using the definition of a logarithm. Since 5^4 = 625, the inner log is exactly 4. Since 2^2 = 4, the outer log base 2 of 4 is exactly 2. No approximations are involved, so the result is exact and clearly correct. Why Other Options Are Wrong: The options 5, 10, and 15 would require either 625 or 4 to be a very different power of 5 or 2, which is not true. For example, log base 2 (4) is definitely not 5 or 10. These incorrect values usually arise from confusing the base and argument or from skipping steps when simplifying powers. Common Pitfalls: Common mistakes include trying to directly combine the logs without first evaluating the inner one, or thinking that log base 2 (log base 5 625) somehow equals log base 10 of 625. Remember that the base of a logarithm matters, and each log must be processed with its own base. Always simplify from the inside out when functions are nested. Final Answer: The value of log base 2 (log base 5 625) is 2.

Question 8

Let a, b, and c be respectively the p-th, q-th, and r-th terms of a geometric progression (G.P.). Evaluate the logarithmic expression: (q − r) log a + (r − p) log b + (p − q) log c.

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Introduction: This question tests your understanding of geometric progressions (G.P.) and logarithm properties. You are asked to simplify an algebraic expression involving logs of three terms of a G.P. with indices p, q, and r. The elegance of this problem lies in recognizing the common ratio structure and using logarithm rules to show that the whole expression simplifies neatly. Given Data / Assumptions: a is the p-th term of a G.P. b is the q-th term of the same G.P. c is the r-th term of the same G.P. We must evaluate (q − r) log a + (r − p) log b + (p − q) log c. We assume the base of the logarithm is fixed and positive, not equal to 1. Concept / Approach: In a geometric progression, each term can be written in terms of the first term and the common ratio. If the first term is A and the common ratio is R, then the n-th term is A * R^(n−1). Taking logs converts products into sums and powers into coefficients. By expressing log a, log b, and log c in terms of log A and log R, we can substitute into the given expression and see large cancellations. Step-by-Step Solution: Let the first term of the G.P. be A and the common ratio be R. Then a = A * R^(p−1), b = A * R^(q−1), c = A * R^(r−1). Take logarithms: log a = log A + (p−1) log R log b = log A + (q−1) log R log c = log A + (r−1) log R Now form the expression E = (q−r) log a + (r−p) log b + (p−q) log c. Substitute logs: E = (q−r)[log A + (p−1) log R] + (r−p)[log A + (q−1) log R] + (p−q)[log A + (r−1) log R] Group coefficients of log A and log R separately. Coefficient of log A: (q−r) + (r−p) + (p−q) = 0. Coefficient of log R: (q−r)(p−1) + (r−p)(q−1) + (p−q)(r−1). This second coefficient also simplifies to 0 after expansion and cancellation. Hence E = 0 * log A + 0 * log R = 0. Verification / Alternative check: You can test with a concrete example. Take a simple G.P. like 2, 4, 8, 16, ... with A = 2 and R = 2, and choose some values for p, q, r. Compute a, b, c and evaluate the expression numerically with logs. You will always obtain zero, confirming the algebraic result. Why Other Options Are Wrong: The values 1, −1, or pqr introduce nonzero constants or dependence on indices that do not appear after simplification. Because both log A and log R cancel out entirely, the expression cannot depend on p, q, r, or any fixed nonzero constant; it must be exactly zero. Common Pitfalls: Many learners forget to write the G.P. terms in the general form A * R^(n−1), or they make sign mistakes when expanding the coefficients. Others attempt to treat log a, log b, and log c as unrelated, which hides the underlying G.P. structure. Always express all related quantities in a common parametric form before simplifying. Final Answer: The expression simplifies to 0 for any geometric progression and any integers p, q, and r.

Question 9

Evaluate the logarithmic expression: Find the exact numerical value of log base 343 of 7, that is, compute log base 343 (7).

Accepted Answer

Introduction: This question focuses on evaluating a simple logarithm where the base and the argument are powers of the same number. Recognizing this power relationship allows you to compute the logarithm exactly without using calculators or approximate values. Given Data / Assumptions: We must find log base 343 (7). 343 is a composite number, and we should try to express it in terms of a simpler base. The logarithm is taken with a valid base greater than 0 and not equal to 1. Concept / Approach: When both the base and the argument of a logarithm are powers of the same number, we can write them as a^m and a^n, respectively. Then log base a^m (a^n) equals n / m. This comes directly from the definition of a logarithm and properties of exponents. Here, both 343 and 7 relate naturally to the base 7. Step-by-Step Solution: Step 1: Express 343 as a power of 7. 343 = 7 × 7 × 7 = 7^3. So the base 343 is 7^3, and the argument is 7^1. Step 2: Rewrite the logarithm using these powers. log base 343 (7) = log base 7^3 (7^1). Step 3: Use the identity log base a^m (a^n) = n / m. Here a = 7, m = 3, and n = 1. Therefore, log base 7^3 (7^1) = 1 / 3. So log base 343 (7) = 1/3. Verification / Alternative check: By definition, log base 343 (7) = x means 343^x = 7. If x = 1/3, then 343^(1/3) is the cube root of 343, which is 7. Since this matches the argument, x = 1/3 is indeed the correct logarithm value. Why Other Options Are Wrong: The value 3 would imply 343^3 = 7, which is false because 343^3 is much larger than 7. The values −3 and −1/3 would imply 343 raised to a negative power equals 7, which would yield a fraction less than 1, not 7. Therefore, none of these can satisfy the logarithm equation. Common Pitfalls: Learners sometimes reverse the roles of base and argument, accidentally evaluating log base 7 (343) instead of log base 343 (7). Others forget the exponent rule and try to approximate using decimal logs. Always rewrite both numbers as powers of the same base and apply the n / m rule carefully. Final Answer: The exact value of log base 343 (7) is 1/3.

Question 10

Solve the linear equation in one variable: Given 5x − 17 = −x + 7, find the value of x that satisfies this equation.

Accepted Answer

Introduction: This problem is a straightforward example of solving a linear equation in one variable. Such equations are a foundation for algebra and appear frequently in aptitude and entrance examinations. The goal is to isolate x on one side of the equation using basic algebraic operations. Given Data / Assumptions: Equation: 5x − 17 = −x + 7 x is a real number. We must find the unique value of x that makes both sides of the equation equal. Concept / Approach: To solve a linear equation, we typically collect all x terms on one side and all constant terms on the other side. Then we combine like terms and divide to isolate x. The operations used must be valid on both sides of the equation to maintain equality. Because this is a first degree equation, we expect exactly one solution. Step-by-Step Solution: Start with: 5x − 17 = −x + 7 Step 1: Move all x terms to the left side by adding x to both sides. 5x − 17 + x = −x + 7 + x This gives: 6x − 17 = 7 Step 2: Move the constant term −17 to the right side by adding 17 to both sides. 6x − 17 + 17 = 7 + 17 This gives: 6x = 24 Step 3: Divide both sides by 6 to isolate x. x = 24 / 6 = 4 Therefore, x = 4 is the solution. Verification / Alternative check: Substitute x = 4 back into the original equation. Left side: 5(4) − 17 = 20 − 17 = 3. Right side: −4 + 7 = 3. Both sides equal 3, so x = 4 satisfies the equation exactly. No other value will do so for a linear equation of this type. Why Other Options Are Wrong: If you substitute x = 5, x = 7, or x = 9 into the equation, the left and right sides will not match. For example, for x = 5, left side is 25 − 17 = 8, while right side is −5 + 7 = 2. Hence these values cannot be solutions. Common Pitfalls: Typical mistakes include sign errors when moving terms across the equal sign, or incorrect arithmetic when adding or dividing. Some students also try to divide too early without first combining like terms. Proceeding step by step and checking each operation avoids these errors. Final Answer: The value of x that satisfies the equation is 4.

Question 11

Let A = log base 32 of 1875 and B = log base 243 of 2187. Based on these definitions, which of the following relationships between A and B is correct?

Accepted Answer

Introduction: This question tests your ability to compare two logarithmic expressions without necessarily computing them to many decimal places. By recognizing the structure of the numbers involved, you can simplify or estimate each logarithm and then decide whether A is less than, equal to, or greater than B. Given Data / Assumptions: A = log base 32 (1875) B = log base 243 (2187) We must determine whether A < B, A = B, A > B, or if the relationship cannot be determined. Concept / Approach: The key idea is to exploit the fact that 32 and 243 are perfect powers and that 2187 is also a perfect power. Specifically, 32 = 2^5, 243 = 3^5, and 2187 = 3^7. For B, this allows an exact and simple calculation. For A, we can express 1875 in terms of prime factors and then obtain a reasonable estimate by comparing exponents, or by using change of base thinking qualitatively. Step-by-Step Solution: Step 1: Compute B exactly. 243 = 3^5 and 2187 = 3^7. Thus B = log base 3^5 (3^7) = 7 / 5 = 1.4. Step 2: Analyze A = log base 32 (1875). We know 32 = 2^5. Factor 1875: 1875 = 3 × 5^4. So A = log base 2^5 (3 × 5^4). This does not simplify to a simple rational like B, but we can compare orders of magnitude. Note that 2^5 = 32 and 32^2 = 1024, 32^3 = 32768. 1875 lies between 32^1 and 32^2, but much closer to 32^2 than to 32. As a result, log base 32 (1875) is between 1 and 2, and closer to 2 than to 1. A rough estimation shows A is a bit above 2, while B is exactly 1.4. Therefore, A is greater than B. Verification / Alternative check: Using approximate common logarithms, log10(1875) is about 3.27 and log10(32) is about 1.51. Then A ≈ 3.27 / 1.51 ≈ 2.16. Since B = 1.4 exactly, this confirms that A > B, with a comfortable margin so that rounding errors cannot change the ordering. Why Other Options Are Wrong: A < B would require A to be less than 1.4, which contradicts the fact that 1875 is far larger than 32 and close to 32^2. A = B is impossible because we have computed B exactly and A approximately as a clearly larger number. Saying the relationship cannot be determined is incorrect because we have enough information to compare them. Common Pitfalls: Some learners attempt to compare the arguments 1875 and 2187 directly, ignoring the different bases. Others may forget that larger bases shrink logarithm values and smaller bases increase them. Always rewrite numbers in terms of prime powers where possible and use the definition of logarithms logically rather than guessing. Final Answer: We conclude that A is greater than B, so A > B is correct.

Question 12

If log 2 = 0.3010 and log 3 = 0.4771 (to base 10), then what is the value of log base 5 of 512, that is log₅ 512?

Accepted Answer

Introduction / Context:This problem tests the use of logarithm properties and change of base formula. We are given common logarithms (base 10) of 2 and 3 and asked to find log base 5 of 512. Since 512 is a power of 2 and 5 is related to 10 through 2, we can express everything in terms of base 10 logarithms, manipulate them using standard rules, and then compute the required value numerically. Given Data / Assumptions: log 2 (base 10) = 0.3010. log 3 (base 10) = 0.4771 (though not directly needed for this question). We need to find log₅ 512. All unspecified logarithms are assumed to be base 10. We use the change of base formula: logₐ b = (log b) / (log a). Concept / Approach:First, note that 512 = 2⁹. So log 512 (base 10) equals 9 times log 2. Also, 5 can be written as 10 / 2, so log 5 can be expressed in terms of log 10 and log 2. The change of base formula tells us that log₅ 512 = (log 512) / (log 5), where these logs are base 10. After substituting the given value of log 2, we compute the numerical answer and compare it with the options given. Step-by-Step Solution:Step 1: Write 512 as a power of 2: 512 = 2⁹. Step 2: Compute log 512 using base 10: log 512 = log(2⁹) = 9 log 2 = 9 × 0.3010 = 2.709. Step 3: Express log 5 in terms of log 10 and log 2. Since 10 = 2 × 5, log 10 = log 2 + log 5. Thus, log 5 = log 10 − log 2 = 1 − 0.3010 = 0.6990. Step 4: Use change of base formula: log₅ 512 = (log 512) / (log 5) = 2.709 / 0.6990. Step 5: Divide 2.709 by 0.6990 to get approximately 3.8755, which rounds to 3.875. Verification / Alternative check:We can check whether 5³·⁸⁷⁵ is approximately equal to 512. Since 5³ = 125 and 5⁴ = 625, the true value should lie between 3 and 4, closer to 4. Our result 3.875 is indeed between 3 and 4 and closer to 4, which is reasonable. Also, using slightly more precise logs would give a value very close to 3.876, reinforcing that 3.875 is the correct option. Why Other Options Are Wrong:Option 2.875 is much too small and would correspond to a number far less than 512. Options 4.875 and 5.875 would lead to huge values, much greater than 512, because the exponential growth of 5 to such powers is very large. Only 3.875 matches the computed ratio of log 512 to log 5 using the given logarithm values. Common Pitfalls:Students sometimes forget to use the change of base formula and attempt to manipulate bases directly. Another common error is to miscalculate log 5; they might mistakenly add instead of subtracting, writing log 5 = 1 + 0.3010. Additionally, rounding too early in the process can lead to slight discrepancies; it is better to maintain at least three decimal places until the final division step. Final Answer:The value of log₅ 512 is 3.875.

Question 13

If log 27 (to base 10) is 1.431, then what is the value of log 9 (to the same base, correct to three decimal places)?

Accepted Answer

Introduction / Context:This question checks understanding of logarithm laws, especially how to relate logs of numbers that are powers of the same base. We are given the common logarithm of 27 and are asked to find the logarithm of 9. Both numbers are powers of 3, so we can express them in terms of log 3 and then calculate log 9 from the given numerical value of log 27. Given Data / Assumptions: log 27 (base 10) = 1.431. We need to find log 9 (base 10). All logarithms are base 10 unless otherwise specified. 27 and 9 can be expressed as powers of 3: 27 = 3³ and 9 = 3². Concept / Approach:Use the rule log aⁿ = n log a. First express 27 and 9 in terms of 3. From log 27 = log 3³ = 3 log 3, we can find log 3. Then use log 9 = log 3² = 2 log 3. Substituting the value of log 3 obtained from the given information yields the required numerical value of log 9. Step-by-Step Solution:Step 1: Write 27 as a power of 3: 27 = 3³. Step 2: Apply the power rule: log 27 = log(3³) = 3 log 3. Step 3: Given that log 27 = 1.431, set 3 log 3 = 1.431, so log 3 = 1.431 / 3. Step 4: Compute log 3: 1.431 / 3 = 0.477. Step 5: Now write 9 as 3² and use the power rule again: log 9 = log(3²) = 2 log 3. Step 6: Substitute log 3 = 0.477 to obtain log 9 = 2 × 0.477 = 0.954. Verification / Alternative check:We can quickly check the consistency of these logs. If log 3 ≈ 0.477, then log 27 = log 3³ = 3 × 0.477 = 1.431, which matches the given data. Similarly, 10^0.954 is between 10^0.9 and 10^1.0, that is between about 7.94 and 10, which is reasonable since 9 lies in this range. This supports that 0.954 is the correct value for log 9. Why Other Options Are Wrong:Options 0.754, 0.854, and 0.654 do not satisfy the relationship between log 27 and log 9 when tested through log 3. Option 1.054 is larger than 1, which would correspond to a number greater than 10, not 9. Only 0.954 preserves the exact relationship log 27 = 1.431 and log 9 = 2 log 3 based on that value. Common Pitfalls:Some learners mistakenly divide the given log 27 by 2 instead of 3 or add or subtract arbitrary values. Others forget that the same logarithm base must be used consistently. Confusing 27 and 9 or misapplying the power rule is also common. Carefully expressing each number as a power of 3 and applying log rules avoids these errors. Final Answer:The value of log 9 (base 10) is 0.954 (approximately).

Question 14

If log 2 = 0.30103 (base 10), then find the number of digits in 2⁵⁶.

Accepted Answer

Introduction / Context:This question uses logarithms to find the number of digits in a large power of 2. Directly calculating 2⁵⁶ is not practical by hand, but the relationship between the logarithm of a number and its number of digits gives a simple and efficient method. This is a standard application of logarithms in number theory and competitive aptitude exams. Given Data / Assumptions: log 2 (base 10) = 0.30103. We need to find the number of digits in 2⁵⁶. All logarithms have base 10. We use the formula: if N is a positive integer, the number of digits in N is equal to floor(log₁₀ N) + 1. Concept / Approach:First express 2⁵⁶ in terms of logs: log₁₀(2⁵⁶) = 56 log₁₀ 2. Once we compute this value, we apply the formula for the number of digits. The integer part of log₁₀ N tells us how many powers of 10 fit into N, and adding one gives the exact number of digits. This avoids any need for direct multiplication of large numbers. Step-by-Step Solution:Step 1: Use the given value log 2 = 0.30103 with base 10. Step 2: Compute log(2⁵⁶): log(2⁵⁶) = 56 log 2 = 56 × 0.30103. Step 3: Multiply 56 × 0.30103 = 16.85768. Step 4: According to the digit formula, if N = 2⁵⁶, the number of digits in N is floor(log N) + 1. Step 5: Here, floor(16.85768) = 16, so the number of digits = 16 + 1 = 17. Verification / Alternative check:We know approximate powers of 2: 2¹⁰ is about 1024 (4 digits), 2²⁰ is about 10⁶ (7 digits), 2³⁰ around 10⁹, and 2⁵⁰ around 10¹⁵. Therefore, 2⁵⁶ will be around 10¹⁶·⁸, which clearly has 17 digits. This matches the result obtained from the logarithm method. The approximation confirms that our calculation is reasonable and consistent with known powers. Why Other Options Are Wrong:If the number of digits were 19, 23, or 25, that would imply exponents near 18, 22, or 24 in base 10, which do not agree with log(2⁵⁶) being 16.85768. Option 21 also does not fit the digit formula. Only 17 matches floor(16.85768) + 1 and is therefore correct. Common Pitfalls:Some learners round log(2⁵⁶) to 17 and then add 1 to get 18, which is incorrect. It is crucial to use the floor of the logarithm, not a rounded value. Others may forget the plus one in the formula and answer 16 instead. Maintaining precision in the multiplication step and strictly applying floor(log N) + 1 is key to solving such problems accurately. Final Answer:The number of digits in 2⁵⁶ is 17.

Question 15

If logₐ b = x (logarithm to base a), then what is the value of log_b a in terms of x?

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Introduction / Context:This problem checks conceptual understanding of logarithms, specifically the relationship between logₐ b and log_b a. It uses the basic definition of a logarithm and the idea that logarithms are inverses of exponentiation. The goal is to express log_b a in terms of x, where x is given as logₐ b. This forms a classic identity that is frequently used in simplifying logarithmic expressions and changing bases. Given Data / Assumptions: logₐ b = x, that is, log base a of b equals x. Both a and b are positive and not equal to 1, which are standard conditions for logarithms. We need to find log_b a in terms of x. All logs have real values and are defined under usual logarithm rules. Concept / Approach:From the definition of logarithm, logₐ b = x means aˣ = b. We want log_b a, which by definition is the exponent to which b must be raised to produce a. Because a and b are related through aˣ = b, the relation between these logs becomes a simple reciprocal. Using either direct exponentiation or the change of base formula, we can show that log_b a is equal to 1 / x. Step-by-Step Solution:Step 1: Start with the given relation logₐ b = x. Step 2: By definition of logarithm, logₐ b = x implies aˣ = b. Step 3: We wish to find log_b a, which is the exponent y such that bʸ = a. Step 4: Substitute b = aˣ into bʸ = a to obtain (aˣ)ʸ = a. Step 5: Simplify the left side: (aˣ)ʸ = a^(x y). Step 6: Thus a^(x y) = a¹, which means x y = 1 because the bases are equal and positive. Step 7: Therefore y = 1 / x. But y is log_b a, so log_b a = 1 / x. Verification / Alternative check:An alternative method uses the change of base formula. We have logₐ b = (log b) / (log a) = x. Then log_b a = (log a) / (log b). Dividing the second by the first, log_b a = 1 / x, since (log a) / (log b) equals 1 divided by (log b / log a). Thus both methods confirm that log_b a is the reciprocal of logₐ b. Why Other Options Are Wrong:Expressions such as x / (x + 1), x / (1 − x), x / (x − 1), or x² do not arise from either the exponent definition or the change of base formula. If any of these were correct, substituting simple values like a = 2 and b = 8 would lead to contradictions. Only the reciprocal 1 / x is consistent with basic logarithm properties. Common Pitfalls:Many learners mistakenly think log_b a equals x or 1 minus x. Others try to manipulate the symbols without referring back to the clear meaning of logₐ b as an exponent. Remembering that logₐ b and log_b a are inverses in this specific way and checking with simple numerical choices for a and b is a good strategy to avoid errors. Final Answer:The required value is log_b a = 1 / x.

Question 16

If logₓ (9/16) = −1/2 (logarithm to base x), then what is the value of x?

Accepted Answer

Introduction / Context:This problem tests the ability to convert a logarithmic equation into its equivalent exponential form and then solve for the base. The equation involves a fractional argument 9/16 and a negative fractional logarithm. Correctly interpreting the log definition and manipulating powers and roots of fractions are key skills here. Given Data / Assumptions: logₓ (9/16) = −1/2. We need to find the value of the base x. Both x and 9/16 are positive, and x is not equal to 1, as required for logarithms. We use the fundamental definition of logarithm: logₐ b = c implies aᶜ = b. Concept / Approach:From the definition logₓ (9/16) = −1/2, we can rewrite this as x^(−1/2) = 9/16. This means 1 / √x equals 9/16. By inverting both sides, we get √x equals 16/9. Squaring both sides then gives x as a rational number. Throughout this process we work carefully with fractions to avoid errors. Step-by-Step Solution:Step 1: Start from the given equation logₓ (9/16) = −1/2. Step 2: Use the definition of logarithm: logₓ (9/16) = −1/2 means x^(−1/2) = 9/16. Step 3: Rewrite x^(−1/2) as 1 / √x. Thus, 1 / √x = 9 / 16. Step 4: Take reciprocals of both sides to obtain √x = 16 / 9. Step 5: Square both sides: x = (16 / 9)² = 256 / 81. Step 6: Therefore x equals 256/81, which matches one of the given options. Verification / Alternative check:Verify by substituting x = 256/81 back into the original logarithm. First, compute x^(−1/2) = 1 / √x. Since √x = 16/9, x^(−1/2) = 9/16. Therefore logₓ (9/16) equals −1/2, because raising 256/81 to the power −1/2 gives exactly 9/16. This confirms that our solution satisfies the original equation. Why Other Options Are Wrong:If x were 3/4 or −3/4, then x^(−1/2) would not equal 9/16; the base must be positive and not equal to 1. Option 81/256 is the reciprocal of the correct answer, which would yield √x = 9/16 and x^(−1/2) equal to 16/9 instead, the inverse of what we need. Option 16/9 is only the square root of the correct base and so does not satisfy the equation when used directly. Common Pitfalls:Some learners mistakenly treat logₓ (9/16) = −1/2 as x^(1/2) = 9/16 instead of x^(−1/2) = 9/16, forgetting that a negative log indicates a reciprocal. Others square the fraction incorrectly or confuse 16/9 with 9/16. Paying careful attention to the sign of the exponent and to operations with fractions is essential for solving such problems. Final Answer:The value of x is 256/81.

Question 17

If log₇ 2 = m (logarithm to base 7), then what is the value of log₄ 28 in terms of m?

Accepted Answer

Introduction / Context:This question tests the ability to express one logarithm in terms of another using properties of logarithms and change of base. We are given log₇ 2 in terms of a parameter m and asked to find log₄ 28 in terms of m. The key is to rewrite 28 and 4 using prime factorisation and then apply log rules carefully to relate them back to m. Given Data / Assumptions: log₇ 2 = m. We need to express log₄ 28 in terms of m. All logs are real and defined for positive arguments and bases not equal to 1. Note that 28 = 4 × 7 and 4 = 2². Concept / Approach:Since 28 = 4 × 7, we can write log₄ 28 = log₄ (4 × 7) = log₄ 4 + log₄ 7 = 1 + log₄ 7, because log₄ 4 is 1. Therefore we only need log₄ 7. Using the change of base formula log₄ 7 = (log 7) / (log 4) and expressing both numerator and denominator in terms of log 2 and log 7, we can then connect back to the given relation log₇ 2 = m and simplify to get an expression involving m only. Step-by-Step Solution:Step 1: Start from log₄ 28. Since 28 = 4 × 7, we have log₄ 28 = log₄ (4 × 7) = log₄ 4 + log₄ 7. Step 2: log₄ 4 equals 1, so log₄ 28 = 1 + log₄ 7. Step 3: Use change of base for log₄ 7: log₄ 7 = (log 7) / (log 4), where logs are base 10 or any common base. Step 4: Express log 4 in terms of log 2: log 4 = log(2²) = 2 log 2. Step 5: From log₇ 2 = m, using change of base, log₇ 2 = (log 2) / (log 7) = m. Therefore, log 2 = m log 7. Step 6: Substitute log 2 = m log 7 into log 4: log 4 = 2 log 2 = 2m log 7. Step 7: Now log₄ 7 = (log 7) / (log 4) = (log 7) / (2m log 7) = 1 / (2m). Step 8: Finally, log₄ 28 = 1 + log₄ 7 = 1 + 1 / (2m) = (2m + 1) / (2m). Verification / Alternative check:As a numerical check, suppose log₇ 2 equals a concrete value by taking 2 and 7 logs in base 10. Then compute m = log 2 / log 7, and separately compute log₄ 28 numerically. Simplifying (2m + 1) / (2m) and comparing with the numeric log₄ 28 confirms that the derived expression is correct, since both evaluations match up to rounding errors. Why Other Options Are Wrong:The other algebraic expressions such as 1/(1 + 2m), (1 + 2m)/2, 2m/(2m + 1), and 4m/(2m + 1) do not follow from the change of base derivation. Substituting specific approximate values for m put them in clear disagreement with the actual value of log₄ 28, while (2m + 1) / (2m) agrees exactly. Common Pitfalls:One common mistake is to attempt to treat log₇ 2 as log 7 divided by log 2, which is the reciprocal of the correct change of base relation. Another error is to misinterpret 28 as 2 × 7² rather than 4 × 7 and then take a longer and more error prone path. Careful use of prime factorisation and attention to base conversions greatly simplifies these problems. Final Answer:The value of log₄ 28 in terms of m is (2m + 1) / (2m).

Question 18

Find the logarithm of 144 to the base 23, that is evaluate log₂₃ 144. Choose the correct value from the given options.

Accepted Answer

Introduction / Context:This question asks for log₂₃ 144, the logarithm of 144 to base 23. The options list a few integer values as well as a general option stating that none of these is correct. To answer correctly, we must reason about the magnitude of 23 raised to different powers in relation to 144, and we can also use the change of base formula for a more precise numerical estimate. Given Data / Assumptions: The base is 23 and the argument is 144. We need to evaluate log₂₃ 144. Available options are 2, 4, 8, 16 and a general option None of these. All logarithms are real with positive bases and positive arguments, bases not equal to 1. Concept / Approach:First, use simple bounds. We know 23¹ = 23 and 23² = 529. Since 144 lies between 23 and 529, log₂₃ 144 must be between 1 and 2. Therefore it cannot be equal to 2, 4, 8, or 16, all of which are outside this interval. For additional confirmation, the change of base formula log₂₃ 144 = (log 144) / (log 23) in base 10 can be used to estimate its exact numerical value, which will clearly not be one of the integer options listed. Step-by-Step Solution:Step 1: Note that 23¹ = 23 and 23² = 529. Step 2: Since 144 is greater than 23 but less than 529, log₂₃ 144 must lie strictly between 1 and 2. Step 3: Observe that the options 2, 4, 8, and 16 are all greater than or equal to 2, so none of them can represent a logarithm value that is strictly between 1 and 2. Step 4: If we wish, we can apply the change of base formula: log₂₃ 144 = (log 144) / (log 23), where these logs are any common base such as 10. Step 5: Approximate values show that this ratio is a non integer real number between 1 and 2, confirming the earlier inequality based reasoning. Verification / Alternative check:We can further check by estimating the value numerically. Since 23² = 529 is much larger than 144, we expect log₂₃ 144 to be significantly closer to 1 than to 2. Using approximate decimal logs (for instance from a calculator) would confirm that log₂₃ 144 is approximately in the range 1.4 to 1.5, clearly not equal to any of the integer values 2, 4, 8, or 16. Why Other Options Are Wrong:Option 2 would imply 23² = 144, which is not true because 23² = 529. Options 4, 8, and 16 would imply enormous values like 23⁴, 23⁸, and 23¹⁶, which are far larger than 144. Therefore they cannot be the logarithm of 144 to base 23. The only reasonable conclusion is that the exact value is not among these integers. Common Pitfalls:Some learners may misunderstand and try to find an approximate integer close to the real value without checking the underlying power relationships. Others may think that since 144 is near 11², they must use 11 somehow, but the base here is 23, not 11. Always consider simple bounding by known powers of the base before attempting precise calculations. Final Answer:The correct choice is None of these, since log₂₃ 144 is a non integer between 1 and 2.

Question 19

If log 64 (base 10) is 1.8061, then what is the approximate value of log 16 (base 10)?

Accepted Answer

Introduction / Context:This question requires using properties of logarithms to relate the log of one power of 2 to the log of another power of 2. We are given log 64 and asked to find log 16. Both numbers are powers of 2, which allows us to express them in terms of log 2 and connect the two values directly. This reinforces understanding of the power rule of logarithms. Given Data / Assumptions: log 64 (base 10) = 1.8061. We need to find log 16 (base 10). 64 = 2⁶ and 16 = 2⁴. We use the rule log aⁿ = n log a. Concept / Approach:First write 64 and 16 as powers of 2: 64 = 2⁶ and 16 = 2⁴. From log 64 = log(2⁶) = 6 log 2, we can find log 2 by dividing the given value by 6. Then use log 16 = log(2⁴) = 4 log 2. Substituting the value of log 2 gives the required value of log 16, which can then be rounded to four decimal places and compared with the given options. Step-by-Step Solution:Step 1: Express 64 as a power of 2: 64 = 2⁶. Step 2: Using the power rule, log 64 = log(2⁶) = 6 log 2. Step 3: Given log 64 = 1.8061, we have 6 log 2 = 1.8061, so log 2 = 1.8061 / 6. Step 4: Compute log 2: 1.8061 / 6 ≈ 0.3010 (more precisely 0.3010167, close to the standard value). Step 5: Express 16 as 2⁴. Then log 16 = log(2⁴) = 4 log 2. Step 6: Substitute log 2 ≈ 0.3010: log 16 ≈ 4 × 0.3010 = 1.2040. Verification / Alternative check:We know from standard log tables that log 2 is about 0.3010, so log 16 = log(2⁴) = 4 × 0.3010 ≈ 1.2040. This matches our computed value. Also, 16 is between 10 and 100, so its log must be between 1 and 2. A value of 1.2040 is reasonable and consistent with 16 being closer to 10 than to 100. Why Other Options Are Wrong:Option 1.9048 is too large and would correspond to a number closer to 80 or 90, not 16. Option 0.9840 is less than 1 and would be for a number below 10. Option 1.4521 would be for a number around 28 to 30, also not 16. Option 1.0000 is exactly log 10 and does not match 16. Only 1.2040 fits both the power rule relationship and the approximate position of 16 on the log scale. Common Pitfalls:Some learners mistakenly divide by 4 instead of 6 when computing log 2, or they directly approximate log 16 without using the connection to log 64. Others confuse 2⁶ with 2⁴ and misapply the power rule. Being systematic, first finding log 2 from log 64 and then finding log 16, helps avoid these errors. Final Answer:The approximate value of log 16 is 1.2040 (base 10).

Question 20

A fast moving superfast express train crosses another passenger train in 20 seconds. The speed of the faster train is 72 km per hour and the speed of the slower train is 27 km per hour. The faster train is 100 metres long. If both trains are moving …

Accepted Answer

Introduction / Context:This is a standard relative speed problem involving two trains moving in the same direction on parallel tracks. The faster train overtakes or crosses the slower train in a given time. We know the speeds of both trains and the length of the faster train. When one train completely crosses the other, the relative speed and the sum of their lengths determine the time taken. From this, we can find the unknown length of the slower train. Given Data / Assumptions: Speed of the faster train = 72 km per hour. Speed of the slower train = 27 km per hour. Length of the faster train = 100 metres. Time taken to cross the slower train completely = 20 seconds. Both trains run in the same direction on parallel tracks. Concept / Approach:When two objects move in the same direction, their relative speed is the difference of their speeds. Here, the faster train effectively moves past the slower one at this relative speed. The distance covered in the crossing process is equal to the sum of the lengths of the two trains, because the faster train has to pass the entire length of the slower train plus its own length. Using the basic relation distance = speed × time, we can solve for the unknown length of the slower train in metres. Step-by-Step Solution:Step 1: Calculate the relative speed when both trains move in the same direction. Step 2: Relative speed = 72 km/h − 27 km/h = 45 km/h. Step 3: Convert this relative speed into metres per second. Since 1 km/h = (5/18) m/s, we have 45 km/h = 45 × 5 / 18 = 225 / 18 = 12.5 m/s. Step 4: Let L be the length of the slower train in metres. The total distance covered during crossing is L + 100 metres. Step 5: Time taken to cross is 20 seconds, so distance = speed × time gives L + 100 = 12.5 × 20. Step 6: Compute the right side: 12.5 × 20 = 250. Step 7: Therefore L + 100 = 250, so L = 250 − 100 = 150 metres. Verification / Alternative check:We can verify by checking if the calculated length is consistent. The total length to be crossed is 100 + 150 = 250 metres. At 12.5 m/s, the time required is 250 / 12.5 = 20 seconds, which matches the given time. This confirms that the length of the slower train has been computed correctly. Why Other Options Are Wrong:If the slower train were 100 metres, the total length would be 200 metres, and time would be 200 / 12.5 = 16 seconds, not 20. For 125 metres, the total length is 225 metres, giving 18 seconds. For 175 metres, the total length is 275 metres, giving 22 seconds. A length of 200 metres would give 300 metres total and 24 seconds. None of these match the required 20 seconds; only 150 metres does. Common Pitfalls:Common mistakes include adding the speeds instead of subtracting them when trains move in the same direction, forgetting to convert speeds from km per hour to metres per second, or using only one train length instead of the sum of both lengths. Carefully distinguishing between opposite and same direction cases and being systematic with units avoids these errors. Final Answer:The length of the slower train is 150 metres.

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