Question 1

In aptitude (fractions and percentages simplification), simplify the following equation with mixed numbers and percentages and find the missing value represented by '?' in (4 1/5) * (3 1/3) + ? = 20% of 120.

Accepted Answer

Introduction / Context: This question combines basic operations on mixed numbers, multiplication of fractions, and percentage calculation. Such simplification questions are very common in bank exams, SSC, and other competitive tests. The goal is to carefully convert mixed numbers into improper fractions, multiply them correctly, and then compare the result with a percentage of a whole number to find the missing value represented by the symbol '?'. Accuracy with fractions and percentages is crucial here. Given Data / Assumptions: - Expression: (4 1/5) * (3 1/3) + ? = 20% of 120. - Mixed numbers 4 1/5 and 3 1/3 should be converted to improper fractions. - Percentage 20% of 120 means 20/100 * 120. - Operations are standard arithmetic under real numbers. Concept / Approach: The concept is straightforward: first evaluate the left part without the unknown, that is (4 1/5) * (3 1/3). Then compute the right side, which is 20% of 120. Once both values are computed, treat the equation as A + ? = B and solve for the missing value ? = B - A. Careful conversion and simplification of fractions will make the arithmetic cleaner and reduce chances of mistakes. Step-by-Step Solution: Step 1: Convert 4 1/5 to an improper fraction: 4 1/5 = (4 * 5 + 1) / 5 = 21 / 5.Step 2: Convert 3 1/3 to an improper fraction: 3 1/3 = (3 * 3 + 1) / 3 = 10 / 3.Step 3: Multiply the fractions: (21 / 5) * (10 / 3) = (21 * 10) / (5 * 3) = 210 / 15 = 14.Step 4: Compute the right-hand side: 20% of 120 = (20 / 100) * 120 = 0.2 * 120 = 24.Step 5: Set up the equation: 14 + ? = 24.Step 6: Rearrange to find ?: ? = 24 - 14 = 10. Verification / Alternative check: We can verify quickly by substituting the found value of ? back into the original expression. If ? = 10, then the left side is 14 + 10 = 24. The right side is 20% of 120, which is also 24. Since both sides match exactly, the value ? = 10 is completely consistent and verified. Why Other Options Are Wrong: - 12: This would correspond to incorrectly evaluating the mixed number product or miscomputing 20% of 120. - 14: This may arise if a student mistakenly thinks the product alone equals the percentage result and ignores the role of the unknown term. - 8: This could result from arithmetic errors such as subtracting in the wrong order (for example, 14 - 24 instead of 24 - 14). - 16: This might come from miscalculating 20% of 120 as 30 or 14 as a different value, leading to a wrong difference. Common Pitfalls: Frequent mistakes include not converting mixed numbers to improper fractions correctly, or incorrectly multiplying the fractions. Some students also compute 20% of 120 as 20 * 120 or 20 / 120 rather than (20 / 100) * 120. Another common error is solving A + ? = B as ? = A - B instead of ? = B - A. Checking each step slowly helps avoid these simple but costly errors in exams. Final Answer: The missing value in the equation is 10.

Question 2

In aptitude (decimal percentage simplification), evaluate 39.96% of 649.85 divided by 13.07 and then use the relation 39.96% of 649.85 / 13.07 = 45.14 - ? to find the value of the unknown ?.

Accepted Answer

Introduction / Context: This question is a good example of decimal and percentage simplification often seen in aptitude tests. It involves converting a percentage into a decimal, multiplying by a non-integer number, dividing by another decimal, and then linking the result to a simple linear expression to find the unknown value denoted by ?. The focus is on handling decimals carefully and understanding that such questions are designed for approximate but accurate arithmetic, not for complex calculators. Given Data / Assumptions: - Expression: 39.96% of 649.85 / 13.07 = 45.14 - ?. - 39.96% means 39.96 / 100 as a decimal multiplier. - Division by 13.07 must be done after calculating the percentage of 649.85. - We solve for the unknown ?, treating all numbers as real decimals. Concept / Approach: The main concept is to first compute the left-hand side as accurately as required: value = (39.96 / 100) * 649.85 / 13.07. Once that numerical value is found, the equation becomes value = 45.14 - ?, which can be rearranged to ? = 45.14 - value. Since the answer options are integers (24, 25, 26, 27, 28), we expect the result to be close to one of these values and can round appropriately if the decimal is near a whole number. Step-by-Step Solution: Step 1: Convert the percentage to decimal: 39.96% = 0.3996.Step 2: Compute 0.3996 * 649.85. This product is approximately 259.80 (more precisely around 259.8, but the exact intermediate is not crucial as long as we keep enough accuracy).Step 3: Now divide by 13.07: 259.8 / 13.07 is approximately 19.87 (more precisely about 19.8684).Step 4: So the left-hand side value is about 19.87.Step 5: Use the equation 19.87 ≈ 45.14 - ?.Step 6: Rearrange to find ?: ? ≈ 45.14 - 19.87 ≈ 25.27.Step 7: Among the integer options, 25 is the closest to 25.27, so ? is taken as 25. Verification / Alternative check: To verify, take ? = 25 and recompute the right-hand side: 45.14 - 25 = 20.14. Our left-hand side estimate was around 19.87, which is reasonably close, considering rounding in intermediate steps. If we use more precise values (for example 259.8684 / 13.07), the left-hand side comes out to around 19.868, and 45.14 - 25.27 exactly matches this, justifying 25 as the nearest integer answer. Exam questions of this style expect the nearest integer rather than an exact non-terminating decimal. Why Other Options Are Wrong: - 24: Too low compared to the precise difference of about 25.27, off by more than one unit from the exact decimal answer. - 26 and 27: These values are too large; they would make 45.14 - ? much smaller than the left-hand side, contradicting the equation. - 28: Even further from the mathematically justified value; it is inconsistent with the computed decimal. Common Pitfalls: Common mistakes include interpreting the expression incorrectly, such as taking 39.96% of (649.85 / 13.07) instead of (39.96% of 649.85) divided by 13.07, or misplacing parentheses mentally. Another source of error is poor handling of decimals, like misplacing the decimal point when converting percentages. Some students also round too early in the computation, leading to large final errors. It is better to keep a few decimal places during intermediate steps and only round to match the answer format at the end. Final Answer: The value of the unknown ? that best satisfies the equation is 25.

Question 3

In aptitude (quadratic comparison and simplification), two quadratic equations are given: I. x^2 + 30x + 81 = 0 and II. y^2 - 9y - 162 = 0. After solving both, compare x and y and choose the correct relationship between their possible values.

Accepted Answer

Introduction / Context: This problem is a classic quadratic comparison question. Instead of asking for explicit numerical roots, the examiner wants to know the nature of the relationship between any possible value of x and any possible value of y. Such questions are common in data comparison sections of aptitude exams, where candidates must reason about all roots of both equations and decide whether one variable is always greater, always smaller, always equal, or if no consistent comparison can be made. Given Data / Assumptions: - Equation I: x^2 + 30x + 81 = 0. - Equation II: y^2 - 9y - 162 = 0. - x and y are real roots of their respective equations. - We must compare all possible combinations of x and y and judge the relationship. Concept / Approach: The concept involves solving both quadratic equations, finding their roots, and then listing possible pairs (x, y). If x is greater than y for every possible combination, we choose x > y, and similarly for other options. If in some cases x > y and in others x < y, the relationship cannot be determined. Factorisation is a convenient method here because both equations have integer coefficients and neat factorisation patterns. Step-by-Step Solution: Step 1: Factor Equation I: x^2 + 30x + 81 = 0.Step 2: The discriminant is 30^2 - 4 * 1 * 81 = 900 - 324 = 576, which is 24^2, so roots are real and rational.Step 3: Factor: x^2 + 30x + 81 = (x + 3)(x + 27) = 0. Thus, x = -3 or x = -27.Step 4: Factor Equation II: y^2 - 9y - 162 = 0.Step 5: We need factors of -162 that add up to -9. These are -18 and +9, giving (y - 18)(y + 9) = 0.Step 6: Therefore, y = 18 or y = -9.Step 7: List all possible (x, y) pairs: (-27, 18), (-27, -9), (-3, 18), (-3, -9).Step 8: Compare: for (-27, 18), x < y; for (-27, -9), x < y; for (-3, 18), x < y; for (-3, -9), x > y. Verification / Alternative check: From the comparison, we see three cases where x < y and one case where x > y. There is no case where x equals y, and there is no consistent ordering. An alternative check is to note that the set of x-values is {-27, -3} and the set of y-values is {-9, 18}. Since -27 is less than both -9 and 18, while -3 is greater than -9 but less than 18, there is no single inequality that holds for all roots. Therefore, no definitive order can be assigned between x and y in general. Why Other Options Are Wrong: - x > y: False, because when x = -27 and y = 18, clearly x < y. - x < y: Also false, because for x = -3 and y = -9, we have x > y. - x = y: There is no pair of equal roots between the two equations. - x >= y: This would require x >= y in all cases, but we have multiple cases where x < y. Common Pitfalls: Students sometimes compare only the larger root of one equation with the larger root of the other equation, or compare only one arbitrary pair, and then assume that relationship holds universally. Another mistake is to forget that quadratic equations can have two distinct roots and that any of them is a possible value of the variable. For comparison questions like this, it is essential to consider all combinations of roots and not rely on partial inspection. Final Answer: The relationship between x and y varies depending on which roots are chosen, so the relationship cannot be determined.

Question 4

In aptitude (nested bracket simplification), evaluate the expression 27 - [38 - {46 - (15 - 13 - 2)}] using the correct order of operations and bracket handling to obtain a single integer value.

Accepted Answer

Introduction / Context: This question checks the ability to handle nested brackets and apply the correct order of operations in arithmetic expressions. Aptitude exams frequently include such problems to assess attention to detail and comfort with basic arithmetic under time pressure. The expression uses different bracket types: parentheses, curly braces, and square brackets, but the rule is the same: always simplify the innermost bracket first and then work outward step by step. Given Data / Assumptions: - Expression: 27 - [38 - {46 - (15 - 13 - 2)}]. - Standard order of operations is followed: first brackets (from innermost to outermost), then multiplication and division, then addition and subtraction from left to right. - All numbers are integers and operations are basic addition and subtraction. Concept / Approach: The key concept is to simplify one layer of brackets at a time. Start from the innermost parentheses, compute that result, substitute it back into the next level of brackets, and continue outward. Carefully track signs, especially when subtracting entire bracketed expressions, because sign mistakes are the most common source of errors in these problems. Step-by-Step Solution: Step 1: Identify the innermost bracket: (15 - 13 - 2).Step 2: Compute inside the parentheses from left to right: 15 - 13 = 2, then 2 - 2 = 0. So (15 - 13 - 2) = 0.Step 3: Substitute back into the curly braces: {46 - (15 - 13 - 2)} becomes {46 - 0} = 46.Step 4: Substitute this into the square brackets: [38 - {46 - (15 - 13 - 2)}] becomes [38 - 46].Step 5: Compute 38 - 46 = -8.Step 6: Now the full expression is 27 - [38 - {46 - (15 - 13 - 2)}] = 27 - (-8).Step 7: Subtracting a negative is the same as adding: 27 - (-8) = 27 + 8 = 35. Verification / Alternative check: We can quickly verify by recomputing the inner part: (15 - 13 - 2) = 0, so {46 - 0} = 46, then [38 - 46] = -8, then 27 - (-8) = 35. Repeating the calculation confirms that there are no sign errors and that the sequence of operations has been followed correctly from inner to outer brackets. Why Other Options Are Wrong: - 31 or 33: These likely arise from mistakes such as miscomputing 38 - 46 as -6 or misinterpreting 27 - (-8) as 27 - 8 instead of 27 + 8. - 30 or 29: These values suggest that either the inner parentheses were not evaluated correctly or that the negative sign in front of the square bracket was mishandled, yielding an incorrect intermediate value. Common Pitfalls: Typical errors involve miscalculating the innermost bracket or forgetting that subtracting a negative yields addition. Some students perform operations in the wrong order, for example, starting from the left without respecting bracket hierarchy, which leads to completely different results. Always remember to go from inside to outside and to rewrite the expression clearly after each simplification step to avoid confusion. Final Answer: The simplified value of the expression is 35.

Question 5

In aptitude (algebraic identities), if a + b + c = 0, then using the identity for the sum of cubes and its square, find the value of (a^3 + b^3 + c^3)^2 in terms of a, b, and c.

Accepted Answer

Introduction / Context: This question tests understanding of algebraic identities involving the sum of cubes and the special case where a + b + c = 0. These identities are widely used in polynomial simplification, factorisation, and competitive exam problems. The key idea is that when the sum of three numbers is zero, the expression a^3 + b^3 + c^3 simplifies significantly. Here, we are asked to find the square of that simplified sum, (a^3 + b^3 + c^3)^2, purely in symbolic form, without substituting any numerical values. Given Data / Assumptions: - a, b, and c are real numbers. - a + b + c = 0. - We need to evaluate (a^3 + b^3 + c^3)^2. - Standard algebraic identities for sums of cubes can be used. Concept / Approach: The fundamental identity used is: a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca). When a + b + c = 0, the right-hand side becomes zero, leading to a^3 + b^3 + c^3 - 3abc = 0. This simplifies to a^3 + b^3 + c^3 = 3abc. After this, we are asked to find (a^3 + b^3 + c^3)^2, which becomes (3abc)^2. This is a direct application of squaring a product. Step-by-Step Solution: Step 1: Recall the identity a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca).Step 2: Substitute the given condition a + b + c = 0 into the right-hand side.Step 3: Then the right-hand side becomes 0 * (a^2 + b^2 + c^2 - ab - bc - ca) = 0.Step 4: So we get a^3 + b^3 + c^3 - 3abc = 0.Step 5: Rearranging gives a^3 + b^3 + c^3 = 3abc.Step 6: Now we need (a^3 + b^3 + c^3)^2, which is (3abc)^2.Step 7: (3abc)^2 = 9a^2b^2c^2. Verification / Alternative check: We can verify using specific values that satisfy a + b + c = 0. For example, choose a = 1, b = 1, c = -2. Then a + b + c = 0 holds. Compute a^3 + b^3 + c^3 = 1^3 + 1^3 + (-2)^3 = 1 + 1 - 8 = -6. Then (a^3 + b^3 + c^3)^2 = (-6)^2 = 36. Now compute 9a^2b^2c^2 = 9 * 1^2 * 1^2 * (-2)^2 = 9 * 4 = 36. Both values match, confirming the identity and the final expression 9a^2b^2c^2. Why Other Options Are Wrong: - 27abc, 9abc, 3abc: These expressions do not match the required squared form and lack the correct exponents on a, b, and c. They represent linear or cubic products, not a squared quantity derived from 3abc. - 27a^2b^2c^2: This would arise from incorrectly assuming a^3 + b^3 + c^3 = 3abc and then squaring it as (3abc)^3 or making a multiplication mistake with 3^3 instead of 3^2. The coefficient must be 9, not 27. Common Pitfalls: Students often misremember the identity and write a^3 + b^3 + c^3 = 3abc directly without stating the condition a + b + c = 0, which is essential. Another common error is to square 3abc incorrectly as 27a^2b^2c^2 instead of 9a^2b^2c^2. Careful handling of exponents and coefficients is important. Also, ignoring the given condition and trying to expand cubes individually wastes time and introduces unnecessary complexity. Final Answer: Under the condition a + b + c = 0, the value of (a^3 + b^3 + c^3)^2 is 9a^2b^2c^2.

Question 6

In aptitude (fractional simplification), compute the value of (1112 / 139) - 5 by first simplifying the fraction 1112 / 139 and then performing the subtraction to obtain the final result.

Accepted Answer

Introduction / Context: This is a straightforward fractional simplification problem. It checks the ability to subtract an integer from a fraction and to recognise when the fraction itself simplifies to an integer plus or minus a smaller fraction. Such questions often appear in the early part of aptitude test papers, where quick mental arithmetic can save valuable time for tougher questions later. Given Data / Assumptions: - Expression: (1112 / 139) - 5. - All numbers are real and standard arithmetic rules apply. - We may convert 5 into a fraction with denominator 139 to combine the terms more easily. Concept / Approach: A simple and effective approach is to express both terms with the same denominator. Since the denominator 139 is already present in the first term, we convert 5 into 5 * 139 / 139 and then subtract numerators. After finding the resulting fraction, we can see whether it simplifies to an integer. Alternatively, we can perform long division to estimate 1112 / 139 and then subtract 5, but the fraction method is cleaner and exact. Step-by-Step Solution: Step 1: Start with the expression (1112 / 139) - 5.Step 2: Write 5 with denominator 139: 5 = 5 * 139 / 139 = 695 / 139.Step 3: Now the expression becomes (1112 / 139) - (695 / 139).Step 4: Combine the fractions by subtracting numerators: (1112 - 695) / 139.Step 5: Compute the numerator: 1112 - 695 = 417.Step 6: So the fraction becomes 417 / 139.Step 7: Notice that 417 = 3 * 139, so 417 / 139 = 3.Step 8: Therefore, (1112 / 139) - 5 simplifies exactly to 3. Verification / Alternative check: An alternative approach is to perform approximate long division. Compute 1112 / 139: since 139 * 8 = 1112, this shows that 1112 / 139 = 8 exactly. Then the expression becomes 8 - 5 = 3. This confirms the result obtained by the fractional method and proves that the answer is exactly 3, not an approximation. Why Other Options Are Wrong: - -3: This would result from reversing the subtraction order and computing 5 - (1112 / 139) instead of (1112 / 139) - 5. - 8: This is simply 1112 / 139 and ignores the subtraction by 5 entirely. - 557/67 and 556/67: These are random-looking fractions that do not correspond to any correct simplification of the given expression. They might arise from misusing denominators or from arithmetic mistakes in subtracting numerators or denominators incorrectly. Common Pitfalls: Students may incorrectly use 5 / 139 instead of 5 * 139 / 139 when trying to combine fractions, which drastically changes the value. Another error is failing to recognise that 417 is exactly divisible by 139, and leaving the answer as 417 / 139 without simplifying. Paying attention to factorisation and basic multiplication tables helps in spotting such simplifications quickly. Final Answer: The expression (1112 / 139) - 5 simplifies to the integer 3.

Question 7

In aptitude (algebraic identities and surds), if x + y = sqrt(3) and x - y = sqrt(2), find the exact value of the expression 8 * x * y * (x^2 + y^2) using algebraic identities without approximating the square roots.

Accepted Answer

Introduction / Context: This problem tests the use of algebraic identities with surds (square roots) and symmetric expressions in two variables x and y. Rather than solving directly for x and y, we use identities such as (x + y)^2 and (x - y)^2 to derive expressions for xy and x^2 + y^2. This is a powerful method in competitive exams, where direct solving may be lengthy or unnecessary, but cleverly using identities leads to a quick and elegant solution. Given Data / Assumptions: - x + y = sqrt(3). - x - y = sqrt(2). - We need to compute 8 * x * y * (x^2 + y^2). - Square roots are exact; we should not approximate them with decimals. Concept / Approach: The approach uses two main identities: 1) (x + y)^2 = x^2 + 2xy + y^2. 2) (x - y)^2 = x^2 - 2xy + y^2. By adding and subtracting these equations, we can find xy and x^2 + y^2 in terms of the given square roots. Once we know xy and x^2 + y^2, we substitute them into the expression 8xy(x^2 + y^2) and simplify to a rational number. Step-by-Step Solution: Step 1: Compute (x + y)^2 using the given value: (x + y)^2 = (sqrt(3))^2 = 3.Step 2: Write (x + y)^2 in identity form: x^2 + 2xy + y^2 = 3.Step 3: Compute (x - y)^2 using the given value: (x - y)^2 = (sqrt(2))^2 = 2.Step 4: Write (x - y)^2 in identity form: x^2 - 2xy + y^2 = 2.Step 5: Subtract the second equation from the first: (x^2 + 2xy + y^2) - (x^2 - 2xy + y^2) = 3 - 2.Step 6: The left side becomes 4xy, and the right side is 1, so 4xy = 1, which gives xy = 1 / 4.Step 7: Now add the two equations: (x^2 + 2xy + y^2) + (x^2 - 2xy + y^2) = 3 + 2.Step 8: The left side becomes 2x^2 + 2y^2 = 5, so x^2 + y^2 = 5 / 2.Step 9: Substitute into the target expression: 8xy(x^2 + y^2) = 8 * (1 / 4) * (5 / 2).Step 10: Simplify: 8 * (1 / 4) = 2, and 2 * (5 / 2) = 5.Step 11: Therefore, 8xy(x^2 + y^2) = 5. Verification / Alternative check: We can also attempt to find x and y explicitly. From x + y = sqrt(3) and x - y = sqrt(2), add the equations to get 2x = sqrt(3) + sqrt(2), so x = (sqrt(3) + sqrt(2)) / 2. Subtract to get 2y = sqrt(3) - sqrt(2), so y = (sqrt(3) - sqrt(2)) / 2. Substituting these into xy and x^2 + y^2 gives the same results derived earlier, confirming the value 5 for the given expression. Why Other Options Are Wrong: - 6 and 4: These are simple integers and might result from miscomputing xy or x^2 + y^2, or from using incorrect coefficients in the identities. - sqrt(6) and sqrt(5): These suggest that the student left surds in the final answer instead of simplifying fully, or did not use symmetric identities properly. The final result here is a rational integer, not a surd. Common Pitfalls: A common error is to attempt to find x and y numerically right away, which can be time consuming and prone to mistakes. Another pitfall is mixing up the identities for (x + y)^2 and (x - y)^2 or forgetting that subtracting them gives 4xy. Some students incorrectly treat sqrt(3) and sqrt(2) as approximate decimals and introduce rounding errors, which is unnecessary because the final answer is a clean integer. Using exact algebraic manipulation is both faster and safer in exam conditions. Final Answer: The value of the expression 8 * x * y * (x^2 + y^2) is 5.

Question 8

In aptitude (HCF / GCD simplification), find the greatest number that will divide 390, 495, and 300 exactly without leaving any remainder, that is, determine the highest common factor of the three given integers.

Accepted Answer

Introduction / Context: This question checks basic number theory skills related to the highest common factor (HCF), also called greatest common divisor (GCD). Many aptitude tests include similar problems where candidates must find the largest integer that divides multiple numbers exactly. Mastery of prime factorisation or repeated division is essential for quick and accurate answers. Given Data / Assumptions: - The three numbers are 390, 495, and 300. - We want the greatest integer that divides all three without leaving any remainder. - This integer is the HCF or GCD of the three numbers. Concept / Approach: The HCF of several numbers is the largest number that divides each of them exactly. There are two popular methods: prime factorisation and successive division (Euclidean algorithm). For these moderate-sized numbers, prime factorisation is straightforward. We factor each number into primes and then multiply the common prime factors with the smallest exponents. Step-by-Step Solution: Step 1: Prime factorise 390.390 = 39 * 10 = 3 * 13 * 2 * 5 = 2 * 3 * 5 * 13.Step 2: Prime factorise 495.495 = 49.5 * 10, but better: 495 = 5 * 99 = 5 * 9 * 11 = 5 * 3^2 * 11.Step 3: Prime factorise 300.300 = 3 * 100 = 3 * 4 * 25 = 3 * 2^2 * 5^2 = 2^2 * 3 * 5^2.Step 4: Identify common prime factors across all three numbers.Common primes: 3 and 5 appear in all three factorizations.Step 5: For prime 3, the smallest exponent among the numbers is 1 (since 390 has 3^1, 495 has 3^2, and 300 has 3^1).Step 6: For prime 5, the smallest exponent among the numbers is 1 (since 390 has 5^1, 495 has 5^1, and 300 has 5^2).Step 7: So HCF = 3^1 * 5^1 = 3 * 5 = 15. Verification / Alternative check: Verify by direct division: 390 / 15 = 26, 495 / 15 = 33, and 300 / 15 = 20. All quotients are integers, and 15 divides each number exactly. Check that no larger option divides all three: for example, 25 does not divide 390 exactly (390 / 25 is not an integer), and 30 does not divide 495 exactly (495 / 30 is not an integer). Therefore, 15 is indeed the greatest common divisor. Why Other Options Are Wrong: - 5: This is a common divisor but not the greatest one, since 15 is larger and still divides all three numbers exactly. - 25: It divides 300 and 495 is not divisible by 25, so it fails the condition. - 35: This does not divide 300 or 390 exactly; it is not common. - 30: It divides 390 and 300, but 495 / 30 is not an integer, so it is not a common divisor of all three numbers. Common Pitfalls: Students may quickly pick a small common factor like 5 or 3 without checking whether it is the greatest. Others might incorrectly factor the numbers or miss a higher common factor such as 15. It is important to systematically list the prime factors and then combine only those that appear in all numbers with the smallest exponent. A quick divisibility check for each option can also help confirm the correct answer in multiple choice settings. Final Answer: The greatest number that divides 390, 495, and 300 exactly is 15.

Question 9

In aptitude (time and money simplification), Mr. Ankit is on tour to Siachen with a total budget of ₹360. If he extends his tour by 4 days, he must reduce his daily expenditure by ₹3 to stay within the same total amount. For how many days was Mr. An…

Accepted Answer

Introduction / Context: This is a classic time and money problem that translates directly into a linear equation. Such questions are common in aptitude exams to check whether candidates can convert a word problem into algebraic expressions. The key idea is that the total expense remains fixed while the number of days and daily expenditure change in a related way. Understanding this trade-off is crucial for solving the equation correctly. Given Data / Assumptions: - Total amount for the tour: ₹360. - Let the original number of days be n. - Original daily expenditure is 360 / n rupees per day. - If he extends the tour by 4 days, the new number of days is n + 4. - In that case, he must reduce his daily expenditure by ₹3. - Total expenditure remains ₹360 in both scenarios. Concept / Approach: The concept is to express both daily expenditures in terms of n and equate them according to the given relationship. The original daily expenditure is 360 / n. The new daily expenditure after extending the tour is 360 / (n + 4), and this is ₹3 less than the original daily expenditure. This gives us a single equation in one variable n, which we can solve using algebraic manipulation and factorisation. Step-by-Step Solution: Step 1: Let n be the original number of days.Step 2: Original daily expenditure = 360 / n.Step 3: New daily expenditure after extension by 4 days = 360 / (n + 4).Step 4: Given condition: new daily expenditure = original daily expenditure - 3.Step 5: Therefore, 360 / (n + 4) = 360 / n - 3.Step 6: Rearrange the equation: 360 / (n + 4) = (360 - 3n) / n.Step 7: Cross multiply: 360n = (n + 4)(360 - 3n).Step 8: Expand the right-hand side: (n + 4)(360 - 3n) = n * 360 - 3n^2 + 4 * 360 - 12n = 360n - 3n^2 + 1440 - 12n.Step 9: So 360n = 360n - 3n^2 + 1440 - 12n.Step 10: Subtract 360n from both sides: 0 = -3n^2 + 1440 - 12n.Step 11: Rearrange into standard quadratic form: 3n^2 + 12n - 1440 = 0 (multiplying by -1).Step 12: Divide by 3: n^2 + 4n - 480 = 0.Step 13: Factorise: n^2 + 4n - 480 = (n - 20)(n + 24) = 0.Step 14: Solutions are n = 20 or n = -24. Negative days are not meaningful, so n = 20. Verification / Alternative check: Check with n = 20: Original daily expenditure = 360 / 20 = 18 rupees per day. If he extends by 4 days, new days = 24. New daily expenditure = 360 / 24 = 15 rupees per day. The decrease in daily expenditure is 18 - 15 = 3 rupees per day, which matches the given condition. Therefore, n = 20 is correct. Why Other Options Are Wrong: - 18, 22, 24, 26: For each of these values, the difference between original and new daily expenditure will not be exactly ₹3 when you compute 360 / n and 360 / (n + 4). They do not satisfy the given condition in the problem statement. Common Pitfalls: Some students incorrectly assume the daily expenditure remains the same when the number of days changes, which contradicts the problem. Others make algebraic errors when cross multiplying or expanding the brackets. Forgetting to discard the negative root when solving the quadratic equation can also lead to confusion. Always check that the final solution makes practical sense (number of days must be positive). Final Answer: Mr. Ankit was originally on tour for 20 days.

Question 10

In aptitude (pattern based simplification), a custom operation * is defined so that 9 * 6 = 45, 7 * 4 = 33, and 6 * 4 = 20. Using the same hidden rule, find the value of 5 * 3.

Accepted Answer

Introduction / Context: This question is a pattern based or rule based operation problem. The operation * is not standard multiplication; instead, it is defined by a hidden pattern which we must discover using the given examples. Such problems are common in reasoning and aptitude tests, designed to check logical thinking and pattern recognition skills rather than pure arithmetic knowledge. Given Data / Assumptions: - 9 * 6 = 45. - 7 * 4 = 33. - 6 * 4 = 20. - The same rule must be used to find 5 * 3. - The operation * is assumed to be symmetric in the examples (but we must infer the rule from the data provided). Concept / Approach: The key idea is to express the result of each operation in terms of the two input numbers a and b and then look for a consistent formula. A good approach is to try combinations such as a * b = (a + b)(a - b), or a * b = a^2 - b^2, or a * b = a + b + ab, and so on. We test each hypothesis against the three given examples. Once we find a rule that works for all given cases, we apply that rule to compute 5 * 3. Step-by-Step Solution: Step 1: Consider the possibility that a * b relates to (a + b)(a - b).Step 2: For a = 9, b = 6: (9 + 6)(9 - 6) = 15 * 3 = 45, which matches 9 * 6 = 45.Step 3: For a = 7, b = 4: (7 + 4)(7 - 4) = 11 * 3 = 33, which matches 7 * 4 = 33.Step 4: For a = 6, b = 4: (6 + 4)(6 - 4) = 10 * 2 = 20, which matches 6 * 4 = 20.Step 5: Since all three examples fit the pattern, we conclude that a * b = (a + b)(a - b).Step 6: Apply the rule to 5 * 3: (5 + 3)(5 - 3) = 8 * 2 = 16.Step 7: Therefore, 5 * 3 = 16 under this custom operation. Verification / Alternative check: We could also recognise that (a + b)(a - b) is equal to a^2 - b^2. Checking with the given examples: for 9 and 6, a^2 - b^2 = 81 - 36 = 45; for 7 and 4, 49 - 16 = 33; for 6 and 4, 36 - 16 = 20. All are consistent. For 5 and 3, a^2 - b^2 = 25 - 9 = 16, matching the result from the previous method. This double confirmation reinforces that the correct rule has been found. Why Other Options Are Wrong: - 24 and 12: These might come from naive multiplication or addition patterns such as a * b = a * b + something, but they do not fit the rule that matches all three given examples. - 13 and 10: These may be guessed values based on partial observation or incorrect pattern discovery, but they do not arise from (a + b)(a - b) or a^2 - b^2 for 5 and 3. Common Pitfalls: One common mistake is to assume the operation is simply a * b = a * b (ordinary multiplication) and then try to fit corrections to get the given results, which usually becomes inconsistent. Another pitfall is picking a pattern that works for only one or two examples without checking all of them. In pattern based questions, it is essential to verify that the discovered rule fits every given instance before applying it to a new pair of numbers. Final Answer: Based on the discovered rule a * b = (a + b)(a - b), the value of 5 * 3 is 16.

Question 11

In aptitude (remainder and modular arithmetic), when the expression 7^21 + 7^22 + 7^23 + 7^24 is divided by 25, what remainder is obtained? Use properties of powers and modular cycles for efficient calculation.

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Introduction / Context: This question is a typical modular arithmetic problem where we must find the remainder when a large power expression is divided by a given number. Direct calculation of 7^21 and higher powers is impractical, so we use patterns in powers of 7 modulo 25. Recognising repeating cycles in the remainders is the main skill tested here. Such problems are widely used in quantitative aptitude and number theory sections of competitive exams. Given Data / Assumptions: - Expression: 7^21 + 7^22 + 7^23 + 7^24. - Divisor: 25. - We need the remainder when the entire sum is divided by 25. - Standard rules of modular arithmetic apply: (a + b) mod m = [(a mod m) + (b mod m)] mod m, and similar for products. Concept / Approach: The approach is to find the pattern in powers of 7 modulo 25. We compute 7^1 mod 25, 7^2 mod 25, and so on until we see a cycle. Once a cycle is identified, we reduce the exponents 21, 22, 23, and 24 modulo the cycle length to find equivalent smaller exponents. Then we compute the corresponding remainders, add them, and finally find the remainder when that sum is divided by 25. Step-by-Step Solution: Step 1: Compute initial powers of 7 modulo 25.7^1 mod 25 = 7.7^2 = 49; 49 mod 25 = 49 - 25 * 1 = 24.7^3 = 7^2 * 7 = 24 * 7 = 168; 168 mod 25 = 168 - 25 * 6 = 168 - 150 = 18.7^4 = 7^3 * 7 = 18 * 7 = 126; 126 mod 25 = 126 - 25 * 5 = 126 - 125 = 1.Step 2: We observe that 7^4 mod 25 = 1. This means powers of 7 repeat every 4 exponents modulo 25.Step 3: Therefore, 7^n mod 25 depends on n mod 4.Step 4: Compute remainders of exponents modulo 4: 21 mod 4 = 1, 22 mod 4 = 2, 23 mod 4 = 3, 24 mod 4 = 0.Step 5: So 7^21 mod 25 = 7^1 mod 25 = 7.Step 6: 7^22 mod 25 = 7^2 mod 25 = 24.Step 7: 7^23 mod 25 = 7^3 mod 25 = 18.Step 8: 7^24 mod 25 = 7^4 mod 25 = 1.Step 9: Now sum these remainders: 7 + 24 + 18 + 1 = 50.Step 10: Finally, 50 mod 25 = 0. Verification / Alternative check: An alternative view is to factor out 7^21: 7^21(1 + 7 + 7^2 + 7^3). Modulo 25, we already know that 7^4 ≡ 1, so 7^0 ≡ 1, 7^1 ≡ 7, 7^2 ≡ 24, 7^3 ≡ 18, and 1 + 7 + 24 + 18 = 50 ≡ 0 mod 25. Therefore, 7^21 * 0 ≡ 0 mod 25, confirming that the entire expression leaves remainder zero when divided by 25. Why Other Options Are Wrong: - 5, 7, 10, 17: These values might appear if one miscalculates powers or stops the remainder calculation prematurely. However, once the full pattern and sum are handled correctly, the remainder is exactly 0. None of these other values can be justified by correct modular arithmetic steps. Common Pitfalls: Common errors include computing large powers directly without using cycles, leading to arithmetic mistakes, or misidentifying the cycle length. Some students incorrectly assume a cycle length of 5 or 10 without checking. Others may forget to reduce each power mod 25 before summing, causing overflow and confusion. Always establish the cycle carefully and then apply it to each exponent. Final Answer: The remainder obtained when 7^21 + 7^22 + 7^23 + 7^24 is divided by 25 is 0.

Question 12

In aptitude (height and distance simplification), the angle of elevation of the top of a tower from point A is 30 degrees. After moving 20 meters towards the foot of the tower to point B, the angle of elevation becomes 60 degrees. Using basic trigon…

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Introduction / Context: This is a standard height and distance problem in trigonometry, frequently tested in aptitude and engineering entrance examinations. The key idea is to form right-angled triangles from the given angles of elevation and to express the height of the tower and horizontal distances in terms of tangent functions. By using two different positions with known separation, we can set up two equations and solve for the unknown height. Given Data / Assumptions: - The angle of elevation from point A to the top of the tower is 30 degrees. - After moving 20 meters towards the tower to point B, the angle of elevation becomes 60 degrees. - The tower stands vertically on level ground, and the lines of sight form right angles with the horizontal ground at the base of the tower. - We need to find the height h of the tower. Concept / Approach: We use the tangent function in right-angled triangles. If h is the height of the tower and x is the horizontal distance from point A to the foot of the tower, then from point A we have tan(30 degrees) = h / x. From point B, the new distance to the tower is (x - 20), and tan(60 degrees) = h / (x - 20). These two equations in h and x can be solved simultaneously to find the height h. Step-by-Step Solution: Step 1: Let h be the height of the tower and x be the distance from point A to the foot of the tower.Step 2: From point A, tan(30 degrees) = h / x. We know tan(30 degrees) = 1 / sqrt(3), so h / x = 1 / sqrt(3), giving h = x / sqrt(3).Step 3: From point B, which is 20 meters closer, the distance to the tower is x - 20. The angle of elevation is 60 degrees.Step 4: So tan(60 degrees) = h / (x - 20). We know tan(60 degrees) = sqrt(3), so h / (x - 20) = sqrt(3), giving h = sqrt(3) * (x - 20).Step 5: Equate the two expressions for h: x / sqrt(3) = sqrt(3) * (x - 20).Step 6: Multiply both sides by sqrt(3): x = 3 * (x - 20).Step 7: Expand the right-hand side: x = 3x - 60.Step 8: Rearrange to solve for x: 3x - x = 60, so 2x = 60, hence x = 30.Step 9: Substitute x back into h = x / sqrt(3): h = 30 / sqrt(3) = 10 * sqrt(3). Verification / Alternative check: Check from point B: distance to tower is x - 20 = 30 - 20 = 10 meters. For height h = 10 * sqrt(3), tan(60 degrees) should be h / 10 = (10 * sqrt(3)) / 10 = sqrt(3), which matches tan(60 degrees). From point A: h / x = (10 * sqrt(3)) / 30 = sqrt(3) / 3 = 1 / sqrt(3), which matches tan(30 degrees). Both checks confirm the correctness of the found height. Why Other Options Are Wrong: - sqrt(3) and 5*sqrt(3): These values are far too small to be consistent with the geometry of the problem, especially considering the 20 meter horizontal shift. - 20*sqrt(3) and 15*sqrt(3): These are larger than required and do not satisfy both tangent equations simultaneously. Substituting them into tan(30 degrees) and tan(60 degrees) relationships would fail either at point A or point B. Common Pitfalls: Common mistakes include mixing up which distance is x and which is x - 20, writing tan(30 degrees) and tan(60 degrees) incorrectly, or forgetting that tan(30 degrees) = 1 / sqrt(3) and tan(60 degrees) = sqrt(3). Some students also assume that the height is numerically equal to one of the given distances without solving the equations properly. Always write the equations explicitly and solve systematically. Final Answer: The height of the tower is 10*sqrt(3) meters.

Question 13

In aptitude (trigonometric simplification), if 5*sec(theta) - 3*tan(theta) = 5 for an acute angle theta, use trigonometric identities to find the exact value of 5*tan(theta) - 3*sec(theta).

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Introduction / Context: This question tests algebraic manipulation of trigonometric functions sec(theta) and tan(theta), combined with the Pythagorean identity sec^2(theta) - tan^2(theta) = 1. Instead of finding theta explicitly, we use the given linear relation between sec and tan to determine a related expression. This style of problem is common in trigonometry based aptitude sections and requires comfort with both identities and simultaneous equations. Given Data / Assumptions: - Given equation: 5*sec(theta) - 3*tan(theta) = 5. - Theta is an acute angle, so sec(theta) > 0 and tan(theta) > 0. - We need to find the value of 5*tan(theta) - 3*sec(theta). - Trigonometric identity: sec^2(theta) - tan^2(theta) = 1. Concept / Approach: We treat sec(theta) and tan(theta) as variables, say s and t respectively. Then the given equation becomes 5s - 3t = 5. Along with this, we use the identity s^2 - t^2 = 1. This gives a system of two equations in two unknowns s and t. Solving this system allows us to find specific values for s and t, and then we substitute them into the target expression 5t - 3s to get the required result. Step-by-Step Solution: Step 1: Let s = sec(theta) and t = tan(theta).Step 2: The given equation becomes 5s - 3t = 5.Step 3: The trigonometric identity gives s^2 - t^2 = 1.Step 4: From 5s - 3t = 5, express one variable in terms of the other. For example, 5s = 5 + 3t, so s = 1 + (3t / 5).Step 5: Substitute s into s^2 - t^2 = 1 and solve for t. However, an easier approach is to solve the system directly using algebraic methods.Step 6: Solving the system 5s - 3t = 5 and s^2 - t^2 = 1, we obtain two solutions: (s, t) = (1, 0) and (s, t) = (17/8, 15/8).Step 7: Since theta is acute and tan(theta) must be positive, we discard the solution (1, 0) because it would imply t = 0.Step 8: Therefore, sec(theta) = 17/8 and tan(theta) = 15/8.Step 9: Now compute the required expression 5*tan(theta) - 3*sec(theta) = 5t - 3s.Step 10: Substitute s = 17/8 and t = 15/8: 5t - 3s = 5*(15/8) - 3*(17/8).Step 11: Compute numerators: 5*(15/8) = 75/8 and 3*(17/8) = 51/8.Step 12: So 5t - 3s = (75/8) - (51/8) = 24/8 = 3. Verification / Alternative check: We can verify the solution by substituting s = 17/8 and t = 15/8 back into the original equations. Check the identity: s^2 - t^2 = (17/8)^2 - (15/8)^2 = (289/64) - (225/64) = 64/64 = 1, which is correct. Check the linear equation: 5s - 3t = 5*(17/8) - 3*(15/8) = (85/8) - (45/8) = 40/8 = 5, which matches the given condition. Therefore, the values are consistent and the derived expression 3 is correct. Why Other Options Are Wrong: - 1, 2, 4, and -3: These values might result from incorrect algebra, sign mistakes, or using the discarded solution where tan(theta) = 0. Only 3 satisfies both the identity and the original equation under the requirement that theta is acute and tan(theta) is positive. Common Pitfalls: Some students forget to use the identity sec^2(theta) - tan^2(theta) = 1 and try to solve using only the linear equation, which is impossible. Others may make errors in squaring and subtracting when substituting into the identity. A frequent conceptual error is not considering the condition that theta is acute, and mistakenly accepting a solution where tan(theta) is zero or negative, which is not allowed here. Final Answer: The value of the expression 5*tan(theta) - 3*sec(theta) is 3.

Question 14

In aptitude (exponential equations simplification), if 4^(x + y) = 256 and 256^(x - y) = 4, where x and y are real numbers, determine the values of x and y using properties of powers of 4.

Accepted Answer

Introduction / Context: This question involves solving a system of exponential equations by expressing all terms in the same base. Exponential equations of this type are common in algebra and aptitude exams, and the key trick is to rewrite 256 and 4 as powers of a common base, here base 4. Once the exponents are rewritten, the equations can be converted to linear equations in x and y and solved using standard algebraic techniques. Given Data / Assumptions: - 4^(x + y) = 256. - 256^(x - y) = 4. - x and y are real numbers. - 256 and 4 can be expressed as powers of 4: 256 = 4^4 and 4 = 4^1. Concept / Approach: The concept is to convert both equations to the same base and then equate exponents. For the first equation, we directly equate x + y to the exponent that produces 256 when base is 4. For the second equation, we write 256 as 4^4, then use rules of exponents to combine powers, and finally equate the resulting exponent to 1 to match 4 = 4^1. This reduces the problem to solving a pair of linear equations in two variables x and y. Step-by-Step Solution: Step 1: Rewrite 256 in terms of base 4: 256 = 4^4.Step 2: From the first equation 4^(x + y) = 256, we get 4^(x + y) = 4^4, so x + y = 4.Step 3: For the second equation, 256^(x - y) = 4.Step 4: Substitute 256 = 4^4: (4^4)^(x - y) = 4.Step 5: Apply the power rule (a^b)^c = a^(b * c): 4^(4(x - y)) = 4.Step 6: Since 4 = 4^1, equate exponents: 4(x - y) = 1.Step 7: Therefore, x - y = 1 / 4.Step 8: Now we have a system of linear equations: x + y = 4 and x - y = 1 / 4.Step 9: Add the two equations: (x + y) + (x - y) = 4 + 1 / 4.Step 10: This gives 2x = 17 / 4, so x = (17 / 4) / 2 = 17 / 8.Step 11: Substitute x into x + y = 4: 17 / 8 + y = 4.Step 12: Write 4 as 32 / 8: 17 / 8 + y = 32 / 8, so y = (32 / 8) - (17 / 8) = 15 / 8. Verification / Alternative check: Check in the original equations. First equation: 4^(x + y) = 4^(17/8 + 15/8) = 4^(32/8) = 4^4 = 256, which is correct. Second equation: 256^(x - y) = 256^(17/8 - 15/8) = 256^(2/8) = 256^(1/4). Since 256 = 4^4, 256^(1/4) = (4^4)^(1/4) = 4, which matches the right-hand side. Thus, the pair (x, y) = (17/8, 15/8) satisfies both equations. Why Other Options Are Wrong: - 17/4, 15/4: These are exactly double the correct values, suggesting a mistake in dividing by 2 when solving for x and y from 2x and 2y equations. - 9/17, 15/17 and 8/17, 8/15: These fractions do not satisfy x + y = 4 and x - y = 1/4 when tested, so they cannot be correct solutions of the system. - 15/4, 17/4: This reverses the roles of x and y and uses wrong magnitudes; checking them fails in the original equations. Common Pitfalls: Common mistakes include confusing exponent rules, such as thinking (4^4)^(x - y) equals 4^(x - y) instead of 4^(4(x - y)), or incorrectly converting 256 to base 4. Some students also mis-handle fractions when solving the linear system, leading to values like 17/4 instead of 17/8. Working step by step and checking each algebraic transformation helps avoid these errors. Final Answer: The values of the variables are x = 17/8 and y = 15/8.

Question 15

In aptitude (symmetric polynomial identities), if (x + y + z) = 12, (xy + yz + zx) = 44, and xyz = 48, use the standard identity for the sum of cubes to find the value of x^3 + y^3 + z^3.

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Introduction / Context: This question focuses on symmetric polynomials in three variables and the use of algebraic identities relating sums, pairwise products, and products of variables to the sum of their cubes. Instead of finding x, y, and z individually, we use the identity for x^3 + y^3 + z^3 in terms of x + y + z, xy + yz + zx, and xyz. Problems of this kind are popular in higher level aptitude and algebra exams, as they test conceptual understanding of identities and symbolic manipulation. Given Data / Assumptions: - x + y + z = 12. - xy + yz + zx = 44. - xyz = 48. - We need to find x^3 + y^3 + z^3. - All variables are real numbers, but their exact values are not required. Concept / Approach: The key identity is: x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx). We already have x + y + z, xy + yz + zx, and xyz. To use the identity, we also need x^2 + y^2 + z^2, which can be found using another identity: (x + y + z)^2 = x^2 + y^2 + z^2 + 2(xy + yz + zx). Combining these identities allows us to find x^3 + y^3 + z^3 directly. Step-by-Step Solution: Step 1: Use the square identity to find x^2 + y^2 + z^2.(x + y + z)^2 = x^2 + y^2 + z^2 + 2(xy + yz + zx).Step 2: Substitute the known values: 12^2 = x^2 + y^2 + z^2 + 2 * 44.Step 3: Compute 12^2 = 144 and 2 * 44 = 88.Step 4: So 144 = x^2 + y^2 + z^2 + 88, which gives x^2 + y^2 + z^2 = 144 - 88 = 56.Step 5: Now use the sum of cubes identity: x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx).Step 6: Compute the expression inside the parentheses: x^2 + y^2 + z^2 - xy - yz - zx = 56 - 44 = 12.Step 7: So the right-hand side becomes (x + y + z) * 12 = 12 * 12 = 144.Step 8: Therefore, x^3 + y^3 + z^3 - 3xyz = 144.Step 9: Substitute xyz = 48: x^3 + y^3 + z^3 - 3 * 48 = 144.Step 10: Compute 3 * 48 = 144.Step 11: So x^3 + y^3 + z^3 - 144 = 144, which gives x^3 + y^3 + z^3 = 288. Verification / Alternative check: We could attempt to construct a cubic polynomial with roots x, y, z using the given symmetric sums. The polynomial would be t^3 - (x + y + z)t^2 + (xy + yz + zx)t - xyz = 0, that is t^3 - 12t^2 + 44t - 48 = 0. If we could factor this polynomial and find explicit roots, we could then cube and sum them to verify 288, but this is more time consuming. Since the symmetric identities are well established and the calculations here are straightforward, the identity based method is both efficient and reliable. Why Other Options Are Wrong: - 104, 144, 196, 240: These values do not satisfy the identity x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx) when we plug in the given data. They are likely results of arithmetic mistakes, such as miscomputing x^2 + y^2 + z^2 or forgetting to multiply xyz by 3. Common Pitfalls: Students often misremember the sum of cubes identity, sometimes omitting the -3xyz term or misplacing signs. Another common error is computing (x + y + z)^2 incorrectly or forgetting to subtract 2(xy + yz + zx) when solving for x^2 + y^2 + z^2. Careful substitution and stepwise calculation are essential. Also, trying to find x, y, z individually is usually unnecessary and wastes time in exam conditions. Final Answer: The value of the sum of cubes is x^3 + y^3 + z^3 = 288.

Question 16

In an algebraic expression involving cubes, if x^3 - y^3 = 81 and the difference x - y is equal to 3, what is the value of the sum of squares x^2 + y^2?

Accepted Answer

Introduction / Context: This algebra question checks your understanding of identities involving cubes and squares, and how to connect different expressions like x^3 - y^3, x - y, x^2 + xy + y^2, and x^2 + y^2. Problems of this kind are very common in aptitude exams because they require both conceptual knowledge of standard identities and careful algebraic manipulation without necessarily solving for x and y individually. Given Data / Assumptions: x^3 - y^3 = 81. x - y = 3. We need the value of x^2 + y^2. x and y are real numbers for which the given relations are valid. Concept / Approach: The key identity is x^3 - y^3 = (x - y)(x^2 + xy + y^2). We can use the given values of x^3 - y^3 and x - y to find x^2 + xy + y^2. Then we relate x^2 + xy + y^2 to x^2 + y^2 and xy using two different identities: the cube identity and the square of a difference identity (x - y)^2 = x^2 + y^2 - 2xy. Solving these small equations will give us xy, and then x^2 + y^2 directly. Step-by-Step Solution: From x^3 - y^3 = (x - y)(x^2 + xy + y^2). Substitute the given values: 81 = 3 * (x^2 + xy + y^2). So x^2 + xy + y^2 = 81 / 3 = 27. We also know (x - y)^2 = x^2 + y^2 - 2xy. Given x - y = 3, so (x - y)^2 = 9. Thus x^2 + y^2 - 2xy = 9. Call this equation (1). From x^2 + xy + y^2 = 27, we get x^2 + y^2 = 27 - xy. Call this (2). Substitute x^2 + y^2 from (2) into (1): (27 - xy) - 2xy = 9. This simplifies to 27 - 3xy = 9. So 3xy = 27 - 9 = 18, giving xy = 6. Now use (2): x^2 + y^2 = 27 - xy = 27 - 6 = 21. Verification / Alternative check: We can quickly check consistency. If x - y = 3 and xy = 6, then x and y are roots of t^2 - (x + y)t + xy = 0, but we do not need explicit values. Instead, compute x^2 + xy + y^2 = (x^2 + y^2) + xy = 21 + 6 = 27, which matches our earlier result. Substituting back into (x - y)(x^2 + xy + y^2) gives 3 * 27 = 81, which is exactly the given x^3 - y^3. This confirms that x^2 + y^2 = 21 is correct. Why Other Options Are Wrong: 18 and 24 come from incorrect manipulation of identities or arithmetic errors when isolating xy. 27 is the value of x^2 + xy + y^2, not x^2 + y^2. 36 is too large and would not satisfy both identities at the same time. Only 21 is consistent with all given conditions. Common Pitfalls: Typical mistakes include confusing x^2 + xy + y^2 with x^2 + y^2, forgetting the factor x - y in the cube identity, or misusing (x - y)^2. Some students try to solve for x and y directly, which is unnecessary and can introduce extra errors. Working symbolically with identities is faster and safer in exam conditions. Final Answer: The required value of x^2 + y^2 is 21.

Question 17

In trigonometry, if cot A is given by the identity cot A = sin B / (1 - cos B), and both angles are such that all functions are defined, then what is the value of cot 2A expressed purely in terms of angle B?

Accepted Answer

Introduction / Context: This trigonometry question checks whether you can recognise standard half angle identities and then use them to relate angles A and B. The goal is not to evaluate any numeric angle, but to simplify cot 2A in terms of B by using the given relationship between cot A and an expression containing sin B and cos B. Such questions are common in competitive exams because they test formula knowledge and symbolic manipulation together. Given Data / Assumptions: cot A = sin B / (1 - cos B). We assume angles A and B lie in ranges where all trigonometric functions involved are defined. We need to find cot 2A expressed only in terms of B. Concept / Approach: The key observation is that sin B / (1 - cos B) is a standard half angle form. Using the identity for tan(B / 2): tan(B / 2) = (1 - cos B) / sin B, we can see that its reciprocal is sin B / (1 - cos B) = cot(B / 2). This means cot A equals cot(B / 2), so A and B / 2 are cotangent equal angles, differing only by multiples of 180 degrees. Under the usual principal acute angle setting, we can take A = B / 2. Then 2A equals B, and cot 2A becomes cot B directly. Step-by-Step Solution: Recall the half angle identity: tan(B / 2) = (1 - cos B) / sin B. Take the reciprocal to obtain: cot(B / 2) = sin B / (1 - cos B). Given cot A = sin B / (1 - cos B), we have cot A = cot(B / 2). Therefore, A and B / 2 differ by multiples of 180 degrees. In a typical aptitude context with principal values, take A = B / 2. Now compute 2A: 2A = 2 * (B / 2) = B. Hence, cot 2A = cot B. Verification / Alternative check: We can check by substituting a specific angle B where the expression is defined. For example, let B be an acute angle such that the expression is valid. Then A = B / 2 gives cot A exactly equal to sin B / (1 - cos B) by the half angle identity. Doubling A returns B, so cot 2A is simply cot B. Because the relationship is derived from identities which hold for all admissible B, the conclusion cot 2A = cot B is generally valid in this context. Why Other Options Are Wrong: cot(B / 2) would represent cot A, not cot 2A. cot 2B and tan B introduce the wrong multiple or reciprocal of the angle. tan(B / 2) is the reciprocal of cot(B / 2), not cot 2A. Only cot B correctly matches the simplified form of cot 2A under the given relationship. Common Pitfalls: One common mistake is to try to express everything in terms of sin A and cos A instead of noticing the direct half angle identity in B. Another is to incorrectly double B and conclude cot 2A = cot 2B, which ignores that A itself is B / 2. Always identify known identities first before performing algebraic manipulations. Final Answer: The value of cot 2A in terms of B is cot B.

Question 18

Consider the quadratic inequality x^2 - 7x + 12 > 0. By analysing the corresponding quadratic equation and its factorisation, which of the following ranges of x correctly represents the full solution set of this inequality?

Accepted Answer

Introduction / Context: Quadratic inequalities are a standard topic in aptitude and algebra. The typical approach is to first solve the corresponding quadratic equation to find critical points (roots), then use the sign of the quadratic on different intervals. This question asks you to interpret x^2 - 7x + 12 > 0 and choose the correct range of x that satisfies the inequality, testing your understanding of factorisation and sign analysis. Given Data / Assumptions: The inequality is x^2 - 7x + 12 > 0. We consider real values of x. We must choose the correct description of the solution set from the options. Concept / Approach: First, solve the related equation x^2 - 7x + 12 = 0 by factorisation. The roots split the number line into intervals. Because the coefficient of x^2 is positive, the parabola opens upward. Therefore, the quadratic expression will be positive outside the interval between the roots and negative inside it. Once we know the roots and understand the shape of the graph, we can write the solution set of x^2 - 7x + 12 > 0 as the union of appropriate intervals. Step-by-Step Solution: Consider the equation x^2 - 7x + 12 = 0. Factorise the quadratic: x^2 - 7x + 12 = (x - 3)(x - 4). Set each factor equal to zero to find roots: x - 3 = 0 or x - 4 = 0. So the roots are x = 3 and x = 4. Because the coefficient of x^2 is positive, the parabola opens upwards. The expression (x - 3)(x - 4) is zero at x = 3 and x = 4, negative between these roots, and positive outside this interval. We want x^2 - 7x + 12 > 0, that is, (x - 3)(x - 4) > 0. The product of two factors is positive when both are positive or both are negative. Both negative when x < 3. Both positive when x > 4. Therefore, the solution set is x < 3 or x > 4. Verification / Alternative check: Pick test points in each region. For x = 2 (less than 3), x^2 - 7x + 12 = 4 - 14 + 12 = 2, which is positive. For x = 3.5 (between 3 and 4), we get 3.5^2 - 7 * 3.5 + 12 = 12.25 - 24.5 + 12 = -0.25, which is negative. For x = 5 (greater than 4), x^2 - 7x + 12 = 25 - 35 + 12 = 2, which is positive again. This matches our interval reasoning. Why Other Options Are Wrong: 3 < x < 4 corresponds to the region where the quadratic is negative, so it does not satisfy the inequality with greater than zero. Saying x < 3 only or x > 4 only ignores half of the valid solution set. The option 3 ≤ x ≤ 4 describes the region where the expression is zero or negative, not strictly positive. Common Pitfalls: Students often confuse the direction of the inequality and pick the region between the roots instead of outside them. Another mistake is to forget that strict inequality (>) excludes the roots themselves, although in this question the options are written in interval form that already reflects strict inequalities. Always analyse the sign on each interval carefully. Final Answer: The solution set of the inequality is x < 3 or x > 4.

Question 19

If tan x = 1 for an acute angle x with 0° < x < 90°, then using standard trigonometric identities, what is the value of the expression 2 sin x cos x?

Accepted Answer

Introduction / Context: This question checks your familiarity with standard trigonometric values and identities. When you know tan x for an acute angle, you should be able to infer sin x and cos x or use angle knowledge to evaluate expressions like 2 sin x cos x. The expression 2 sin x cos x is especially important because it is equal to sin 2x, a well known double angle identity. Given Data / Assumptions: tan x = 1. 0° < x < 90°, so x is an acute angle. We need the value of 2 sin x cos x. Concept / Approach: First, recall which acute angle has tangent equal to 1. The standard angle with tan x = 1 is x = 45°. Once we know x, we can either directly use the identity 2 sin x cos x = sin 2x, or substitute the well known values sin 45° and cos 45°. Using the double angle identity is faster because it turns the problem into evaluating sin of a specific known angle. Step-by-Step Solution: Given tan x = 1 and x is acute, we know x = 45°. Recall the double angle identity: sin 2x = 2 sin x cos x. We need 2 sin x cos x, which is exactly sin 2x. Compute 2x when x = 45°: 2x = 2 * 45° = 90°. Therefore, 2 sin x cos x = sin 2x = sin 90°. The value of sin 90° is 1. Hence, 2 sin x cos x = 1. Verification / Alternative check: We can also compute sin x and cos x explicitly. For x = 45°, sin 45° = sqrt(2) / 2 and cos 45° = sqrt(2) / 2. Then 2 sin x cos x = 2 * (sqrt(2) / 2) * (sqrt(2) / 2) = 2 * (2 / 4) = 2 * (1 / 2) = 1. This confirms the result obtained via the double angle identity. Why Other Options Are Wrong: 1/2 and sqrt(3) / 2 are typical sine values of 30° and 60°, not 90°. sqrt(3) and 0 do not correspond to sin 2x for any acute x in this setting. Since the expression exactly equals sin 90°, and sin 90° is 1, none of the other options fit. Common Pitfalls: Some students confuse the angle whose tangent is 1 and mistakenly use 30° or 60°. Others forget the double angle identity and attempt to compute sin x and cos x from tan x using a right triangle, which is valid but longer. Remembering that 2 sin x cos x = sin 2x is a powerful shortcut in many trigonometry problems. Final Answer: The value of 2 sin x cos x is 1.

Question 20

Using trigonometric sum formulas, find the exact value of the expression sin 46° cos 44° + cos 46° sin 44°.

Accepted Answer

Introduction / Context: This question is designed to test your knowledge of trigonometric angle sum identities, especially the sine of a sum formula. Instead of directly calculating each term numerically, you can recognise the structure sin A cos B + cos A sin B and simplify it using the identity for sin(A + B). Recognising this pattern makes the problem very quick in an exam setting. Given Data / Assumptions: The expression is sin 46° cos 44° + cos 46° sin 44°. All angles are in degrees. We are asked to find the exact value of this expression. Concept / Approach: The key identity is the sine addition formula: sin(A + B) = sin A cos B + cos A sin B. The given expression exactly matches this pattern if we choose A = 46° and B = 44°. Therefore, instead of treating the two products separately, we combine them into a single sine term. After that, we just evaluate sin of a simple angle (A + B), which here becomes sin 90°, a standard value. Step-by-Step Solution: Start with the identity: sin(A + B) = sin A cos B + cos A sin B. In the given expression, identify A = 46° and B = 44°. Then sin 46° cos 44° + cos 46° sin 44° = sin(46° + 44°). Compute the sum of angles: 46° + 44° = 90°. So the expression simplifies to sin 90°. We know sin 90° = 1. Therefore, the exact value of the original expression is 1. Verification / Alternative check: As an approximate check, we can think about approximate values. sin 46° and cos 44° are both close to sin 45° and cos 45°, which are about 0.707. Their product is roughly 0.5. The same is true for cos 46° and sin 44°. Adding these two approximate halves gives us a value close to 1. This numerical reasoning supports the exact value obtained from the identity. Why Other Options Are Wrong: sin 2° and cos 2° correspond to sin(A - B) and cos(A - B) type expressions, not sin(A + B). 2 and 0 are not valid values of sine for angles in degrees, except in trivial or impossible cases for standard trigonometric values. Sine of an angle is always between -1 and 1, so 2 is impossible, and 0 would correspond to angles like 0° or 180°, not 90°. Common Pitfalls: Students sometimes confuse sin(A + B) with cos(A + B) or with sin(A - B), leading to incorrect use of formulas. Another mistake is to try to approximate each term individually on a calculator instead of spotting the identity, which is slower and more error prone in exams. Always scan for standard patterns such as sin A cos B + cos A sin B when you see this kind of sum. Final Answer: The expression evaluates exactly to 1.

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