Question 1

The sum of two real numbers is 7 and the sum of their squares is 23. Using algebraic identities, what is the value of the product of the two numbers?

Accepted Answer

Introduction / Context: This algebra question involves two unknown real numbers that are related through their sum and the sum of their squares. You are required to find the product of these two numbers using a known identity that links sums, products, and squares. Such problems are common in aptitude tests and help reinforce the use of algebraic identities. Given Data / Assumptions: Let the two numbers be x and y. x + y = 7. x^2 + y^2 = 23. We assume x and y are real numbers. Concept / Approach: We use the identity relating the sum of squares to the square of the sum and the product: x^2 + y^2 = (x + y)^2 − 2xy. The given information provides both x + y and x^2 + y^2, which allows us to solve directly for xy (the product). Once xy is found, we simply choose the matching value from the options. Step-by-Step Solution: Step 1: Start from the identity x^2 + y^2 = (x + y)^2 − 2xy. Step 2: Substitute the known values: x + y = 7 and x^2 + y^2 = 23. Step 3: Compute (x + y)^2: 7^2 = 49. Step 4: Use the identity: 23 = 49 − 2xy. Step 5: Rearrange to solve for xy: 2xy = 49 − 23. Step 6: Simplify the right side: 49 − 23 = 26, so 2xy = 26. Step 7: Divide by 2 to find the product: xy = 26 / 2 = 13. Verification / Alternative check: To verify, suppose the two numbers are roots of a quadratic equation with sum 7 and product 13. The quadratic would be t^2 − 7t + 13 = 0. The discriminant is 7^2 − 4 * 1 * 13 = 49 − 52 = −3, which is negative, indicating complex roots. However, the question only asks for the product xy and does not require the individual numbers to be real for the identity to hold. The algebraic identity remains valid, and the arithmetic checks out correctly. Why Other Options Are Wrong: The values 9, 10, 11 and 12 do not satisfy the equation 23 = 49 − 2xy when substituted for xy. For example, if xy = 10, then 49 − 2 * 10 = 29, which is not equal to 23. Only xy = 13 gives the correct relationship. Common Pitfalls: Students sometimes try to guess the two numbers or set up a more complicated system of equations, which is unnecessary. Another mistake is misremembering the identity, for instance writing x^2 + y^2 = (x + y)^2 + 2xy instead of subtracting 2xy. Carefully recalling the correct form of the identity avoids these errors. Final Answer: The product of the two numbers is 13.

Question 2

Rs 7,750 is divided among three people X, Y and Z such that three times the share of X equals five times the share of Y, which also equals two times the share of Z. Under this condition, what is the share in rupees received by Z?

Accepted Answer

Introduction / Context: This question tests ratio and proportion concepts in the context of dividing a total amount of money among three people according to a given relationship. The relations between the shares of X, Y and Z are given in terms of multiples, and you must convert these into a consistent ratio to find the share of Z. Given Data / Assumptions: Total sum to be divided = Rs 7,750. Let the shares be X, Y and Z respectively. 3X = 5Y = 2Z. All values are non-negative and represent real monetary amounts. Concept / Approach: If 3X, 5Y and 2Z are all equal to the same constant value, say k, then we can express X, Y and Z in terms of k. Specifically, X = k / 3, Y = k / 5 and Z = k / 2. The total sum X + Y + Z must equal Rs 7,750. This gives an equation in k that we can solve, then use to obtain Z. Step-by-Step Solution: Step 1: Let 3X = 5Y = 2Z = k. Step 2: Then X = k / 3, Y = k / 5 and Z = k / 2. Step 3: The total sum is X + Y + Z = Rs 7,750. Step 4: Substitute the expressions in terms of k: k / 3 + k / 5 + k / 2 = 7,750. Step 5: Find the common denominator of 3, 5 and 2, which is 30. Step 6: Rewrite each term: k / 3 = 10k / 30, k / 5 = 6k / 30, k / 2 = 15k / 30. Step 7: Add them: (10k + 6k + 15k) / 30 = 31k / 30. Step 8: Set this equal to 7,750: 31k / 30 = 7,750. Step 9: Solve for k: k = 7,750 * 30 / 31 = 7,500. Step 10: Now find Z: Z = k / 2 = 7,500 / 2 = 3,750. Verification / Alternative check: Check the individual shares. X = k / 3 = 7,500 / 3 = 2,500. Y = k / 5 = 7,500 / 5 = 1,500. Z = 3,750. Sum: 2,500 + 1,500 + 3,750 = 7,750, which matches the given total. Also verify the ratio condition: 3X = 7,500, 5Y = 7,500, 2Z = 7,500, so the condition 3X = 5Y = 2Z is satisfied. Why Other Options Are Wrong: Other values such as 4,250, 3,875, 4,475 or 3,500 do not produce a set of X, Y and Z values that both sum to 7,750 and satisfy 3X = 5Y = 2Z. They fail either the total-sum check or the ratio relationship when back-substituted. Common Pitfalls: A frequent error is to misinterpret 3X = 5Y = 2Z as simple ratios like 3 : 5 : 2, which is incorrect in this context. You must treat each expression as equal to the same constant and then derive the shares accordingly. Another pitfall is arithmetic mistakes when finding the common denominator or solving for k. Final Answer: The share received by Z is Rs 3,750.

Question 3

In this comparison question, two quadratic equations are given: I. 5x^2 + 28x = -15 (that is, 5x^2 + 28x + 15 = 0) and II. 3y^2 + 11y + 6 = 0. After solving both equations, what is the correct relationship between x and y?

Accepted Answer

Introduction / Context: This is a standard quadratic comparison problem often seen in bank and aptitude exams. You are given two quadratic equations in variables x and y. Each equation can have two roots. You must determine the relationship between any possible values of x and y (derived from their respective equations). Given Data / Assumptions: Equation I: 5x^2 + 28x = −15, which can be rewritten as 5x^2 + 28x + 15 = 0. Equation II: 3y^2 + 11y + 6 = 0. x and y can take any root from their respective equations. We must compare all combinations of roots. Concept / Approach: First, solve each quadratic equation to find its real roots. Then, consider the possible pairs (x, y), where x is any root of Equation I and y is any root of Equation II. If x is always greater than y, you choose x > y, and similarly for other relations. If sometimes x is greater and sometimes smaller, then the relationship cannot be determined. Step-by-Step Solution: Step 1: Rewrite Equation I as 5x^2 + 28x + 15 = 0. Step 2: Factor Equation I: 5x^2 + 28x + 15 = (x + 5)(5x + 3) = 0. Step 3: Roots of Equation I: x = −5 and x = −3 / 5 (−0.6). Step 4: Factor Equation II: 3y^2 + 11y + 6 = (y + 3)(3y + 2) = 0. Step 5: Roots of Equation II: y = −3 and y = −2 / 3 (approximately −0.6667). Step 6: Compare all combinations of roots. Case 1: x = −5, y = −3. Then x < y. Case 2: x = −5, y = −2 / 3. Then x < y again. Case 3: x = −0.6, y = −3. Then x > y. Case 4: x = −0.6, y = −2 / 3 (≈ −0.6667). Then x > y. Verification / Alternative check: From the comparisons, we see that sometimes x is less than y (when x = −5) and sometimes x is greater than y (when x = −0.6). Because the problem does not restrict x or y to a particular root, all these combinations are possible. Therefore, there is no single consistent relationship that holds for all allowed values of x and y. Why Other Options Are Wrong: x > y is not always true because when x = −5 and y = −3, x is clearly less than y. x ≥ y and x ≤ y both require a consistent inequality in one direction or equality, which does not occur for all pairs of roots. x < y is not always true because when x = −0.6 and y = −3, x is greater than y. Common Pitfalls: A common mistake is to compare only one pair of roots (for example, the larger root of each equation) and draw a conclusion without checking all possible combinations. Another error is misfactorising the quadratic equations, which leads to incorrect roots and hence incorrect comparisons. Final Answer: The correct conclusion is that the relationship between x and y cannot be determined uniquely.

Question 4

In the following quadratic comparison question, two equations are given: I. 2x^2 − 21x + 54 = 0 and II. y^2 − 14y + 49 = 0. After solving both equations, what is the correct relationship between x and y?

Accepted Answer

Introduction / Context: This is another quadratic comparison problem. Here, two quadratic equations in variables x and y are given, and you must determine the relationship between any possible root of the first equation and any possible root of the second equation. These questions strengthen understanding of quadratic roots and inequality comparisons. Given Data / Assumptions: Equation I: 2x^2 − 21x + 54 = 0. Equation II: y^2 − 14y + 49 = 0. x can be any root of Equation I, and y can be any root of Equation II. We look for a relationship that holds for all possible combinations. Concept / Approach: We first solve each quadratic to find its roots. Then we compare the numerical values of the roots. If every possible x is less than every possible y, then x < y. If the relationships vary, the conclusion would be that the relationship cannot be determined. Here, carefully comparing all roots is key. Step-by-Step Solution: Step 1: Factor Equation I: 2x^2 − 21x + 54 = 0. Step 2: Factorisation gives (x − 6)(2x − 9) = 0. Step 3: Roots for x are x = 6 and x = 9 / 2 = 4.5. Step 4: Factor Equation II: y^2 − 14y + 49 = 0. Step 5: This is a perfect square: (y − 7)^2 = 0. Step 6: So y has a single repeated root y = 7. Step 7: Now compare x and y for all combinations. Case 1: x = 6, y = 7. Then x < y. Case 2: x = 4.5, y = 7. Then x < y again. Step 8: In both possible cases, x is strictly less than y. Verification / Alternative check: We can check quickly: the smallest y value possible is 7, and the largest x value possible is 6 (since 6 > 4.5). Because even the maximum value of x is still less than the minimum value of y, we can confidently say that for every possible pair (x, y), x < y holds true. Why Other Options Are Wrong: x > y is impossible because both roots of x are less than 7. x ≥ y and x ≤ y imply that x could equal y in at least one case, but there is no root of x equal to 7. The combined statement “x = y or relation can’t be established between x and y” is incorrect because a clear consistent relationship does exist: x is always less. Common Pitfalls: Some students may only compare one root of x with y and incorrectly choose a non-committal option. Others may fail to recognise the perfect square form of Equation II and miscalculate the roots. Always determine all roots carefully and compare the extreme values to see if a consistent inequality exists. Final Answer: For all possible values, x < y.

Question 5

What approximate value should come in place of the question mark (?) in the expression √580 × ∛510 + 49.999 × 3.999, if you are not expected to calculate the exact value but only a close numerical approximation?

Accepted Answer

Introduction / Context: This question tests your ability to work with approximations and to estimate the value of expressions involving square roots, cube roots and decimals. Instead of requiring an exact evaluation, it asks for a close approximate value that can be matched with one of the given options, a common pattern in competitive exams. Given Data / Assumptions: Expression: √580 × ∛510 + 49.999 × 3.999. We are allowed to approximate square roots, cube roots, and decimals. We need a reasonably accurate estimate, not the exact value. Concept / Approach: To approximate the expression, we replace difficult numbers by nearby friendly values for which mental calculations are easier. For instance, √580 is close to √576, and ∛510 is close to ∛512. Likewise, 49.999 is almost 50, and 3.999 is almost 4. After approximating each component, we compute the expression and then select the closest option. Step-by-Step Solution: Step 1: Approximate √580. Since 580 is close to 576 and √576 = 24, we take √580 ≈ 24. Step 2: Approximate ∛510. Since 510 is close to 512 and ∛512 = 8, we take ∛510 ≈ 8. Step 3: Multiply these approximations: √580 × ∛510 ≈ 24 × 8 = 192. Step 4: Approximate 49.999 as 50 and 3.999 as 4. Step 5: Multiply: 49.999 × 3.999 ≈ 50 × 4 = 200. Step 6: Add the two approximate parts: 192 + 200 = 392. Step 7: Compare with the options: 392 is one of the given choices and matches the approximate value well. Verification / Alternative check: If we used a calculator, we would get a more precise value close to 392, confirming the correctness of this approximation strategy. Even if our approximations for the roots are slightly off, the final answer remains closest to 392 among the given options, because the other options (384, 410, 372, 360) are either significantly lower or higher than the reasonable estimated value. Why Other Options Are Wrong: 384 and 372 are too low, given that the second term alone is already about 200 and the first term is clearly above 180. 410 and 360 are on the higher or much lower side compared to our careful estimate. None of these match the sum of 192 and 200 as closely as 392 does. Common Pitfalls: A major pitfall is over-approximating or under-approximating the roots or decimals without checking the effect on the total. Some students may approximate √580 as 23 or 25 incorrectly or mis-handle the cube root of 510. Always pick the nearest perfect square or cube to base your approximation, and remember to approximate all parts consistently. Final Answer: The approximate value of the expression is 392.

Question 6

What approximate value should come in place of the question mark (?) in the expression (55.01 + 16.0003) × 22.01 ÷ 10.998, if you are expected to find a close numerical estimate rather than the exact value?

Accepted Answer

Introduction / Context: This problem checks your ability to approximate operations involving decimals. Instead of doing lengthy exact calculations, you are expected to gently round the numbers to convenient values and then compute a close estimate, which is a common technique in time-pressured competitive exams. Given Data / Assumptions: Expression: (55.01 + 16.0003) × 22.01 ÷ 10.998. We are looking for a reasonable approximation, not exact arithmetic. All numbers are close to convenient integers. Concept / Approach: The idea is to round each number to a nearby simple value, as long as the rounding does not introduce too much error. We then perform the simplified operations to get an approximate result. Since the options are well separated, a reasonably good approximation will clearly indicate the correct choice. Step-by-Step Solution: Step 1: Approximate 55.01 as 55. Step 2: Approximate 16.0003 as 16. Step 3: Compute the sum inside the brackets: 55.01 + 16.0003 ≈ 55 + 16 = 71. Step 4: Approximate 22.01 as 22. Step 5: Approximate 10.998 as 11. Step 6: Substitute these approximations into the expression: (55.01 + 16.0003) × 22.01 ÷ 10.998 ≈ 71 × 22 ÷ 11. Step 7: First divide: 22 ÷ 11 = 2. Step 8: Multiply: 71 × 2 = 142. Step 9: Therefore, the approximate value is about 142. Verification / Alternative check: Because all the original decimals are extremely close to the integers we used (differences are around 0.01 or less), the resulting product and quotient will not deviate much from our approximate result. A more precise calculation would yield a value very close to 142, confirming that 142 is the best available approximation among the choices. Why Other Options Are Wrong: Values like 190 or 160 are significantly higher than our estimate, and 130 or 110 are noticeably lower. Since our approximations are very tight and involve only minimal rounding, any large deviation from 142 is not justified by the arithmetic. Common Pitfalls: Some students try to perform full precise calculations, which is time consuming. Others incorrectly round in a way that changes the structure of the expression, such as rounding both numerator and denominator in opposite directions. A balanced rounding, where all numbers are rounded sensibly to the nearest simple values, produces an accurate estimate efficiently. Final Answer: The approximate value of the expression is 142.

Question 7

What approximate value should come in place of the question mark (?) in the expression 499.99 + 1999 ÷ 39.99 × 50.01, if you are required to estimate the value without calculating it exactly?

Accepted Answer

Introduction / Context: This question involves estimating the value of an expression that includes division, multiplication, and addition with decimals. The goal is to find a good approximation that matches one of the provided options, a typical skill tested in competitive exams to save time during calculations. Given Data / Assumptions: Expression: 499.99 + 1999 ÷ 39.99 × 50.01. We are allowed to approximate 499.99, 39.99 and 50.01 by nearby integers. The objective is a close but not exact numerical answer. Concept / Approach: Since 499.99 is very close to 500, 39.99 is very near 40, and 50.01 is essentially 50, we can replace these decimals with their closest integers. This simplifies the calculation dramatically: we first handle the division, then the multiplication, and finally the addition, in accordance with the correct order of operations. Step-by-Step Solution: Step 1: Approximate 499.99 ≈ 500. Step 2: Approximate 39.99 ≈ 40. Step 3: Approximate 50.01 ≈ 50. Step 4: Rewrite the expression with these approximations: 499.99 + 1999 ÷ 39.99 × 50.01 ≈ 500 + 1999 ÷ 40 × 50. Step 5: Perform the division first: 1999 ÷ 40 ≈ 50 (since 40 × 50 = 2000, which is very close). Step 6: Then multiply by 50: 50 × 50 = 2,500. Step 7: Finally add the approximate 500: 500 + 2,500 = 3,000. Step 8: Thus, the approximate value is about 3,000. Verification / Alternative check: Even if you are slightly more precise, 1999 ÷ 40 is 49.975, extremely close to 50, and using 49.975 × 50 gives roughly 2,498.75. Adding 500 yields around 2,998.75, which is still closest to 3,000 among the given options. Therefore, the approximation is robust and consistent with the exact arithmetic. Why Other Options Are Wrong: Options like 2,500 or 2,700 underestimate the result, since the multiplication term alone is around 2,500 and then you still add roughly 500 more. Options like 3,200 or 2,800 are not as close to the more accurate approximate total of about 3,000. Common Pitfalls: A common mistake is to round some numbers up and others down in a biased way that skews the result, or to accidentally change the order of operations. Always remember that division and multiplication take precedence over addition, and when approximating, keep the rounding consistent and moderate. Final Answer: The approximate value of the expression is 3,000.

Question 8

Evaluate the integer expression −261 + (−380) − (−521) + 821 − (−121) and find its exact numerical value.

Accepted Answer

Introduction / Context: This is a straightforward arithmetic problem involving positive and negative integers. It tests your ability to handle addition and subtraction of signed numbers correctly, especially when multiple minus signs and brackets appear in the same expression. Given Data / Assumptions: Expression: −261 + (−380) − (−521) + 821 − (−121). All numbers are integers. Standard rules of integer addition and subtraction apply. Concept / Approach: When dealing with signed numbers, we interpret addition of a negative number as subtraction and subtraction of a negative number as addition. We can simplify the series of operations by removing brackets and rewriting the expression in terms of plus and minus signs clearly, then combining the positive and negative contributions. Step-by-Step Solution: Step 1: Start with the given expression: −261 + (−380) − (−521) + 821 − (−121). Step 2: Remove brackets, keeping track of signs: = −261 − 380 + 521 + 821 + 121. Step 3: Group positive and negative terms separately. Positive terms: 521 + 821 + 121. Negative terms: −261 and −380. Step 4: Add positive terms: 521 + 821 = 1,342. Step 5: Then add 121: 1,342 + 121 = 1,463. Step 6: Add negative terms: −261 − 380 = −641. Step 7: Combine positive and negative totals: 1,463 + (−641) = 1,463 − 641. Step 8: Subtract: 1,463 − 641 = 822. Step 9: Therefore, the value of the expression is 822. Verification / Alternative check: You can also compute step by step from left to right: −261 − 380 = −641; then −641 − (−521) becomes −641 + 521 = −120; then −120 + 821 = 701; and finally 701 − (−121) becomes 701 + 121 = 822. This alternative calculation again yields 822, confirming the result. Why Other Options Are Wrong: Values like 833, 825, 800 and 812 result from arithmetic mistakes such as misadding the positives, forgetting to change a minus minus to plus, or mishandling one of the intermediate sums. A correct and careful calculation as shown leads uniquely to 822. Common Pitfalls: Typical errors include forgetting that subtracting a negative number is equivalent to adding a positive number, or incorrectly combining numbers with different signs. Keeping track of signs by rewriting the entire expression without brackets and then grouping positive and negative terms separately helps avoid mistakes. Final Answer: The exact value of the expression is 822.

Question 9

Solve the expression −1/4 × { −45 − [ (−96) ÷ (−32) ] } using the correct order of operations and determine its exact numerical value.

Accepted Answer

Introduction / Context: This question is a simplification problem involving fractions, negative numbers and nested brackets. It tests your ability to follow the correct order of operations (division before subtraction, and brackets first) and to handle negative signs carefully. Given Data / Assumptions: Expression: −1/4 × { −45 − [ (−96) ÷ (−32) ] }. Standard arithmetic rules apply, with division done before addition or subtraction within the brackets. Multiplication by a negative fraction appears outside the brackets. Concept / Approach: We simplify from the inside out. First compute the inner division (−96) ÷ (−32), then handle the subtraction inside the curly braces, and finally multiply the result by −1/4. Being careful with signs at every step is crucial, because a negative multiplied by a negative yields a positive. Step-by-Step Solution: Step 1: Focus on the innermost bracket: (−96) ÷ (−32). Step 2: Division of two negative numbers gives a positive result: (−96) ÷ (−32) = 3. Step 3: Substitute this into the expression inside the curly braces: −45 − [ (−96) ÷ (−32) ] becomes −45 − 3. Step 4: Compute −45 − 3 = −48. Step 5: Now the full expression becomes −1/4 × { −48 }. Step 6: Multiplying a negative by a negative gives a positive result. Step 7: Compute the magnitude: 1/4 of 48 is 48 / 4 = 12. Step 8: Therefore, −1/4 × (−48) = +12. Step 9: The final value of the expression is 12. Verification / Alternative check: We can quickly check the calculation: the inner division 96 / 32 is clearly 3, and the signs cancel because both numerator and denominator are negative. Then we have −45 − 3 = −48. Multiplying −48 by −1/4 should yield a positive number with magnitude 48 / 4 = 12. No step violates the order of operations, so the result 12 is consistent. Why Other Options Are Wrong: 10.5 and −10.5 would imply that the fraction applied was not 1/4 or that the subtraction inside the braces was done incorrectly. −12 would mean that the effect of one of the negative signs was ignored. 11 is simply a miscalculation of 48 / 4. Only 12 matches careful application of the arithmetic rules. Common Pitfalls: Students may forget that subtracting a positive 3 from −45 gives −48, or they might mishandle the negative signs when multiplying by −1/4. Another common mistake is to divide −96 by 32 instead of −32, giving the wrong sign in the intermediate step. Writing each step clearly and checking the sign at each operation helps avoid these errors. Final Answer: The exact value of the expression is 12.

Question 10

If the polynomial 3x^2 + ax + 4 is exactly divisible by x − 5, meaning that x − 5 is a factor of the polynomial, what is the value of the constant a?

Accepted Answer

Introduction / Context: This algebra question involves the factor theorem for polynomials. If x − 5 is a factor of the quadratic polynomial 3x^2 + ax + 4, then substituting x = 5 into the polynomial must give zero. This condition allows us to find the unknown coefficient a. Given Data / Assumptions: Polynomial: P(x) = 3x^2 + ax + 4. x − 5 is a factor of P(x). Therefore, P(5) = 0 by the factor theorem. Concept / Approach: The factor theorem states that if x − r is a factor of a polynomial P(x), then P(r) = 0. Here, r is 5. We substitute x = 5 into the polynomial and set the result equal to zero, then solve the resulting linear equation for a. This method avoids long division and directly uses the property of polynomial factors. Step-by-Step Solution: Step 1: Write down the condition from the factor theorem: P(5) = 0. Step 2: Compute P(5) using P(x) = 3x^2 + ax + 4. Step 3: Substitute x = 5: P(5) = 3 * (5^2) + a * 5 + 4. Step 4: Calculate 5^2 = 25, so 3 * 25 = 75. Step 5: The expression becomes P(5) = 75 + 5a + 4. Step 6: Simplify: P(5) = 79 + 5a. Step 7: Set P(5) equal to zero because x − 5 is a factor: 79 + 5a = 0. Step 8: Solve for a: 5a = −79, so a = −79 / 5. Step 9: Convert the fraction to decimal: −79 / 5 = −15.8. Step 10: Therefore, the required value of a is −15.8. Verification / Alternative check: We can quickly verify by substituting a = −15.8 back into the polynomial and evaluating at x = 5. P(5) = 3 * 25 + (−15.8) * 5 + 4 = 75 − 79 + 4 = 0. Since P(5) is zero, x − 5 is indeed a factor, confirming that a = −15.8 is correct. Why Other Options Are Wrong: Values like −5, −12, −15.6 and −20 do not satisfy the condition that P(5) = 0. Substituting any of these into P(x) and evaluating at x = 5 yields a nonzero result, meaning x − 5 would not be a factor of the polynomial. Common Pitfalls: Students sometimes forget to use the factor theorem and attempt polynomial long division, which is more time consuming and can lead to arithmetic mistakes. Others may incorrectly simplify the expression 3 * 25 + 4 or mis-handle the sign when solving 79 + 5a = 0. Always write the intermediate step clearly as 79 + 5a and solve carefully for a. Final Answer: The value of the constant is a = −15.8.

Question 11

Solve the expression 80 ÷ [48 − 56 − (60 − 36 ÷ 12 × 4)] by carefully applying the correct order of operations (BODMAS or PEMDAS) and find its exact value in simplest fractional form.

Accepted Answer

Introduction / Context: This problem is designed to test your understanding of the order of operations in arithmetic expressions involving multiple nested brackets. You must correctly apply the rules for brackets, division and multiplication before subtraction, and then perform the final division to express the answer as a simple fraction. Given Data / Assumptions: Expression: 80 ÷ [48 − 56 − (60 − 36 ÷ 12 × 4)]. Standard order of operations applies (brackets first, then division and multiplication from left to right, followed by addition and subtraction). The final answer should be a simplified fraction. Concept / Approach: We evaluate from the innermost bracket outward. Inside the parentheses (60 − 36 ÷ 12 × 4), we must first perform 36 ÷ 12 and then multiply by 4, before subtracting from 60. The result is then used inside the square bracket [48 − 56 − (result)], which we simplify step by step. Finally, we divide 80 by the simplified bracket value. Step-by-Step Solution: Step 1: Consider the inner parentheses: (60 − 36 ÷ 12 × 4). Step 2: Perform the division: 36 ÷ 12 = 3. Step 3: Then multiply by 4: 3 × 4 = 12. Step 4: Now the inner parentheses become: 60 − 12 = 48. Step 5: Substitute this result into the square bracket: [48 − 56 − (60 − 36 ÷ 12 × 4)] becomes [48 − 56 − 48]. Step 6: Compute from left to right: 48 − 56 = −8. Step 7: Then −8 − 48 = −56. Step 8: So the full expression reduces to 80 ÷ (−56). Step 9: Compute the division: 80 ÷ (−56) = 80 / (−56) = −80 / 56. Step 10: Simplify −80 / 56 by dividing numerator and denominator by 8: (−80 ÷ 8) / (56 ÷ 8) = −10 / 7. Step 11: Therefore, the value of the expression is −10 / 7. Verification / Alternative check: We can verify by checking the intermediate bracket value. If the bracket equals −56, then 80 divided by −56 should give a negative fraction with magnitude 80 / 56. Simplifying 80 / 56 by the greatest common divisor 8 indeed gives 10 / 7, so the magnitude is correct and the sign is negative because we divided by a negative quantity. Why Other Options Are Wrong: −5 / 7 and −20 / 7 have the wrong magnitude, indicating mistakes in simplifying 80 / 56 or miscomputing the bracket. The positive fractions 5 / 7 and 10 / 7 ignore the negative sign from the bracket value, which is a sign error. Only −10 / 7 matches the correct step-by-step evaluation. Common Pitfalls: Frequent errors include performing 36 ÷ 12 × 4 as 36 ÷ (12 × 4) instead of handling division and multiplication from left to right, and miscalculating the bracket expression 48 − 56 − 48. Another common issue is forgetting to simplify the final fraction or dropping the negative sign when dividing by a negative number. Final Answer: The exact value of the expression is −10 / 7.

Question 12

Simplify the fractional expression 3/4 + {3/4 + (3/4 ÷ (3/4 + 3/4))} using the correct order of operations and find its numerical value.

Accepted Answer

Introduction / Context: This question tests simplification of a nested fractional expression using the correct order of operations. Many aptitude and competitive exams include such problems to check comfort with fractions, brackets, and division. The expression has the same fraction 3/4 repeated in different places, which can tempt a candidate to rush. A systematic approach based on BODMAS or PEMDAS avoids mistakes and gives the exact numerical value without needing a calculator. Given Data / Assumptions: • The given expression is 3/4 + {3/4 + (3/4 ÷ (3/4 + 3/4))}. • All operations are ordinary real number operations on fractions. • We must respect the standard order: brackets first, then division and multiplication, then addition and subtraction. Concept / Approach: The key idea is to simplify from the innermost brackets outward. First we add the fractions inside the innermost bracket, then perform the division 3/4 ÷ something, and finally add the outer 3/4 terms. Division of fractions is performed by multiplying by the reciprocal of the divisor. Keeping everything as exact fractions rather than decimals prevents rounding errors and makes checking easier. Step-by-Step Solution: Step 1: Compute the inner sum: 3/4 + 3/4 = 6/4 = 3/2. Step 2: Evaluate the division: 3/4 ÷ (3/2) = 3/4 × 2/3 = (3 × 2) / (4 × 3) = 6/12 = 1/2. Step 3: Now the bracket becomes 3/4 + 1/2. Convert 1/2 to a denominator of 4: 1/2 = 2/4. Step 4: Add these: 3/4 + 2/4 = 5/4. Step 5: Finally add the first 3/4: 3/4 + 5/4 = 8/4 = 2. Verification / Alternative check: We can convert each fraction to decimals as a rough check. We have 3/4 = 0.75. Inside the denominator, 3/4 + 3/4 = 1.5, so 0.75 ÷ 1.5 = 0.5. Then the whole expression is 0.75 + (0.75 + 0.5) = 0.75 + 1.25 = 2.00. This matches the exact fraction result of 2, so the calculation is consistent. Any other order that respects brackets will also lead to the same value. Why Other Options Are Wrong: Option 3/4 is obtained if someone mistakenly adds only the outer 3/4 terms and ignores the division part. Option 1 arises if 3/4 ÷ (3/4 + 3/4) is wrongly treated as 3/4 ÷ 3/4 = 1 without adding the second 3/4. Option 5/4 is the value of the expression before adding the very first 3/4, so it is an intermediate step, not the final answer. Option 0 would require negative or cancelling terms which do not appear here, so it is clearly incorrect. Common Pitfalls: A common mistake is to ignore the curly brackets and perform operations from left to right without respecting nesting. Another frequent error is to treat division of fractions as dividing numerators and denominators separately rather than multiplying by the reciprocal. Some learners also simplify 3/4 + 3/4 incorrectly as 3/8 instead of 6/4. Writing each step clearly and converting to a common denominator whenever fractions are added helps to avoid these issues in any similar aptitude question. Final Answer: The simplified numerical value of the given fractional expression is 2.

Question 13

Using the approximation √42 ≈ 6.480, estimate the value of the square root √(6/7) correct to three decimal places.

Accepted Answer

Introduction / Context: This question checks understanding of square roots, manipulation of fractions under a square root, and use of a given approximate value to estimate another root. Such problems are common in exams where calculators are not allowed, and candidates must use number sense and algebraic relationships between related square roots. Here we are given an approximate value of √42 and asked to estimate √(6/7). Given Data / Assumptions: • Given approximation: √42 ≈ 6.480. • Target expression: √(6/7). • We assume positive square roots only, since we are dealing with magnitudes. • Required answer is an approximate decimal value, chosen from the options. Concept / Approach: The idea is to rewrite 6/7 in a form that involves 42, because we are given √42. Notice that 6/7 = 36/42. This allows us to write √(6/7) as √(36/42). Then we can separate numerator and denominator using the property √(a/b) = √a / √b for positive a and b. Since √36 is exactly 6 and we know an approximation for √42, we can form a ratio 6 / √42 and use the given decimal to estimate the value. Step-by-Step Solution: Step 1: Rewrite the fraction: 6/7 = 36/42. Step 2: Express the root: √(6/7) = √(36/42). Step 3: Use the property of roots: √(36/42) = √36 / √42 for positive numbers. Step 4: Compute √36 = 6 exactly. Step 5: Substitute the approximation √42 ≈ 6.480. Step 6: Form the ratio: √(6/7) ≈ 6 ÷ 6.480. Step 7: Evaluate 6 ÷ 6.480 ≈ 0.9259, which rounds to approximately 0.926. Verification / Alternative check: We know that 6/7 is less than 1, so its square root must also be less than 1. Among the options, only 0.915 and 0.925 are significantly below 1. A quick mental calculation can bound the value. Since √(6/7) is close to √(0.857), and √0.81 = 0.9 while √0.9 ≈ 0.949, we expect the answer somewhere in the low 0.9 range. The approximate calculation 6 ÷ 6.480 gives about 0.926, which is closer to 0.925 than to 0.935 or other options. This confirms the chosen option. Why Other Options Are Wrong: Option 0.935 and 0.945 are somewhat larger than the calculated value 0.926 and would correspond to a slightly larger fraction under the root. Option 0.955 is even larger and would match a value much closer to 1 or above. Option 0.915 is a little smaller than 0.926 and therefore underestimates the root. Given the approximation provided, 0.925 is the most accurate among the listed answers. Common Pitfalls: Learners may directly divide 6 by 7 and then try to approximate the square root of the resulting decimal, which is slower and more error prone. Others may forget to use the relationship between 6/7 and 36/42, losing the benefit of the given √42 approximation. Rounding too early, for example using 6.5 instead of 6.480, can push the estimate away from the correct option. Careful algebraic manipulation followed by precise division avoids these mistakes. Final Answer: The approximate value of √(6/7) using √42 ≈ 6.480 is 0.925.

Question 14

If sin x = 4/5 for an acute angle x in a right triangle, find the value of cosec x + cot x.

Accepted Answer

Introduction / Context: This trigonometry question tests the ability to move between the basic trigonometric ratios sin, cosec and cot using a right triangle. When one ratio is known, the others can be found by using Pythagoras theorem and the definitions of each function. The angle x is acute, so all primary trigonometric functions are positive, which simplifies the reasoning. Given Data / Assumptions: • sin x = 4/5 for an acute angle x. • x lies in the first quadrant, so sin x, cos x, cosec x and cot x are all positive. • We need to calculate cosec x + cot x in exact fractional form. Concept / Approach: For any right triangle with acute angle x, sin x = opposite / hypotenuse and cos x = adjacent / hypotenuse. If sin x is known, we can reconstruct the triangle sides in a convenient ratio and then compute cos x, cosec x and cot x. The reciprocal relation gives cosec x = 1 / sin x. The cotangent function is cot x = cos x / sin x. Substituting the specific values and simplifying yields the required sum. Step-by-Step Solution: Step 1: Interpret sin x = 4/5 as opposite side = 4k and hypotenuse = 5k for some positive k. Step 2: Use Pythagoras theorem to find the adjacent side: adjacent^2 = hypotenuse^2 − opposite^2 = (5k)^2 − (4k)^2 = 25k^2 − 16k^2 = 9k^2. Step 3: Therefore adjacent side = 3k. Step 4: Compute cosec x. Since cosec x = 1 / sin x, we get cosec x = 1 / (4/5) = 5/4. Step 5: Compute cot x. Since cot x = cos x / sin x and cos x = adjacent / hypotenuse = 3k / 5k = 3/5, we have cot x = (3/5) / (4/5) = 3/4. Step 6: Add the two values: cosec x + cot x = 5/4 + 3/4 = 8/4 = 2. Verification / Alternative check: We can verify numerically by assigning a specific triangle with sides 3, 4 and 5. Then sin x = 4/5 is consistent with the given condition. For this triangle, cosec x = 5/4 and cot x = 3/4, so their sum is 2. Any scaling of the triangle by factor k keeps the ratios unchanged, so the result is independent of k. This confirms that the exact answer 2 is correct and not an approximation. Why Other Options Are Wrong: Option 31/12 and 35/12 are larger than 2 and would arise from incorrect side lengths or algebraic errors when adding fractions. Option 1/2 is far too small and would require both cosec x and cot x to be less than 1, which cannot happen when sin x = 4/5. Option 9/5 equals 1.8 and could appear if a candidate misreads cot x as 1/tan x without computing properly. None of these match the correct, systematically derived value. Common Pitfalls: Some learners confuse cot x with tan x, leading to inverted ratios. Others forget to apply Pythagoras theorem correctly and may mistakenly treat the hypotenuse as the longest side without checking the arithmetic. Errors in fraction addition, such as adding numerators but keeping the same denominator, are also common. Drawing the triangle, labelling sides, and writing each trig ratio explicitly helps avoid these mistakes and reinforces conceptual understanding of right triangle trigonometry. Final Answer: The required value of cosec x + cot x is 2.

Question 15

Evaluate the decimal expression 0.55 × 4.5 ÷ 0.81 by correctly applying multiplication and division operations.

Accepted Answer

Introduction / Context: This numerical aptitude question checks the ability to handle multiplication and division with decimal numbers. Instead of immediately reaching for a calculator, a strong candidate converts decimals to fractions and simplifies the expression systematically. Doing so often leads to a neat rational result, which can be compared directly with fractional options and decimal options. Given Data / Assumptions: • Expression: 0.55 × 4.5 ÷ 0.81. • All numbers are real, and the operations follow the usual left to right rule for multiplication and division. • The final answer should match one of the provided options, which include fractions and decimals. Concept / Approach: A convenient method is to convert each decimal into a fraction with a power of 10 in the denominator. This converts the problem to a fraction multiplication and division task. Multiplication of fractions is straightforward: multiply numerators and denominators separately, then simplify. Division by a fraction is handled as multiplication by its reciprocal. After simplification, we can compare our fraction with the given fraction options or convert it to a decimal as a check. Step-by-Step Solution: Step 1: Express 0.55 as a fraction: 0.55 = 55/100 = 11/20 after simplification. Step 2: Express 4.5 as a fraction: 4.5 = 45/10 = 9/2 after simplification. Step 3: Express 0.81 as a fraction: 0.81 = 81/100 = 81/100. Step 4: Rewrite the expression: 0.55 × 4.5 ÷ 0.81 = (11/20) × (9/2) ÷ (81/100). Step 5: First multiply the fractions: (11/20) × (9/2) = 99/40. Step 6: Now divide by 81/100, which is the same as multiplying by 100/81, so we get (99/40) × (100/81). Step 7: Simplify: 99 and 81 have common factor 9, so 99/81 = 11/9. The product becomes (11/40) × 100/9. Step 8: Simplify further: 100/40 = 5/2, so we have 11 × (5/2) × (1/9) = 55 / 18. Verification / Alternative check: We can roughly check the magnitude by computing approximate decimals. 0.55 × 4.5 ≈ 2.475. Dividing by 0.81 gives about 2.475 ÷ 0.81 ≈ 3.0556. Converting 55/18 to decimal form: 55 ÷ 18 ≈ 3.0556, which perfectly matches. This confirms that the fraction 55/18 is indeed the exact simplified value of the original expression. Why Other Options Are Wrong: Option 3.555 is larger than 3.0556 and would require a larger numerator or smaller denominator in the fraction. Option 3.05 is close but slightly smaller than the true value and likely comes from premature rounding during intermediate steps. Option 55/81 is a completely different fraction whose decimal value is about 0.679, far from the correct magnitude. Option 3.25 equals 13/4 and does not match the exact computation. Only 55/18 matches both the exact fraction and the decimal approximation. Common Pitfalls: A common error is to multiply and divide decimals directly without carefully tracking place values, which leads to misplaced decimal points. Another mistake is treating the division by 0.81 as dividing by 81 instead of 81/100. Also, some students incorrectly cancel digits rather than factors when simplifying fractions. Converting all decimals to fractions and simplifying step by step is a reliable approach that avoids these issues. Final Answer: The correct value of the expression 0.55 × 4.5 ÷ 0.81 is 55/18.

Question 16

Find the value of [12/(√5 + √3)] + [18/(√5 − √3)] after rationalising the denominators and simplifying the surds to simplest radical form.

Accepted Answer

Introduction / Context: This question involves simplifying expressions with surds (square roots) in the denominator. Rationalising denominators is a standard algebraic technique taught in school mathematics, often used in aptitude tests to check comfort with irrational numbers. Here, we must simplify a sum of two rationalised terms and express the final answer in the neatest radical form. Given Data / Assumptions: • Expression: [12/(√5 + √3)] + [18/(√5 − √3)]. • The square roots represent positive real numbers. • We must simplify completely and match the result with one of the structured options. Concept / Approach: The main idea is to multiply numerator and denominator of each fraction by the conjugate of its denominator. For terms of the form a/(√5 + √3), the conjugate is (√5 − √3). Multiplying conjugates produces a difference of squares, eliminating the square roots from the denominator. After rationalising both fractions, we expand the numerators, combine like terms, and then factor if possible to match the answer pattern. Step-by-Step Solution: Step 1: Rationalise the first fraction: 12/(√5 + √3) × (√5 − √3)/(√5 − √3). Step 2: Denominator becomes (√5)^2 − (√3)^2 = 5 − 3 = 2. Numerator becomes 12(√5 − √3). Step 3: So the first term simplifies to [12(√5 − √3)] / 2 = 6(√5 − √3). Step 4: Rationalise the second fraction: 18/(√5 − √3) × (√5 + √3)/(√5 + √3). Step 5: Denominator becomes (√5)^2 − (√3)^2 = 5 − 3 = 2. Numerator becomes 18(√5 + √3). Step 6: So the second term simplifies to [18(√5 + √3)] / 2 = 9(√5 + √3). Step 7: Add the two simplified expressions: 6(√5 − √3) + 9(√5 + √3). Step 8: Expand and combine: 6√5 − 6√3 + 9√5 + 9√3 = (6√5 + 9√5) + (−6√3 + 9√3) = 15√5 + 3√3. Step 9: Factor out 3: 15√5 + 3√3 = 3(5√5 + √3). Verification / Alternative check: We can check correctness by approximating the square roots numerically. Take √5 ≈ 2.236 and √3 ≈ 1.732. Then 12/(√5 + √3) ≈ 12 / (3.968) ≈ 3.025 and 18/(√5 − √3) ≈ 18 / (0.504) ≈ 35.714. Their sum is around 38.739. Evaluate 3(5√5 + √3) with the same approximations: inside the bracket we have 5 × 2.236 + 1.732 ≈ 11.18 + 1.732 = 12.912. Multiply by 3 to get about 38.736, which matches the earlier approximate sum very closely. This confirms that the expression simplifies to 3(5√5 + √3). Why Other Options Are Wrong: Options of the form 15(√5 − √3) and 3(5√5 − √3) involve negative coefficients for √3 and would yield much smaller numerical values. Option 15(√5 + √3) has too large a multiplier and gives a value roughly five times the simplified expression. Option 3(3√5 + √3) does not match the exact combination of coefficients obtained. Only 3(5√5 + √3) fits both the algebraic simplification and numerical verification. Common Pitfalls: Common mistakes include multiplying only numerator or only denominator by the conjugate, forgetting to use the difference of squares formula correctly, or mismanaging signs when expanding (√5 − √3) and (√5 + √3). Another pitfall is adding the two original fractions by finding a common denominator without first rationalising, which makes the task more complicated. Rationalising each term first and then simplifying is a clean and efficient strategy for surd expressions like this. Final Answer: The simplified value of the given surd expression is 3(5√5 + √3).

Question 17

If l + m + n = 9 and l^2 + m^2 + n^2 = 31 for real numbers l, m and n, find the value of the pairwise product sum lm + mn + nl.

Accepted Answer

Introduction / Context: This algebra question tests understanding of symmetric expressions and the identity that links the square of a sum to the sum of squares and pairwise products. Problems like this appear frequently in aptitude tests to check comfort with algebraic manipulation without directly solving for individual variables. The goal is to find lm + mn + nl using the given values of l + m + n and l^2 + m^2 + n^2. Given Data / Assumptions: • l, m and n are real numbers. • l + m + n = 9. • l^2 + m^2 + n^2 = 31. • We need lm + mn + nl. Concept / Approach: The key identity is (l + m + n)^2 = l^2 + m^2 + n^2 + 2(lm + mn + nl). This identity expresses the square of a sum as the sum of squares plus twice the sum of pairwise products. We already know both the left side and the sum of squares on the right side, so we can directly solve for lm + mn + nl. This avoids any need to find the individual values of l, m and n. Step-by-Step Solution: Step 1: Write the identity: (l + m + n)^2 = l^2 + m^2 + n^2 + 2(lm + mn + nl). Step 2: Substitute the given values: (9)^2 = 31 + 2(lm + mn + nl). Step 3: Compute 9^2 = 81. Step 4: Set up the equation: 81 = 31 + 2(lm + mn + nl). Step 5: Subtract 31 from both sides: 81 − 31 = 2(lm + mn + nl). Step 6: Compute 81 − 31 = 50, so 50 = 2(lm + mn + nl). Step 7: Divide by 2: lm + mn + nl = 50 / 2 = 25. Verification / Alternative check: We can check the identity by expanding (l + m + n)^2 directly: (l + m + n)^2 = l^2 + m^2 + n^2 + 2lm + 2mn + 2nl. This matches the identity used. Since the algebraic manipulation is straightforward and depends only on substitution and rearranging, the result lm + mn + nl = 25 is reliable. There is no need to solve for l, m and n individually, which might lead to unnecessary complexity or multiple possible triples, while the symmetric sum remains fixed. Why Other Options Are Wrong: Option 22 and 31 are values that do not satisfy the equation 81 = 31 + 2S, where S represents lm + mn + nl. Option 50 incorrectly treats 50 as the final sum without dividing by 2. Option −25 would require a negative symmetric sum, yet 81 = 31 + 2(−25) would give only 81 = −19, which is clearly false. Therefore, only 25 is consistent with the identity and the given numerical values. Common Pitfalls: Students sometimes forget the factor 2 in the identity and write (l + m + n)^2 = l^2 + m^2 + n^2 + (lm + mn + nl), which leads to the incorrect answer 50. Another common error is substituting incorrectly, for example squaring each term individually without including cross terms. Remembering the full expansion and carefully isolating the required sum helps avoid these mistakes. Practising similar problems reinforces the use of this powerful identity in algebra. Final Answer: The value of the pairwise product sum lm + mn + nl is 25.

Question 18

If 3^10 × 27^2 = 9^2 × 3^n, express both sides as powers of 3 and find the value of the exponent n.

Accepted Answer

Introduction / Context: This problem tests exponent rules and the ability to express different bases as powers of a common base. In this case all numbers involved, 3, 27 and 9, can be written as powers of 3. Once everything is converted to 3 raised to some exponent, comparing exponents on both sides of the equation directly yields the value of n. Questions like this are standard in quantitative aptitude and algebra topics on indices and powers. Given Data / Assumptions: • Equation: 3^10 × 27^2 = 9^2 × 3^n. • All quantities are positive real numbers. • We need to determine n so that the equality holds. Concept / Approach: The main ideas are: write 27 and 9 as powers of 3, then apply the law a^m × a^n = a^(m+n). On the left side, we express 27^2 in terms of 3. On the right side, we express 9^2 in terms of 3. With everything written as 3 raised to some exponent, equality of powers implies equality of exponents because the base is the same and non zero. Solving the resulting simple linear equation gives n. Step-by-Step Solution: Step 1: Note that 27 = 3^3. Therefore 27^2 = (3^3)^2 = 3^(3×2) = 3^6. Step 2: Compute the exponent on the left: 3^10 × 3^6 = 3^(10 + 6) = 3^16. Step 3: Note that 9 = 3^2. Therefore 9^2 = (3^2)^2 = 3^(2×2) = 3^4. Step 4: On the right side, we have 9^2 × 3^n = 3^4 × 3^n = 3^(4 + n). Step 5: Since 3^16 = 3^(4 + n) and the base 3 is the same and positive, the exponents must be equal: 16 = 4 + n. Step 6: Solve for n: n = 16 − 4 = 12. Verification / Alternative check: As a quick check, substitute n = 12 back into the equation. The right side becomes 9^2 × 3^12 = 3^4 × 3^12 = 3^16. The left side we already simplified to 3^16, so both sides match. No other value of n can satisfy the equality because exponents of the same positive base are unique. Therefore n = 12 is confirmed as correct. Why Other Options Are Wrong: If n were 10, the right side exponent would be 4 + 10 = 14, giving 3^14, which is smaller than 3^16. For n = 15, we would get 3^19, which is larger than 3^16. Values 20 and 8 similarly yield exponents 24 and 12, neither equal to 16. Only n = 12 makes the exponents match exactly and thus satisfies the equation. Common Pitfalls: Common mistakes include adding exponents incorrectly, for example writing 3^10 × 3^6 as 3^60, or miswriting 27 as 3^2 instead of 3^3. Another error is failing to convert 9 and 27 into powers of 3 and attempting to compare products numerically, which is time consuming and prone to error. Remembering the key identities and always expressing terms with a common base makes exponent comparison problems straightforward. Final Answer: The value of the exponent n that satisfies the equation is 12.

Question 19

If x + 1/x ≠ 0 and x^3 + 1/x^3 = 0 for a non zero number x, find the value of the fourth power (x + 1/x)^4.

Accepted Answer

Introduction / Context: This question probes algebraic identities involving x + 1/x and x^3 + 1/x^3. Such identities are common in algebra and aptitude exams because they allow high powers to be expressed in terms of simpler expressions. Given a condition on x^3 + 1/x^3, we are asked to find the fourth power of x + 1/x, without solving explicitly for x. Given Data / Assumptions: • x is a non zero number so that 1/x is defined. • x + 1/x ≠ 0. • x^3 + 1/x^3 = 0. • We need to evaluate (x + 1/x)^4. Concept / Approach: We use the identity for the cube of a sum involving reciprocals: (x + 1/x)^3 = x^3 + 3x + 3/x + 1/x^3 = x^3 + 1/x^3 + 3(x + 1/x). This can be written compactly as (x + 1/x)^3 = (x^3 + 1/x^3) + 3(x + 1/x). Since x^3 + 1/x^3 is given as zero, this identity simplifies dramatically. Once we find x + 1/x, we can compute its fourth power using the relation (x + 1/x)^4 = [(x + 1/x)^2]^2. Step-by-Step Solution: Step 1: Use the identity: (x + 1/x)^3 = x^3 + 1/x^3 + 3(x + 1/x). Step 2: Substitute x^3 + 1/x^3 = 0 into the identity to get (x + 1/x)^3 = 0 + 3(x + 1/x) = 3(x + 1/x). Step 3: Let t = x + 1/x. Then the equation becomes t^3 = 3t. Step 4: Rearrange: t^3 − 3t = 0, so t(t^2 − 3) = 0. Step 5: Since x + 1/x ≠ 0, t ≠ 0. Therefore t^2 − 3 = 0 and t^2 = 3. Step 6: We need (x + 1/x)^4, which is t^4. With t^2 = 3, we have t^4 = (t^2)^2 = 3^2 = 9. Verification / Alternative check: We can check consistency by working backwards. If t^2 = 3, then t = √3 or t = −√3. In each case t^3 = t × t^2 = t × 3 = 3t, so t^3 − 3t = 0, which matches the identity derived from x^3 + 1/x^3 = 0. Therefore the value t^2 = 3 and t^4 = 9 is consistent with the condition. The exact sign of t is not needed because the fourth power t^4 is the same for both √3 and −√3. Why Other Options Are Wrong: Option 12 and 15 would require t^2 to be non integer or would not follow from t^2 = 3. Option 16 corresponds to t^2 = 4, which would make t^3 − 3t equal to 16t − 3t = 13t, not zero unless t = 0, contradicting x + 1/x ≠ 0. Option 3 is equal to t^2 and not t^4. Only 9 matches the correct computation of the fourth power based on the given condition. Common Pitfalls: A frequent mistake is to try to solve for x directly from x^3 + 1/x^3 = 0, which is unnecessarily complex and may lead to confusion with complex numbers. Another pitfall is to misapply the identity for (x + 1/x)^3, forgetting the middle term 3(x + 1/x). Some candidates also plug in numerical guesses for x instead of using algebra. Focusing on the symmetric form t = x + 1/x and applying identities in terms of t is a much cleaner method. Final Answer: The value of the fourth power (x + 1/x)^4 is 9.

Question 20

If the ratio (x + 1/x) : (x − 1/x) is 5 : 3 for a real number x ≠ 0, find the possible values of x.

Accepted Answer

Introduction / Context: This problem tests algebraic manipulation involving expressions of the form x + 1/x and x − 1/x and a given ratio between them. Such questions are classic in algebra practice and aptitude tests. By converting the ratio relation into an equation and simplifying, we can derive a quadratic equation whose roots are the possible values of x. Given Data / Assumptions: • (x + 1/x) : (x − 1/x) = 5 : 3. • x is a real number with x ≠ 0 (so that 1/x is defined). • We must determine the real values of x consistent with the ratio. Concept / Approach: A ratio a : b = c : d implies a/b = c/d when all quantities are defined and b, d are non zero. Here we can write (x + 1/x) / (x − 1/x) = 5/3. Cross multiplication will give a linear equation in x and 1/x. Multiplying through by x to clear denominators turns this into a polynomial equation in x. Rearranging produces a quadratic equation which can be solved using standard techniques. The roots of this quadratic give the required values of x, provided they are real and non zero. Step-by-Step Solution: Step 1: Start from the ratio equation: (x + 1/x) / (x − 1/x) = 5 / 3. Step 2: Cross multiply: 3(x + 1/x) = 5(x − 1/x). Step 3: Expand both sides: 3x + 3/x = 5x − 5/x. Step 4: Bring like terms together: move terms with x to one side and terms with 1/x to the other side. Step 5: Subtract 3x from both sides: 3/x = 2x − 5/x. Step 6: Add 5/x to both sides: 3/x + 5/x = 2x, giving 8/x = 2x. Step 7: Multiply both sides by x (x ≠ 0): 8 = 2x^2. Step 8: Divide both sides by 2: x^2 = 4. Step 9: Taking square roots gives x = ±2. Verification / Alternative check: Check x = 2. Then x + 1/x = 2 + 1/2 = 2.5 and x − 1/x = 2 − 1/2 = 1.5. The ratio is 2.5 : 1.5 which simplifies to 5 : 3 after multiplying numerator and denominator by 2. For x = −2, we have x + 1/x = −2 − 1/2 = −2.5 and x − 1/x = −2 + 1/2 = −1.5. The ratio (−2.5) : (−1.5) again simplifies to 5 : 3 since the negatives cancel. Hence both x = 2 and x = −2 satisfy the given ratio. Why Other Options Are Wrong: Option +1 or −1 would give x − 1/x equal to zero, making the ratio undefined. Option +3 or −3 and option +4 or −4 do not satisfy the equation 8 = 2x^2 and produce different ratios. The option stating no real value is incorrect because we have found two real solutions. Only x = +2 or x = −2 produce the correct 5 : 3 ratio between x + 1/x and x − 1/x. Common Pitfalls: A common error is to forget that division by x − 1/x requires this quantity to be non zero, but this is automatically ensured at the correct solutions. Some students incorrectly cross multiply or mishandle the terms with 1/x, leading to a wrong equation. Others may forget to consider both positive and negative square roots when solving x^2 = 4. Writing each step carefully and explicitly resolving signs prevents these mistakes. Final Answer: The possible real values of x that satisfy the given ratio are +2 or −2.

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