Question 1

Simplify the algebraic expression 1/[(p − n)(n − q)] + 1/[(n − q)(q − p)] + 1/[(q − p)(p − n)] and find its value in terms of p, q and n.

Accepted Answer

Introduction / Context: This question tests simplification of a symmetric rational expression involving three variables p, q and n. Such expressions often appear complicated but simplify nicely when denominators are examined carefully. The structure shows products of differences of the variables, and the sum has a cyclic pattern, which suggests that many terms may cancel out when a common denominator is used. Given Data / Assumptions: • Expression: 1/[(p − n)(n − q)] + 1/[(n − q)(q − p)] + 1/[(q − p)(p − n)]. • p, q and n are distinct real numbers so that none of the denominators is zero. • We must simplify the sum and express it as a single number or simple expression. Concept / Approach: The three terms share related factors in the denominators. The natural approach is to bring them over a common denominator, which is the product (p − n)(n − q)(q − p), and then combine numerators. Because of symmetry and alternating signs, there is a strong chance that the combined numerator simplifies to zero. Carefully accounting for signs of (q − p) versus (p − q) and similar pairs is crucial to avoid errors. Step-by-Step Solution: Step 1: Identify the common denominator as D = (p − n)(n − q)(q − p). Step 2: Rewrite the first term with denominator D. Multiply numerator and denominator by (q − p), giving (q − p) / D. Step 3: Rewrite the second term. Its denominator is (n − q)(q − p), so multiply numerator and denominator by (p − n), giving (p − n) / D. Step 4: Rewrite the third term. Its denominator is (q − p)(p − n), so multiply numerator and denominator by (n − q), giving (n − q) / D. Step 5: Combine the three numerators: (q − p) + (p − n) + (n − q). Step 6: Simplify the combined numerator: q − p + p − n + n − q = 0. Step 7: Therefore the entire expression becomes 0 / D = 0 (provided D ≠ 0). Verification / Alternative check: As a numerical check, choose specific values such as p = 1, q = 2 and n = 3. Then compute each term. We have (p − n)(n − q) = (1 − 3)(3 − 2) = (−2)(1) = −2, so the first term is −1/2. Next, (n − q)(q − p) = (3 − 2)(2 − 1) = (1)(1) = 1, so the second term is 1. Finally, (q − p)(p − n) = (2 − 1)(1 − 3) = (1)(−2) = −2, so the third term is −1/2. The sum is −1/2 + 1 − 1/2 = 0. This matches the algebraic result and confirms the simplification. Why Other Options Are Wrong: Option 1 would require the combined numerator to equal the common denominator, which does not happen. Options p + q + n and pq + qn + np would be non zero in general, contradicting both the algebraic simplification and numerical checks. The expression 2np + q is unrelated to the structure of the original sum. Only the value 0 is consistent for all allowed values of p, q and n that keep denominators non zero. Common Pitfalls: The most common errors are sign mistakes when handling factors such as (q − p) and (p − q), which are negatives of each other. Some learners also forget to multiply the numerators properly while forming the common denominator. Another pitfall is trying to plug in arbitrary values without considering whether denominators become zero. Following a systematic common denominator approach and simplifying the numerator step by step avoids these issues. Final Answer: The simplified value of the expression is 0.

Question 2

The angle of elevation of the top of a vertical tower from two points on opposite sides of its base, at distances 25 m and 64 m from the foot, are x and 90° − x respectively. Find the height of the tower in metres.

Accepted Answer

Introduction / Context: This is a standard height and distance problem in trigonometry. The height of a vertical tower is to be found using angles of elevation measured from two points on opposite sides of the tower. The angles of elevation are complementary, x and 90° − x, and the distances from the base are known. Such problems test understanding of tangent and cotangent relationships in right triangles. Given Data / Assumptions: • The tower is vertical and its height is h metres. • One point is 25 m from the base of the tower and the angle of elevation there is x. • Another point is 64 m from the base on the opposite side and the angle of elevation there is 90° − x. • Ground between the points and the tower base is assumed to be horizontal. Concept / Approach: For a right triangle with opposite side h and adjacent side d, tan of the angle at the ground is h/d. From the first point, tan x = h/25. From the second point, where the angle is 90° − x, we have tan(90° − x) = h/64. Using the identity tan(90° − x) = cot x = 1 / tan x, we obtain two equations involving h and tan x. Equating the expressions gives a simple relation that allows us to solve for h by eliminating the trigonometric function entirely. Step-by-Step Solution: Step 1: From the first point, tan x = h / 25. Step 2: From the second point on the opposite side, tan(90° − x) = h / 64. Step 3: Use the identity tan(90° − x) = cot x = 1 / tan x. Step 4: Therefore h / 64 = 1 / tan x. Step 5: But from Step 1, tan x = h / 25, so 1 / tan x = 25 / h. Step 6: Equate the expressions for tan(90° − x): h / 64 = 25 / h. Step 7: Cross multiply: h^2 = 64 × 25. Step 8: Compute 64 × 25 = 1600, so h^2 = 1600. Step 9: Hence h = √1600 = 40 (taking the positive root since height is positive). Verification / Alternative check: We can check consistency by computing tan x and cot x. From h = 40, we have tan x = h/25 = 40/25 = 8/5. Then cot x = 1 / tan x = 5/8. At the second point, h/64 = 40/64 = 5/8, which matches cot x, confirming that tan(90° − x) indeed equals h/64. This confirms that h = 40 m satisfies both angle conditions and distances given in the problem. Why Other Options Are Wrong: Option 39 m and 89 m are close in magnitude but do not satisfy h^2 = 64 × 25 and would produce inconsistent values for tan x at the two points. Option 1.6 m is far too small for the given distances and angles. Option 64 m would give tan x = 64/25 and tan(90° − x) = 64/64 = 1, so tan x and cot x would not be reciprocals. Only h = 40 m makes the geometry and trigonometric identities work together correctly. Common Pitfalls: A common mistake is to forget that the angles are complementary and to treat tan(90° − x) as tan x instead of cot x. Another pitfall is to misplace the distances and write tan x = 25/h rather than h/25. Some candidates attempt to assign a numerical value to x unnecessarily, even though it cancels out. Focusing directly on tan x and cot x and eliminating the angle leads quickly to the required height. Final Answer: The height of the tower is 40 m.

Question 3

Simplify and evaluate the expression 213 × 213 + 187 × 187 using suitable algebraic identities or direct calculation.

Accepted Answer

Introduction / Context: This question checks basic arithmetic with large numbers and recognition of patterns involving squares. The expression 213 × 213 + 187 × 187 is the same as 213^2 + 187^2. While it can be computed directly, one may also look for algebraic shortcuts or numeric patterns. However, even without a special identity, systematic multiplication and addition can quickly yield the exact value. Given Data / Assumptions: • Expression: 213 × 213 + 187 × 187. • All numbers are integers and we seek an exact integer result. • No approximations or calculators are required; accurate manual computation or structured reasoning is expected. Concept / Approach: We can evaluate each square separately and then add the results. For 213^2, write it as (200 + 13)^2 or perform standard column multiplication. The same applies for 187^2. Alternatively, we can use the identity (a + b)^2 = a^2 + 2ab + b^2 to reduce mental workload. After finding both squares, we add them to get the final value and then match it against the options. Step-by-Step Solution: Step 1: Compute 213^2. Write 213 as 200 + 13. Step 2: Using the identity (a + b)^2 = a^2 + 2ab + b^2, we get 213^2 = 200^2 + 2 × 200 × 13 + 13^2. Step 3: Calculate these terms: 200^2 = 40000, 2 × 200 × 13 = 5200, and 13^2 = 169. Step 4: Add them: 40000 + 5200 + 169 = 45369. Step 5: Compute 187^2. Write 187 as 200 − 13. Step 6: Apply (a − b)^2 = a^2 − 2ab + b^2: 187^2 = 200^2 − 2 × 200 × 13 + 13^2. Step 7: Calculate: 200^2 = 40000, −2 × 200 × 13 = −5200, and 13^2 = 169. Step 8: Add them: 40000 − 5200 + 169 = 350? Wait carefully: 40000 − 5200 = 34800, plus 169 = 34969. Step 9: Now add the two squares: 213^2 + 187^2 = 45369 + 34969 = 80338. Verification / Alternative check: We can check the addition: 45369 + 34969. Add units: 9 + 9 = 18, write 8 carry 1. Tens: 6 + 6 + 1 = 13, write 3 carry 1. Hundreds: 3 + 9 + 1 = 13, write 3 carry 1. Thousands: 5 + 4 + 1 = 10, write 0 carry 1. Ten thousands: 4 + 3 + 1 = 8. The result is 80338, which matches the earlier sum. This confirms that the calculation is accurate. Why Other Options Are Wrong: The options 80438 and 81338 are close but differ by 100 or 1000, errors that could come from misplacing carries during addition. The options 60338 and 60438 are much smaller and would require significantly smaller squares, which is inconsistent with 213^2 and 187^2 both being near 40000. Only 80338 matches the exact computed value of the expression. Common Pitfalls: A typical mistake is mismanaging carries during multi digit addition or subtraction, especially when mental calculation is attempted too quickly. Some learners may attempt to compute the squares by repeated multiplication without using identities, increasing the chance of error. Using (200 ± 13)^2 reduces mental load and makes each step explicit. Double checking intermediate sums is a good habit for avoiding small arithmetic slips in exam conditions. Final Answer: The value of 213 × 213 + 187 × 187 is 80338.

Question 4

If the inequalities 2x + 2(4 + 3x) < 2 + 3x and 2 + 3x > 2x + x/2 both hold simultaneously, determine which given value of x satisfies both conditions.

Accepted Answer

Introduction / Context: This question involves solving two linear inequalities in one variable and then finding a value of x from the given options that satisfies both simultaneously. Inequalities are a core algebra topic in aptitude tests, and combining several inequality conditions is a common way to check deeper understanding. Careful step by step manipulation is needed to avoid sign errors. Given Data / Assumptions: • Inequality 1: 2x + 2(4 + 3x) < 2 + 3x. • Inequality 2: 2 + 3x > 2x + x/2. • x is a real number, and we must select the correct value from the options −3, 1, 0, −1 and −5. Concept / Approach: We simplify each inequality separately to obtain an interval of allowed x values. Inequality 1 will produce one condition (such as x less than some number), and inequality 2 will produce another condition (for example x greater than some number). The values of x that satisfy both conditions must lie in the intersection of the two intervals. Finally, we test each option to see which falls in this intersection. Step-by-Step Solution: Step 1: Simplify Inequality 1: 2x + 2(4 + 3x) < 2 + 3x. Step 2: Expand: 2x + 8 + 6x < 2 + 3x, so 8x + 8 < 2 + 3x. Step 3: Subtract 3x from both sides: 5x + 8 < 2. Step 4: Subtract 8: 5x < −6, hence x < −6/5 = −1.2. Step 5: Simplify Inequality 2: 2 + 3x > 2x + x/2. Step 6: Combine terms on the right: 2x + x/2 = (4x/2 + x/2) = 5x/2. Step 7: So 2 + 3x > 5x/2. Multiply every term by 2 to clear the denominator: 4 + 6x > 5x. Step 8: Subtract 5x: 4 + x > 0, so x > −4. Step 9: Combine both conditions: from Inequality 1, x < −1.2 and from Inequality 2, x > −4. Therefore −4 < x < −1.2. Verification / Alternative check: We now test the options. x = −3 lies between −4 and −1.2, so it is in the allowed interval. Plugging x = −3 into Inequality 1 gives 2(−3) + 2(4 + 3(−3)) = −6 + 2(4 − 9) = −6 + 2(−5) = −16. The right side is 2 + 3(−3) = 2 − 9 = −7. Since −16 < −7, Inequality 1 holds. For Inequality 2, 2 + 3(−3) = −7 and 2(−3) + (−3)/2 = −6 − 1.5 = −7.5, and −7 > −7.5, so Inequality 2 also holds. None of the other options fall strictly between −4 and −1.2, so they cannot satisfy both inequalities together. Why Other Options Are Wrong: x = 1 and x = 0 are greater than −1.2, so they fail Inequality 1 which requires x < −1.2. x = −1 is greater than −1.2 and also fails Inequality 1. x = −5 is less than −4 and fails Inequality 2, which requires x > −4. Only x = −3 lies in the intersection of both solution intervals and satisfies both inequalities when substituted explicitly. Common Pitfalls: Students sometimes make sign errors when moving terms across the inequality or forget to reverse the inequality sign when multiplying by a negative number. In this problem no sign reversal is needed, but careful handling of each arithmetic step is still essential. Another pitfall is to stop after solving only one inequality without checking the second, which can lead to incorrect conclusions about which options are valid. Final Answer: The value of x that satisfies both inequalities is −3.

Question 5

Evaluate 52000 ÷ 40 ÷ 65 × 30 step by step and hence find the value of ? in the equation 52000 ÷ 40 ÷ 65 × 30 = ? − √400.

Accepted Answer

Introduction / Context: This question combines multi step arithmetic with a simple equation involving a square root. The left side involves division and multiplication of whole numbers, and the result is related to the unknown symbol ? through subtraction by √400. Many aptitude problems follow this pattern, where a straightforward numerical calculation is linked to a small algebraic step. Given Data / Assumptions: • Expression: 52000 ÷ 40 ÷ 65 × 30. • Equation: 52000 ÷ 40 ÷ 65 × 30 = ? − √400. • All operations are on real numbers and follow left to right rules for division and multiplication. Concept / Approach: First we simplify the left side expression in the natural order: divide 52000 by 40, then by 65, and finally multiply by 30. This gives a single numerical value. Then we recognise that √400 is 20, and substitute this into the equation. Solving for ? involves adding 20 to the calculated left side value. The resulting number is then matched to the options provided. Step-by-Step Solution: Step 1: Compute 52000 ÷ 40. Since 52000/40 = 5200/4 = 1300, the result is 1300. Step 2: Divide this result by 65: 1300 ÷ 65. Step 3: Note that 65 × 20 = 1300, so 1300 ÷ 65 = 20. Step 4: Multiply by 30: 20 × 30 = 600. Therefore 52000 ÷ 40 ÷ 65 × 30 = 600. Step 5: Compute the square root term: √400 = 20, since 20^2 = 400. Step 6: Substitute into the equation: 600 = ? − 20. Step 7: Solve for ?: add 20 to both sides to obtain ? = 600 + 20 = 620. Verification / Alternative check: We can verify by substituting ? = 620 back into the original equation. The right side becomes 620 − √400 = 620 − 20 = 600. This matches exactly the computed left side value. The arithmetic operations were simple multiples of 10 and common factors, so there is little room for error when carefully executed. This confirms that ? must equal 620. Why Other Options Are Wrong: Option 580 would give 580 − 20 = 560, which does not equal 600. Option 720 would give 720 − 20 = 700, again incorrect. Option 780 results in 780 − 20 = 760, and option 600 gives 600 − 20 = 580. None of these match the left side value 600. Only 620 produces the correct equality between both sides of the equation. Common Pitfalls: Common mistakes include performing operations in the wrong order, such as multiplying by 30 before completing both divisions, or miscomputing √400 as 400/2 instead of 20. Another error is forgetting to add the square root term back when solving for ?. Following left to right evaluation for division and multiplication, and treating the equation transformation step separately, helps avoid these issues. Final Answer: The value of the unknown symbol in the equation is 620.

Question 6

Compute the product 12.5 × 3.2 × 8.8 by converting decimals if needed and select the correct result.

Accepted Answer

Introduction / Context: This question checks fluency with multiplication of decimal numbers. Products like 12.5 × 3.2 × 8.8 often appear in quantitative tests to see whether candidates can handle place values and decimals confidently. A common strategy is to convert decimals into fractions with denominators that are powers of 10 and then multiply and simplify. Given Data / Assumptions: • Expression: 12.5 × 3.2 × 8.8. • All numbers are terminating decimals, so they can be expressed as simple fractions. • We seek an exact integer result that matches one of the options. Concept / Approach: We write each decimal as a fraction: 12.5 as 125/10, 3.2 as 32/10 and 8.8 as 88/10. Then we multiply these fractions, combine numerators and denominators, and simplify step by step. Another method is to group factors cleverly to make round numbers like 10 or 100, reducing the chance of errors. Both methods should lead to the same final integer value. Step-by-Step Solution: Step 1: Convert decimals to fractions: 12.5 = 125/10, 3.2 = 32/10 and 8.8 = 88/10. Step 2: Multiply: 12.5 × 3.2 × 8.8 = (125/10) × (32/10) × (88/10). Step 3: Combine numerators and denominators: (125 × 32 × 88) / (10 × 10 × 10) = (125 × 32 × 88) / 1000. Step 4: Multiply stepwise. First 125 × 32 = 4000. Step 5: Next multiply 4000 × 88 = 352000. Step 6: Now divide by 1000: 352000 / 1000 = 352. Verification / Alternative check: We can estimate the magnitude by approximating each factor. 12.5 is about 12, 3.2 is about 3, and 8.8 is about 9. The rough product 12 × 3 × 9 = 324, which is in the same ballpark as 352. A more careful mental grouping is also possible: 12.5 × 8.8 = (25/2) × (88/10) = (25 × 88) / 20. Since 25 × 88 = 2200, we have 2200 / 20 = 110. Then 110 × 3.2 = 110 × (32/10) = (110 × 32) / 10 = 3520 / 10 = 352. Both methods confirm the same result. Why Other Options Are Wrong: Values like 358, 355, 354 and 356 are close to 352 but do not match the exact product. These options represent typical rounding or small arithmetic errors. Since the arithmetic above is exact and uses integer operations after converting decimals to fractions, there is no ambiguity. Thus only 352 correctly represents the product 12.5 × 3.2 × 8.8. Common Pitfalls: Common errors include misplacing the decimal point after multiplication, for example counting the total number of decimal places incorrectly. Another mistake is partially rounding one factor before multiplying, which may shift the result away from the exact value. Converting to fractions or strategically grouping factors to form tens and hundreds reduces reliance on decimal place counting and improves accuracy. Final Answer: The exact value of 12.5 × 3.2 × 8.8 is 352.

Question 7

Solve for the unknown ? in the percentage equation 41% of 600 − 250 = ? − 77% of 900.

Accepted Answer

Introduction / Context: This question checks understanding of percentage calculations and simple linear equations. The expression involves computing percentages of given numbers, then forming and solving an equation to find an unknown value represented by ?. Problems like this are common in arithmetic sections of aptitude exams and in everyday financial reasoning. Given Data / Assumptions: • Equation: 41% of 600 − 250 = ? − 77% of 900. • All percentages are simple fractions of the given base numbers. • We must find the value of ? that balances the equation. Concept / Approach: First we compute 41% of 600 and 77% of 900 by converting percentages to decimals or fractions. After simplifying the left hand side, we express the right hand side in terms of ? and the computed percentage. Equating both sides and solving for ? yields a simple addition. Finally, we choose the matching option from the list provided. Step-by-Step Solution: Step 1: Compute 41% of 600. Since 41% = 41/100, we have (41/100) × 600 = 41 × 6 = 246. Step 2: Evaluate the left side: 41% of 600 − 250 = 246 − 250 = −4. Step 3: Compute 77% of 900. Since 77% = 77/100, we have (77/100) × 900 = 77 × 9 = 693. Step 4: Substitute into the right side: ? − 77% of 900 = ? − 693. Step 5: Set the equation: −4 = ? − 693. Step 6: Add 693 to both sides: −4 + 693 = ?, so ? = 689. Verification / Alternative check: Verify by substituting ? = 689 back into the original equation. The right side becomes 689 − 77% of 900 = 689 − 693 = −4. The left side was already found to be −4. Since both sides are equal, the value ? = 689 satisfies the equation exactly. No other value will balance the equation because this is a simple one variable linear equation. Why Other Options Are Wrong: If ? were 589, the right side would be 589 − 693 = −104, not −4. For ? = 789, the right side becomes 789 − 693 = 96. For ? = 889, we get 889 − 693 = 196. For ? = 609, we get 609 − 693 = −84. None of these match the left side value −4. Only 689 produces equality between left and right sides. Common Pitfalls: Common mistakes include miscalculating the percentages, for example computing 41% of 600 as 2460, or mixing up subtraction signs when forming the equation. Some students also move terms across the equals sign without changing their sign correctly. Writing each step clearly and checking intermediate values helps to avoid these errors, especially when time pressure is high in exam conditions. Final Answer: The value of the unknown ? that satisfies the equation is 689.

Question 8

Evaluate 65% × 700 + √196 − 9 × 3 by carefully applying percentage, square root and multiplication operations.

Accepted Answer

Introduction / Context: This arithmetic problem combines percentage calculation, a square root, and integer multiplication. The aim is to evaluate the expression step by step: 65% × 700 + √196 − 9 × 3. Such mixed operations are common in aptitude tests and require careful attention to order of operations and basic numerical skills. Given Data / Assumptions: • Expression: 65% × 700 + √196 − 9 × 3. • Percentages are interpreted as parts per hundred. • √196 denotes the positive square root of 196. • Operations follow the usual order: roots and multiplications first, then additions and subtractions. Concept / Approach: We first convert 65% into a fraction or decimal and multiply by 700 to find that term. Next, we evaluate the square root √196. We also compute 9 × 3 as a simple product. Finally, we perform the addition and subtraction in the correct order. Doing each step separately with clear intermediate results helps avoid sign mistakes and misplacement of decimal points. Step-by-Step Solution: Step 1: Compute 65% of 700. Since 65% = 65/100, we have (65/100) × 700 = 0.65 × 700. Step 2: Multiply 0.65 × 700 = 65 × 7 = 455. So 65% × 700 = 455. Step 3: Evaluate the square root: √196 = 14, because 14^2 = 196. Step 4: Compute the product 9 × 3 = 27. Step 5: Substitute these values into the expression: 65% × 700 + √196 − 9 × 3 = 455 + 14 − 27. Step 6: Add 455 and 14: 455 + 14 = 469. Step 7: Subtract 27: 469 − 27 = 442. Verification / Alternative check: We can check by re grouping. First compute 455 − 27 = 428, then add the root term 14 to get 428 + 14 = 442. This is consistent with the earlier calculation. Another mental check is to approximate: 65% of 700 is a little less than 70% of 700, which would be 490, so a value in the mid 400s after subtracting and adding small integers is reasonable. The exact arithmetic confirms 442 without ambiguity. Why Other Options Are Wrong: Option 522 would require 455 + 14 − 27 to be 522, which would mean adding instead of subtracting 27. Options 402, 362 and 472 result from different miscalculations, such as subtracting both 14 and 27 from 455, or miscomputing the percentage. Since the step by step arithmetic clearly leads to 442, these alternatives are incorrect. Only 442 matches the correctly evaluated expression. Common Pitfalls: A frequent error is confusing 65% with 0.65 and misplacing the decimal when multiplying by 700, sometimes giving 45.5 or 4550 instead of 455. Another mistake is miscomputing √196 as 196/2 or 98. Finally, some candidates may forget the order of operations and perform the subtraction before the multiplication, changing the result. Handling each operation in the correct sequence avoids these mistakes and leads directly to the correct answer. Final Answer: The value of 65% × 700 + √196 − 9 × 3 is 442.

Question 9

In aptitude (simplification: quadratic comparison), two quadratic equations in real variables x and y are given. First solve each equation to find all possible values of x and y, and then compare these values to decide the correct relationship betwe…

Accepted Answer

Introduction / Context: This question tests comparison of roots of two quadratic equations in aptitude level simplification. Instead of directly guessing the relationship between x and y, we must first find all possible roots of each quadratic, then carefully compare every possible pair of x and y values. Only after that analysis can we decide whether x is always greater than y, always less, always equal, or if no single consistent relationship exists. Given Data / Assumptions: We have two equations in real variables x and y. I. 2x^2 - 13x - 189 = 0 II. 2y^2 - 3y - 189 = 0 Assume x and y are real numbers and any root of the equations is a valid value. Concept / Approach: For a quadratic ax^2 + bx + c = 0, roots are obtained by factorisation or by the quadratic formula. Once we compute the two roots of the x equation and the two roots of the y equation, we form all root pairs (x, y) and compare them. If we can find even one pair where x > y and another pair where x < y, then no unique relationship can be fixed and the correct conclusion is that the relationship cannot be determined in a strict sense. Step-by-Step Solution: For equation I: 2x^2 - 13x - 189 = 0. Factor 2x^2 - 13x - 189 as (2x + 21)(x - 9) = 0. Hence x = -21/2 or x = 9. For equation II: 2y^2 - 3y - 189 = 0. Factor 2y^2 - 3y - 189 as (2y + 27)(y - 7) = 0. Hence y = -27/2 or y = 7. Now compare pairs: for example x = -21/2 and y = -27/2 gives x > y, but x = -21/2 and y = 7 gives x < y. Because different choices of roots lead to different inequalities, no single global relation between x and y can be fixed. Verification / Alternative check: We can quickly approximate the roots as decimals to cross check. For x: -21/2 is -10.5 and 9 is 9. For y: -27/2 is -13.5 and 7 is 7. By pairing 9 with -13.5, x is greater. By pairing -10.5 with 7, x is smaller. This confirms that both x > y and x < y situations are possible, so there is no unique comparison that holds for all combinations of x and y that satisfy the equations. Why Other Options Are Wrong: Options x > y and x < y each claim a strict direction for all pairs, which is not true because we found counterexamples. The option x = y would require every root pair to be equal, which is clearly false because the sets of roots are different. The option x >= y also fails because we have at least one pair where x < y. Therefore none of these directional statements holds for every valid combination of x and y. Common Pitfalls: A common mistake in such comparison questions is to compare only one root pair, for example taking the larger root of x with the smaller root of y and concluding x > y. Another pitfall is to ignore the fact that quadratics generally have two roots, not one, so conclusions must be based on all combinations. Students sometimes also approximate too roughly and miscompare numbers, so it is safer to work with exact fractions or clearly written decimals. Final Answer: The only safe conclusion is that no single consistent inequality describes all possible pairs. Hence the correct option is Relationship cannot be determined.

Question 10

In aptitude (simplification: quadratic comparison), two quadratic equations in x and y are given. Solve both equations to obtain all real roots, then compare every possible pair of x and y values and choose the option that best describes the relatio…

Accepted Answer

Introduction / Context: This question again involves comparison of roots of two quadratic equations, which is a common pattern in banking and management aptitude exams. Instead of computing a single numeric answer, we must determine an inequality relation between x and y that holds for every possible root of the two equations. Such questions test understanding of quadratic roots, multiplicity, and careful logical comparison rather than simple arithmetic alone. Given Data / Assumptions: Equation I: x^2 + 2x - 195 = 0. Equation II: y^2 + 30y + 225 = 0. Both equations are standard quadratics in real variables x and y. All their real roots are considered possible values. Concept / Approach: The plan is to factorise each quadratic to obtain its roots. If equation II factors into a perfect square, then y will take only one repeated value. Once all roots are known, we compare every possible x value with that single y value. If all comparisons give x greater than or equal to y, the correct relation is x >= y. We must be careful to include equality cases as well as strict inequality cases while choosing the option. Step-by-Step Solution: For equation I: x^2 + 2x - 195 = 0. Factorise: x^2 + 2x - 195 = (x + 15)(x - 13) = 0. So x = -15 or x = 13. For equation II: y^2 + 30y + 225 = 0. Recognise a perfect square: y^2 + 30y + 225 = (y + 15)^2 = 0. So y = -15 (a repeated root). Now compare: when x = -15, y = -15 gives x = y. When x = 13, y = -15 gives x > y. Thus for all possible roots, x is always greater than or equal to y, and never less than y. Verification / Alternative check: We can verify using the sum and product of roots. For equation II, sum of roots is -30 and product is 225. This is consistent with a double root at -15 since -15 plus -15 is -30 and the product is 225. For equation I, the roots -15 and 13 clearly satisfy the factorisation found. Comparing numerically, the smallest x value equals y and the larger x is greater than y, so the relationship x >= y is firmly established. Why Other Options Are Wrong: Option x > y ignores the equality case when both x and y are -15. Option x < y is impossible because y is never greater than any x root. Option x = y fails because x can also be 13, which is strictly greater than y. The option relationship cannot be determined would be correct only if different combinations gave conflicting orderings, which does not happen here. Common Pitfalls: Many learners forget to consider both roots of the first quadratic and only compare a single pair. Others may misfactor the quadratics or overlook that equation II has a repeated root. Some students also confuse x <= y with x >= y when selecting options, so it is important to read the inequality direction carefully. Writing down all root values and comparing them systematically helps avoid such errors. Final Answer: For every possible pair, x is never less than y and is sometimes equal and sometimes greater. Therefore the correct relationship is x >= y.

Question 11

In this aptitude (simplification) question, two quadratic equations in real variables x and y are given. You must solve both equations completely and then compare all possible values of x and y to decide the correct overall relationship. I. 2x^2 + 1…

Accepted Answer

Introduction / Context: This is another quadratic comparison question that appears frequently in quantitative aptitude. The candidate must find all real roots for each quadratic equation and then analyse how these sets of roots compare. Such questions are designed to test conceptual understanding of quadratic equations, factorisation skills, and logical reasoning about inequalities rather than pure calculation speed alone. Given Data / Assumptions: Equation I: 2x^2 + 19x + 42 = 0. Equation II: 4y^2 + 43y + 30 = 0. All real roots of each equation are valid values that x and y can take respectively. Concept / Approach: We factorise each quadratic to obtain its two real roots. Then we list the set of x values and the set of y values. To determine a global comparison such as x > y, x < y, x = y, or x >= y, the relationship must hold for every possible pairing of roots. If we can find at least two pairs with conflicting order (for example one pair where x > y and another where x < y), then the relationship cannot be determined uniquely. Step-by-Step Solution: Factor equation I: 2x^2 + 19x + 42 = 0. We can factor this as (2x + 7)(x + 6) = 0. Thus x = -7/2 or x = -6. Now factor equation II: 4y^2 + 43y + 30 = 0. Factorisation gives (4y + 5)(y + 6) = 0. Thus y = -5/4 or y = -6. Now compare different combinations. When x = -6 and y = -6 we get x = y. When x = -7/2 (that is -3.5) and y = -5/4 (that is -1.25) we have x < y. Hence we have at least equality and x < y. By taking x = -7/2 and y = -6 we get x > y. So all three types of relations occur depending on the chosen roots. Verification / Alternative check: To verify, convert all roots to decimals. For x we have approximately -3.5 and -6. For y we have approximately -1.25 and -6. Now form pairs: (-6, -6) gives equality. (-3.5, -1.25) gives x < y. (-3.5, -6) gives x > y. Because the relative order changes across different valid root pairs, no single inequality or equality can summarise the relationship between x and y across all possibilities. Why Other Options Are Wrong: Options x > y, x < y, and x = y all demand a single strict relation for every pair, which is not true. Option x >= y also fails because we have a pair where x < y. Therefore none of these fixed relationships is valid for all combinations. The only correct conclusion is that the relationship cannot be determined in a unique way. Common Pitfalls: A frequent error is to choose only one convenient pair of roots and draw a global conclusion, ignoring other possible combinations. Another mistake is incorrect factorisation, which leads to wrong roots and then a wrong comparison. Learners should also be cautious when working with negative numbers, as sign confusion can easily invert the inequality. Always list all roots clearly before comparing them. Final Answer: Because x can be less than, equal to, or greater than y depending on the chosen roots, the correct option is Relationship cannot be determined.

Question 12

In this aptitude (simplification: quadratic comparison) question, two quadratic equations in x and y are given. Solve both equations, list all possible values of x and y, and then compare them to decide the correct overall relationship between x and…

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Introduction / Context: Here we again compare the roots of two quadratic equations, but in this case a consistent inequality does exist. These types of questions appear in quantitative aptitude tests to assess algebraic manipulation, root comparison, and careful reasoning about all possible values of variables, not just one solution pair. Given Data / Assumptions: Equation I: 3x^2 - 14x + 16 = 0. Equation II: 5y^2 - 16y + 12 = 0. All real roots of each equation are considered valid values of x and y respectively. Concept / Approach: We factorise each quadratic to find explicit roots. Then we compare every x root with every y root. If the smallest x value is still greater than or equal to the largest y value, then x >= y. If instead there are mixed comparisons, the relationship would not be determined. Thus the key is to compute roots accurately and inspect their order on the number line. Step-by-Step Solution: Consider 3x^2 - 14x + 16 = 0. Factorise: 3x^2 - 14x + 16 = (3x - 8)(x - 2) = 0. Therefore x = 8/3 or x = 2. Now consider 5y^2 - 16y + 12 = 0. Factorise: 5y^2 - 16y + 12 = (5y - 6)(y - 2) = 0. Therefore y = 6/5 or y = 2. Now order the values: 6/5 is 1.2, 2 is 2, and 8/3 is approximately 2.67. Thus y takes values 1.2 and 2, while x takes values 2 and about 2.67. So each x value is greater than or equal to each y value. Verification / Alternative check: We can verify by forming all pairs. For x = 2 and y = 2, we get x = y. For x = 2 and y = 6/5, we have 2 > 1.2, so x > y. For x = 8/3 and y = 2, x > y again. For x = 8/3 and y = 6/5, x is still greater. In no valid pair is x less than y. Hence x is always greater than or equal to y, which confirms the relationship x >= y. Why Other Options Are Wrong: Option x > y ignores the equality case where both variables take value 2. Option x < y is never true because x is never smaller than y. Option x = y fails because there exist pairs with strict inequality. Relationship cannot be determined is not correct because there is a clear consistent inequality once all root pairs are examined carefully. Common Pitfalls: Learners sometimes forget to convert fractions to decimals for easy comparison or they misfactor the quadratic expressions, leading to wrong roots. Another common mistake is to compare only one root pair rather than all possible combinations. To avoid errors, always factor step by step and place all roots in increasing order on the number line before deciding the inequality relation. Final Answer: All valid x values are greater than or equal to all valid y values. Therefore the correct choice is x >= y.

Question 13

In this aptitude (simplification: quadratic comparison) question, two quadratic equations for x and y are given. Solve each equation to find all roots of x and y, then compare these values and select the option that correctly describes the relations…

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Introduction / Context: This problem gives two quadratic equations and asks for the relationship between x and y. Unlike earlier questions where sets of roots overlapped, here the roots are positioned in a way that delivers a clear inequality. Such tasks check understanding of factorisation and the standard relationships between roots and coefficients while also training students to compare multiple values correctly. Given Data / Assumptions: Equation I: x^2 - 11x + 28 = 0. Equation II: y^2 - 18y + 81 = 0. We treat all real roots of each equation as possible values of x and y. Concept / Approach: We first factor each quadratic to get explicit roots. After that we compare every x root with every y root. If every possible x is less than every possible y, the correct relation is x < y. This approach guarantees that we do not miss any combination and supports a logically sound inequality conclusion. Step-by-Step Solution: Factor equation I: x^2 - 11x + 28 = 0. We get (x - 7)(x - 4) = 0. So x = 7 or x = 4. Now factor equation II: y^2 - 18y + 81 = 0. This is a perfect square: (y - 9)^2 = 0. Hence y = 9 is the only (repeated) root. Now compare values: x can be 4 or 7, while y is always 9. Both 4 and 7 are strictly less than 9, so for all valid roots we have x < y. Verification / Alternative check: Using the sum and product of roots confirms our factorisation. For equation I, sum of roots is 11 and product is 28, matching 7 and 4. For equation II, sum is 18 and product is 81, which matches a double root at 9. When we place the values on the number line, we see 4 and 7 lying to the left of 9. Therefore in every allowable combination, x is less than y and no combination violates this ordering. Why Other Options Are Wrong: Option x > y is wrong because no x value exceeds 9. Option x = y fails as neither 4 nor 7 equals 9. Option x >= y would require at least equality or greater than in some case, but that never occurs. Relationship cannot be determined would apply only if there were inconsistent comparisons across pairs, which does not happen here. Common Pitfalls: Students sometimes misfactor the second quadratic and overlook that it is a perfect square, leading to incorrect roots. Another pitfall is to forget to test all roots for x, comparing only one of them with y. Writing down the complete sets {4, 7} and {9} and then checking each combination avoids such mistakes. Drawing a quick number line diagram with marks at 4, 7, and 9 can further support intuitive understanding. Final Answer: Since both possible values of x are strictly less than the single value of y, the correct relationship is x < y.

Question 14

In this aptitude (simplification) question, you must find the value that replaces the question mark to keep the numerical equation balanced. Compute 115 / 5 + 12 * 6 on the left side and ? + 64 / 4 - 35 on the right side, using correct order of oper…

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Introduction / Context: This is a straightforward arithmetic simplification question from the simplification topic in aptitude. Candidates need to apply the correct order of operations, sometimes remembered as BODMAS or PEMDAS, to evaluate both sides of an equation. The aim is to find the number that should replace the question mark so that the equality holds exactly. Given Data / Assumptions: We are given the equation 115 / 5 + 12 * 6 = ? + 64 / 4 - 35. All operations are standard arithmetic in real numbers. We must find the value of the unknown represented by ?. Concept / Approach: The key concept is the order of operations. Multiplication and division are performed before addition and subtraction. We first simplify each arithmetic block step by step, working separately on the left side and on the right side. Once we obtain a numeric value on one side, we can solve the resulting linear equation in the unknown ? by simple addition or subtraction. Step-by-Step Solution: Compute the left side: 115 / 5 + 12 * 6. 115 / 5 = 23, and 12 * 6 = 72, so the left side is 23 + 72 = 95. Now simplify the right side: ? + 64 / 4 - 35. 64 / 4 = 16, so the right side is ? + 16 - 35. Combine constants: 16 - 35 = -19, so the right side becomes ? - 19. Set the two sides equal: 95 = ? - 19. Add 19 to both sides: ? = 95 + 19 = 114. Verification / Alternative check: Substitute ? = 114 back into the original expression to verify. The right side becomes 114 + 64 / 4 - 35. We already know 64 / 4 = 16, so this is 114 + 16 - 35 = 130 - 35 = 95. This matches the left side, which we computed as 95. Since both sides are equal, the solution ? = 114 is confirmed as correct. Why Other Options Are Wrong: If we try 95, the right side would be 95 + 16 - 35 = 76, which is not equal to 95. Option 102 gives 102 + 16 - 35 = 83. Option 136 gives 136 + 16 - 35 = 117. Option 74 gives 74 + 16 - 35 = 55. None of these values makes the right side equal to the left side, so they are all incorrect. Common Pitfalls: A common mistake is to perform addition or subtraction before division or multiplication, which changes the result. Another error is incorrect handling of negative numbers when combining constants, for example miscomputing 16 - 35. Carefully following the order of operations and writing intermediate results helps prevent such issues. Final Answer: The equation balances only when the missing number is 114.

Question 15

In this aptitude (simplification) question, you must evaluate a simple arithmetic expression involving four integers. Compute 1234 + 2345 - 3456 + 4567 carefully and choose the correct simplified result from the options provided.

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Introduction / Context: This problem is a classic example of numerical simplification that appears frequently in aptitude tests. The candidate must carry out addition and subtraction for relatively large four digit numbers without making place value errors. Even though the calculation is straightforward, carelessness can easily lead to a wrong answer, so a systematic approach is important. Given Data / Assumptions: We need to simplify the expression 1234 + 2345 - 3456 + 4567. All quantities are integers and the operations are standard addition and subtraction. Concept / Approach: We simply apply integer arithmetic, but we can choose an order that reduces mental effort. Grouping positive terms together and negative terms together, or pairing terms that almost cancel, can help. The result must be computed exactly, since options are close to one another and a small mistake in one digit can lead to choosing the wrong option. Step-by-Step Solution: Start with 1234 + 2345 = 3579. Now compute 3579 - 3456. This gives 123. Then add the final term: 123 + 4567 = 4690. So the overall value of the expression is 4690. Verification / Alternative check: As a cross check, we can regroup terms differently. Combine 2345 and 4567 first: 2345 + 4567 = 6912. Now compute 1234 - 3456 = -2222. Finally add: 6912 - 2222 = 4690, which matches the earlier result. Using two different routes yet arriving at the same answer gives strong confidence that 4690 is correct. Why Other Options Are Wrong: Option 4680 and 4670 are very close to the correct answer and likely represent errors of ten or twenty due to misaligned subtraction. Option 4590 reflects a larger mistake while carrying or borrowing. Option 4700 may come from rounding or incomplete subtraction. Since we have carefully verified the exact sum, only 4690 satisfies the expression. Common Pitfalls: The main pitfalls include incorrectly subtracting 3456, mixing up hundreds and tens, or failing to carry over digits correctly. Another issue is rushing without checking the result with an alternative grouping. Always rewrite long expressions clearly and perform operations step by step, checking intermediate totals as needed. Final Answer: The correctly simplified value of the expression is 4690.

Question 16

In this aptitude (simplification and analytic geometry) question, you must find the y intercept of a line. For the linear equation 59x + 14y - 112 = 0, set x = 0 and determine the corresponding value of y, which is the y intercept of the line on the…

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Introduction / Context: This question combines basic algebra with coordinate geometry. The goal is to find the y intercept of a straight line given in standard form. The y intercept is the point where the line crosses the y axis, and it is an essential concept in slope intercept representation and in graph sketching for linear functions. Given Data / Assumptions: The line is given by the equation 59x + 14y - 112 = 0. The y intercept is defined as the value of y when x equals 0. We work in the usual Cartesian coordinate system with x and y real valued. Concept / Approach: To find the y intercept of any line, we set x = 0 in the equation and solve for y. This is because any point on the y axis has x coordinate equal to zero. The resulting value gives the vertical coordinate at which the line meets the y axis. The process requires simple substitution and solution of a linear equation in one variable. Step-by-Step Solution: Start from the equation 59x + 14y - 112 = 0. Set x = 0 to represent a point on the y axis. Substitute: 59 * 0 + 14y - 112 = 0, which simplifies to 14y - 112 = 0. Add 112 to both sides: 14y = 112. Divide by 14: y = 112 / 14 = 8. So the y intercept is 8, and the point is (0, 8). Verification / Alternative check: We can also rearrange the equation into slope intercept form y = mx + c. Starting from 59x + 14y - 112 = 0, move terms: 14y = -59x + 112. Divide by 14 to get y = (-59/14)x + 112/14. The constant term 112/14 simplifies to 8. In y = mx + c form, c is the y intercept, so again we obtain y intercept equal to 8, which confirms the result. Why Other Options Are Wrong: Option -8 would arise if a sign mistake is made when moving 112 across the equals sign. Option 14 results from incorrectly dividing or ignoring the coefficient of y. Option 28 and 59 are unrelated to the derived intercept because they correspond to other coefficients in the equation rather than the constant term divided by the coefficient of y. Only the value 8 satisfies the definition of the y intercept for this line. Common Pitfalls: Students often forget to set x exactly to zero or mishandle the algebraic rearrangement, especially with signs. Another pitfall is to confuse the slope with the intercept or to read off numbers incorrectly from the original equation. Writing the intermediate step 14y - 112 = 0 clearly and then solving carefully helps avoid these issues. Final Answer: The y intercept of the line 59x + 14y - 112 = 0 is 8.

Question 17

In this aptitude (simplification and trigonometry) question, evaluate the expression: sin(60 degrees) + 2 / sqrt(3), using exact standard trigonometric values and rational simplification, then choose the correct simplified result.

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Introduction / Context: This question checks familiarity with standard trigonometric values and basic algebraic simplification. Instead of using decimal approximations, we are expected to use exact values for special angles and combine them into a single simplified expression. Such problems are common in quantitative aptitude and engineering entrance examinations. Given Data / Assumptions: We must evaluate sin(60 degrees) + 2 / sqrt(3). Use the exact value sin(60 degrees) = sqrt(3) / 2. We treat sqrt(3) as an exact irrational quantity and combine terms algebraically. Concept / Approach: First substitute the exact trigonometric value for sin(60 degrees). Then we need to add two fractions that have different denominators. A common strategy is to express both terms with a common denominator, then sum numerators. Finally, we simplify the resulting fraction as much as possible without rationalising the denominator unless required. Here the main aim is to match the given exact form among the options. Step-by-Step Solution: Start from sin(60 degrees) + 2 / sqrt(3). Replace sin(60 degrees) with sqrt(3) / 2. So the expression becomes sqrt(3) / 2 + 2 / sqrt(3). To add these, use common denominator 2 * sqrt(3). Rewrite sqrt(3) / 2 as (sqrt(3) * sqrt(3)) / (2 * sqrt(3)) = 3 / (2 * sqrt(3)). Rewrite 2 / sqrt(3) as (2 * 2) / (2 * sqrt(3)) = 4 / (2 * sqrt(3)). Now add: 3 / (2 * sqrt(3)) + 4 / (2 * sqrt(3)) = 7 / (2 * sqrt(3)). Verification / Alternative check: We can check numerically to gain confidence. Approximate sqrt(3) as about 1.732. Then sin(60 degrees) is approximately 0.866. The second term 2 / sqrt(3) is about 1.155. Their sum is roughly 2.021. Now compute 7 / (2 * sqrt(3)) using the same approximation: 2 * sqrt(3) is about 3.464, and 7 divided by 3.464 is again close to 2.021. This confirms that 7 / (2 * sqrt(3)) is the correct exact representation of the sum. Why Other Options Are Wrong: Option 3 is much larger than the approximate value near 2. Option (2*sqrt(3)+1)/sqrt(3) simplifies to 2 + 1 / sqrt(3), which does not match. Option 2 + sqrt(3) is clearly larger than 2 plus about 1.732, so it is far above the required value. Option 5 / (2 * sqrt(3)) is smaller than 7 / (2 * sqrt(3)) and does not match the computed sum. Common Pitfalls: Typical mistakes include using the wrong standard value for sin(60 degrees), confusing degrees with radians, or using decimal approximations too early and then trying to match them with complicated exact options. Another error is mishandling fraction addition with radical denominators. Working systematically with a common denominator avoids these issues. Final Answer: The exact simplified value of the expression is 7/(2*sqrt(3)).

Question 18

In this aptitude (simplification and quadratic equations) question, solve the equation: (x - 2)^2 - 36 = 0, find the possible real values of x, and then choose the value of x that belongs to the set of natural numbers (x in N).

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Introduction / Context: This question involves a simple quadratic equation written in shifted square form. The candidate must solve for x, identify all real solutions, and then select the solution that is a natural number. It reinforces basic algebraic manipulation and understanding of number sets such as integers and natural numbers. Given Data / Assumptions: We are given the equation (x - 2)^2 - 36 = 0. We know x belongs to the natural numbers, so x must be a positive integer. We must find all real solutions and then filter those that are natural numbers. Concept / Approach: The equation is of the form (x - a)^2 - b^2 = 0, which resembles a difference of squares. We can either expand and solve as a standard quadratic or directly interpret it as (x - 2)^2 = 36. Taking square roots gives two real solutions, because both positive and negative roots are possible. Finally, we use the natural number condition to pick the appropriate value. Step-by-Step Solution: Start with (x - 2)^2 - 36 = 0. Rearrange to (x - 2)^2 = 36. Take square roots on both sides: x - 2 = 6 or x - 2 = -6. From x - 2 = 6 we get x = 8. From x - 2 = -6 we get x = -4. So the real solutions are x = 8 and x = -4. Verification / Alternative check: Substitute x = 8 into the original equation: (8 - 2)^2 - 36 = 6^2 - 36 = 36 - 36 = 0, so x = 8 is valid. Substitute x = -4: (-4 - 2)^2 - 36 = (-6)^2 - 36 = 36 - 36 = 0, so x = -4 is also valid as a real root. However we must respect the condition that x is a natural number, which usually means positive integers 1, 2, 3, and so on, so we keep only x = 8. Why Other Options Are Wrong: Options -4 and -8 are not natural numbers because they are negative. Option 4 fails because (4 - 2)^2 - 36 = 4 - 36 = -32, which is not zero. Option 2 also fails because (2 - 2)^2 - 36 = 0 - 36 = -36. Therefore these values do not satisfy the equation or do not belong to the required number set. Common Pitfalls: A common mistake is to forget the negative square root and assume only x - 2 = 6, losing one valid root. Another is to overlook the natural number condition and select both positive and negative solutions. Students should always check the domain or set specified in the question and verify solutions against it. Final Answer: The only solution that both satisfies the equation and lies in the natural numbers is 8.

Question 19

In this aptitude (simplification and polynomials) question, the cubic polynomial 2x^3 - x^2 - 2x + 1 has zeros a, b, and c. Using the relationship between roots and coefficients for a cubic, determine the value of a + b + c (the sum of the roots).

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Introduction / Context: This problem tests knowledge of standard relationships between the roots and coefficients of a polynomial, specifically a cubic. Instead of explicitly solving for each root, which may be messy, we can directly use the sum of roots formula. This is a powerful shortcut that saves time in aptitude exams and higher mathematics problems involving symmetric expressions in roots. Given Data / Assumptions: The cubic polynomial is 2x^3 - x^2 - 2x + 1. Its zeros (roots) are a, b, and c, which may be real or complex. We are asked to find a + b + c, the sum of the roots. Concept / Approach: For a general cubic equation of the form ax^3 + bx^2 + cx + d = 0 with roots r1, r2, r3, the sum of the roots is given by -b / a. This result comes from comparing coefficients after factoring the polynomial into a(x - r1)(x - r2)(x - r3). We can apply this directly without finding individual roots, provided we correctly identify the coefficients a and b. Step-by-Step Solution: Write the polynomial in standard form: 2x^3 - x^2 - 2x + 1 = 0. Here a = 2, b = -1, c = -2, and d = 1. Let the roots be a, b, and c as given in the question statement. By the sum of roots formula for cubics, a + b + c = -b / a. Substitute b = -1 and a = 2: a + b + c = -(-1) / 2. This simplifies to a + b + c = 1 / 2. Verification / Alternative check: We can verify the formula quickly by recalling the factorised form for a cubic: 2(x - a)(x - b)(x - c). When we expand this product, the coefficient of x^2 is -2(a + b + c). This must match the given polynomial where the x^2 coefficient is -1. Equating -2(a + b + c) with -1 gives a + b + c = 1 / 2, which confirms the earlier calculation using the standard formula. Why Other Options Are Wrong: Option -1/2 would arise from incorrectly using b / a instead of -b / a. Option 1 and option -1 correspond to errors in equating coefficients, for example mismatching the factor 2 from the leading coefficient. Option 0 would imply that the x^2 coefficient is zero, which is not the case. Therefore only 1/2 is consistent with the actual polynomial coefficients. Common Pitfalls: Learners sometimes confuse formulas for sum of roots of quadratic and cubic equations or forget to divide by the leading coefficient. Another common error is to misread the sign of the x^2 coefficient. To avoid mistakes, always rewrite the polynomial in standard form and clearly label a, b, c, and d before applying the formula for the sum of roots. Final Answer: The sum of the zeros a, b, and c of the polynomial 2x^3 - x^2 - 2x + 1 is 1/2.

Question 20

In this aptitude (simplification and quadratic equations) question, you are given that the two roots of a quadratic equation are x = 1/7 and x = -1/8. Using the factor form (x - r1)(x - r2) = 0 and clearing denominators, determine which option repre…

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Introduction / Context: This question tests understanding of how to reconstruct a quadratic equation from its roots. Rather than starting from a polynomial and finding its zeros, we are given the roots and must build the factor form of the equation. This is a standard skill in algebra and is useful for many aptitude and competitive exam problems. Given Data / Assumptions: The roots are given as x = 1/7 and x = -1/8. We need the corresponding factor form of the quadratic equation in x. The equation is assumed to be set equal to zero, that is, of the form something equals zero. Concept / Approach: If r1 and r2 are roots of a quadratic, then the factors are (x - r1) and (x - r2). Their product equals zero gives the quadratic. When the roots are fractions, we often multiply both sides by the denominators to eliminate fractions and obtain a neat factorised form with integer coefficients. We then match the resulting factors with the options provided. Step-by-Step Solution: Let r1 = 1/7 and r2 = -1/8. The basic factor form is (x - r1)(x - r2) = 0. So the equation is (x - 1/7)(x + 1/8) = 0. Now clear denominators by multiplying each factor by the denominator of its corresponding root. For x - 1/7, multiply by 7 to get 7x - 1. For x + 1/8, multiply by 8 to get 8x + 1. Thus the factorised equation with integer coefficients is (7x - 1)(8x + 1) = 0. Verification / Alternative check: We can verify by checking that each given root satisfies the factorised equation. For x = 1/7, the factor 7x - 1 becomes 7 * (1/7) - 1 = 1 - 1 = 0, so the product is zero and the root is valid. For x = -1/8, the factor 8x + 1 becomes 8 * (-1/8) + 1 = -1 + 1 = 0, again giving zero. Therefore the factor pair (7x - 1) and (8x + 1) correctly corresponds to the two given roots. Why Other Options Are Wrong: Option (7x - 1)(8x - 1) = 0 would have roots 1/7 and 1/8, both positive, which does not match the negative second root. Option (7x + 1)(8x - 1) = 0 would give roots -1/7 and 1/8. Option (7x + 1)(8x + 1) = 0 would give roots -1/7 and -1/8. The final option (56x - 1)(x + 1) = 0 has roots 1/56 and -1, again inconsistent with the given pair. Only the chosen factorisation produces exactly 1/7 and -1/8 as roots. Common Pitfalls: Learners often forget that the factor is x minus the root, not x plus the root, leading to sign errors. Another mistake is to ignore denominators and try to write factors directly with fractional constants, making comparison with the options difficult. Multiplying each factor by the denominator removes fractions and leads to cleaner expressions that are easier to match. Final Answer: The correct factor form of the quadratic equation with roots 1/7 and -1/8 is (7x - 1)(8x + 1) = 0.

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