Q: The value of $\frac{1 - x^4}{1 + x} \div \frac{1 + x^2}{x} \times \frac{1}{x(1 - x)}$ is

Concept & Formula: This problem involves the simplification of rational algebraic expressions. Key techniques include factoring the difference of squares (a^2 - b^2) and converting division by a fraction into multiplication by its reciprocal. Step-by-Step Solution: • Given Expression: (1 - x^4/1 + x) \div (1 + x^2/x) × (1/x(1 - x)) • Step 1: Factorize terms Use the difference of squares formula a^2 - b^2 = (a-b)(a+b) on 1 - x^4: 1 - x^4 = (1 - x^2)(1 + x^2) Factor (1 - x^2) again: 1 - x^4 = (1 - x)(1 + x)(1 + x^2) • Step 2: Convert division to multiplication \div (1 + x^2/x) becomes × (x/1 + x^2) • Step 3: Combine and simplify Substitute everything back into the main expression: ((1 - x)(1 + x)(1 + x^2)/1 + x) × (x/1 + x^2) × (1/x(1 - x)) Cancel (1 + x) from the first fraction: = (1 - x)(1 + x^2) × (x/1 + x^2) × (1/x(1 - x)) Cancel (1 + x^2): = (1 - x) × x × (1/x(1 - x)) Cancel x and (1 - x): = 1 Exam Strategy & Shortcut: Use the value substitution method to bypass algebra completely. Pick a simple, permissible value for x, like x = 2 (avoid x=0, 1, -1 to prevent zero denominators). Substitute x=2: • Term 1: (1 - 16/1 + 2) = (-15/3) = -5 • Term 2: (1 + 4/2) = (5/2). Division by this is × (2/5). • Term 3: (1/2(1-2)) = (1/-2). Combine: -5 × (2/5) × (1/-2) = -2 × (1/-2) = 1. Since it equals a constant 1, it matches Option (a). Common Pitfall: Performing the multiplication before the division due to misinterpreting BODMAS/PEMDAS rules. Multiplication and division have the exact same precedence and must be evaluated strictly left-to-right unless grouped by parentheses. Final Answer: Therefore, the correct answer is 1.

Q: What is $$ \frac{\frac{7}{8} \times \frac{7}{8} + \frac{5}{6} \times \frac{5}{6} + \frac{7}{8} \times \frac{5}{3}}{\frac{7}{8} \times \frac{7}{8} - \frac{5}{6} \times \frac{5}{6}} $$ equal to?

Concept & Formula: This problem masks fundamental algebraic identities inside complex-looking fraction multiplication. The two governing formulas to recognize here are: (a + b)^2 = a^2 + b^2 + 2ab a^2 - b^2 = (a - b)(a + b) Step-by-Step Solution: Let's substitute the repeating fractions with variables to spot the pattern more easily: • Let a = (7/8) • Let b = (5/6) Look closely at the numerator: a × a + b × b + a × (5/3) The trick lies in the fraction (5/3) . Notice that (5/3) is exactly double (5/6) . Therefore, (5/3) = 2 × b . Substitute this back into the numerator: Numerator = a^2 + b^2 + a(2b) = a^2 + b^2 + 2ab Which simplifies to (a + b)^2 . Now look at the denominator: Denominator = a × a - b × b = a^2 - b^2 Which expands to (a - b)(a + b) . Combine the simplified numerator and denominator: Expression = ((a + b)^2/(a - b)(a + b)) Cancel out one (a + b) term from top and bottom: Expression = (a + b/a - b) Now, evaluate (a + b) and (a - b) by substituting the fractions back. Find a common denominator for 8 and 6, which is 24. • a = (7/8) = (21/24) • b = (5/6) = (20/24) Calculate the sum and difference: • a + b = (21/24) + (20/24) = (41/24) • a - b = (21/24) - (20/24) = (1/24) Finally, divide the sum by the difference: (41/24/1/24) = 41 × (24/1) = 41 Exam Strategy & Shortcut: Whenever you see a a^2 + b^2 + \dots structure, actively look for how the remaining term represents 2ab. Here, seeing 5/3 alongside 5/6 immediately signals 2b. You can then instantly simplify the whole block of math down to (a+b)/(a-b) , requiring only one basic fraction addition and subtraction. Common Pitfall: Failing to recognize 5/3 as 2 × 5/6 causes students to abandon the algebraic approach entirely and attempt brute-force arithmetic, which under exam conditions almost always results in a miscalculation given the large mixed denominators. Final Answer: Therefore, the correct answer is 41.

Q: $ \frac{(469 + 174)^2 - (469 - 174)^2}{469 \times 174} = x $

Concept & Formula: This is another classic aptitude exam pattern testing your knowledge of derived algebraic formulas. In this case, it utilizes the subtraction of two expanded binomial squares. The specific derived formula is: (a + b)^2 - (a - b)^2 = 4ab Step-by-Step Solution: Let's replace the large numbers with variables to expose the underlying structure: • Let a = 469 • Let b = 174 Substitute these variables into the expression given in the question: ((a + b)^2 - (a - b)^2/a × b) Let's prove the numerator's behavior by expanding both terms: Numerator = (a^2 + b^2 + 2ab) - (a^2 + b^2 - 2ab) Carefully distribute the negative sign into the second set of parentheses: = a^2 + b^2 + 2ab - a^2 - b^2 + 2ab The a^2 and b^2 terms cancel out, leaving only the ab terms: = 2ab + 2ab = 4ab Now, place this simplified numerator back over the denominator: = (4ab/ab) The ab terms cancel out completely from the top and the bottom: = 4 As demonstrated, the actual numerical values (469 and 174) play no role in determining the final answer. Exam Strategy & Shortcut: Much like the previous problem, you should memorize this format: ((a+b)^2 - (a-b)^2/ab) . Any equation following this exact mathematical structure will ALWAYS evaluate to 4. Recognizing this instantly saves you minutes of calculation. Common Pitfall: If you mistakenly write the denominator as (a^2 + b^2) rather than ab, you might confuse this identity with the previous one and answer 2. Pay very close attention to whether the operator connecting the terms is a plus or a minus, and what the denominator consists of. Final Answer: Therefore, the correct answer is 4.

Q: If $ \frac{x^2 - 1}{x + 1} = 4 $, $ x = $

Concept & Formula: This problem tests your knowledge of basic algebraic factorization. Specifically, it uses the difference of two squares identity to simplify a rational expression. The core formula is: a^2 - b^2 = (a - b)(a + b) Step-by-Step Solution: Examine the numerator of the given fraction: x^2 - 1 . Since 1 = 1^2 , this can be written as x^2 - 1^2 . Apply the difference of squares formula to factor the numerator: x^2 - 1 = (x - 1)(x + 1) Substitute this back into the original equation: ((x - 1)(x + 1)/x + 1) = 4 Cancel the common term (x + 1) from the numerator and denominator: x - 1 = 4 Add 1 to both sides to solve for x : x = 4 + 1 x = 5 Exam Strategy & Shortcut: Whenever you see a quadratic term like x^2 - y^2 divided by a linear term like x + y or x - y , immediately factor the quadratic. The test maker intentionally designed the problem so the denominator will cancel out. Bypassing the cross-multiplication step saves time and reduces the risk of errors. Common Pitfall: A common mistake is to immediately cross-multiply, resulting in x^2 - 1 = 4(x + 1) \Rightarrow x^2 - 1 = 4x + 4 \Rightarrow x^2 - 4x - 5 = 0 . While factoring this quadratic equation will eventually give you x = 5 and x = -1 (which is extraneous), it is significantly slower and more prone to calculation mistakes. Final Answer: Therefore, the correct answer is 5.

Q: $ \frac{\left(3 \frac{2}{3}\right)^2 - \left(2 \frac{1}{2}\right)^2}{\left(4 \frac{3}{4}\right)^2 - \left(3 \frac{1}{3}\right)^2} \div \frac{3 \frac{2}{3} - 2 \frac{1}{2}}{4 \frac{3}{4} - 3 \frac{1}{3}} = x $

Concept & Formula: This complex fraction relies on recognizing that the algebraic structure can be simplified entirely before any arithmetic on the fractions is performed. The primary identity used is the difference of squares: x^2 - y^2 = (x - y)(x + y) Step-by-Step Solution: First, to make the structure clear, assign variables to the mixed fractions: • Let a = 3 (2/3) • Let b = 2 (1/2) • Let c = 4 (3/4) • Let d = 3 (1/3) Substitute these variables into the original equation: (a^2 - b^2/c^2 - d^2) \div (a - b/c - d) Expand the squared terms using the difference of squares formula: ((a - b)(a + b)/(c - d)(c + d)) \div (a - b/c - d) Recall that dividing by a fraction is the same as multiplying by its reciprocal: ((a - b)(a + b)/(c - d)(c + d)) × (c - d/a - b) Cancel out the common (a - b) and (c - d) terms from the numerator and denominator: = (a + b/c + d) Now, evaluate a + b and c + d by converting to improper fractions: a + b = 3 (2/3) + 2 (1/2) = (11/3) + (5/2) = (22 + 15/6) = (37/6) c + d = 4 (3/4) + 3 (1/3) = (19/4) + (10/3) = (57 + 40/12) = (97/12) Divide the sum (a + b) by the sum (c + d) : (37/6/97/12) = (37/6) × (12/97) = (37 × 2/97) = (74/97) Exam Strategy & Shortcut: Whenever you see a structure matching (x^2 - y^2/z^2 - w^2) \div (x - y/z - w) , you should immediately recognize that the negative components will cancel out. You can instantly skip all middle steps and jump to evaluating (x + y/z + w) . Common Pitfall: The most dangerous pitfall is attempting to square the mixed fractions right at the beginning (e.g., trying to calculate (11/3)^2 = 121/9 ). This creates massive, unwieldy fractions that will consume minutes of exam time and almost certainly lead to arithmetic errors. Final Answer: Therefore, the correct answer is (74/97) .

Q: The simplified value of $$ \frac{\left(1 + \frac{1}{1 + \frac{1}{100}}\right)\left(1 + \frac{1}{1 + \frac{1}{100}}\right) - \left(1 - \frac{1}{1 + \frac{1}{100}}\right)\left(1 - \frac{1}{1 + \frac{1}{100}}\right)}{\left(1 + \frac{1}{1 + \frac{1}{10…

Concept & Formula: Despite the intimidating nested fractions, this is fundamentally a simple algebraic simplification problem based on the difference of squares identity. The governing formula is: (a^2 - b^2/a + b) = a - b Step-by-Step Solution: First, simplify the visual complexity by abstracting the repeating expressions into variables: • Let a = 1 + (1/1 + \frac{1){100}} • Let b = 1 - (1/1 + \frac{1){100}} Rewrite the given expression using these variables. Notice that the numerator consists of two identical terms multiplied together, minus another two identical terms multiplied together: (a \cdot a - b \cdot b/a + b) = (a^2 - b^2/a + b) Apply the difference of squares formula to expand the numerator: = ((a - b)(a + b)/a + b) The (a + b) terms cancel each other out, leaving only: = a - b Now, substitute the actual expressions back in and evaluate a - b : a - b = ( 1 + (1/1 + \frac{1){100}} ) - ( 1 - (1/1 + \frac{1){100}} ) Distribute the negative sign: = 1 + (1/1 + \frac{1){100}} - 1 + (1/1 + \frac{1){100}} The 1s cancel out, leaving two identical fractions: = 2 × ( (1/1 + \frac{1){100}} ) Simplify the inner denominator: 1 + (1/100) = (101/100) Substitute this back: = 2 × ( (1/\frac{101){100}} ) = 2 × (100/101) = (200/101) Exam Strategy & Shortcut: Don't let nested fractions bait you into immediate calculation. Notice the (x^2 - y^2/x + y) pattern right away. This allows you to deduce that the answer is simply x - y . When calculating x - y , the "1"s cancel out immediately, leaving 2 × the nested fraction . Common Pitfall: Many students attempt to fully evaluate the value of 1 + (1/1 + 1/100) and square it before looking at the rest of the equation. This leads to squaring fractions like 201/101 , which is an enormous waste of time. Always look for algebraic structure before doing arithmetic. Final Answer: Therefore, the correct answer is (200/101) .

Q: If $ x - y = 1 $ and $ x^2 + y^2 = 41 $, then the value of $ x + y $ will be

Concept & Formula: This problem links the sum and difference of two variables via their squared terms. You need to use the expansion formulas for both (x - y)^2 and (x + y)^2 to bridge the gap using the product xy . The key identities are: (x - y)^2 = x^2 + y^2 - 2xy (x + y)^2 = x^2 + y^2 + 2xy Step-by-Step Solution: Step 1: Find the value of 2xy We are given x - y = 1 and x^2 + y^2 = 41 . Use the squared difference formula: (x - y)^2 = x^2 + y^2 - 2xy Substitute the given values into the formula: (1)^2 = 41 - 2xy 1 = 41 - 2xy Rearrange to solve for 2xy : 2xy = 41 - 1 2xy = 40 Step 2: Find the value of (x + y)^2 Now, use the squared sum formula: (x + y)^2 = x^2 + y^2 + 2xy Substitute the known value of x^2 + y^2 (which is 41) and our calculated value for 2xy (which is 40): (x + y)^2 = 41 + 40 (x + y)^2 = 81 Step 3: Solve for x + y Take the square root of both sides. Remember that squaring either a positive or a negative number yields a positive result. x + y = \pm \sqrt{81} x + y = \pm 9 Exam Strategy & Shortcut: For small integer squares, look for Pythagorean triplets or familiar square sums. We need two numbers whose squares add to 41. Testing small squares: 1, 4, 9, 16, 25, 36 . Notice that 16 + 25 = 41 . These are the squares of 4 and 5. Check the first condition: 5 - 4 = 1 . This matches! So, x = 5 and y = 4 . Their sum is 5 + 4 = 9 . Since variables can also be negative (e.g., -4 and -5 where -4 - (-5) = 1 ), the sum can also be -9 . Thus, \pm 9 . Common Pitfall: The most common trap is calculating (x + y)^2 = 81 and simply selecting 9 as the answer, forgetting that the square root of 81 is algebraically \pm 9 . Exam setters often provide both options to catch this exact oversight. Final Answer: Therefore, the correct answer is \pm 9 .

Q: $ \left( \frac{785 \times 785 \times 785 + 435 \times 435 \times 435}{785 \times 785 + 435 \times 435 - 785 \times 435} \right) $ simplifies to

Concept & Formula: This problem is a classic aptitude question designed to test your recognition of the "Sum of Cubes" algebraic identity. The formula is: a^3 + b^3 = (a + b)(a^2 - ab + b^2) When rearranged as a fraction, it becomes a powerful simplification tool: (a^3 + b^3/a^2 - ab + b^2) = a + b Step-by-Step Solution: To make the algebraic structure obvious, let's replace the large numbers with variables: • Let a = 785 • Let b = 435 Now, substitute these variables into the expression given in the problem: Numerator = a \cdot a \cdot a + b \cdot b \cdot b = a^3 + b^3 Denominator = a \cdot a + b \cdot b - a \cdot b = a^2 + b^2 - ab Putting it all together, the expression is exactly: (a^3 + b^3/a^2 - ab + b^2) Using the algebraic identity, we can expand the numerator: = ((a + b)(a^2 - ab + b^2)/a^2 - ab + b^2) The entire trinomial block (a^2 - ab + b^2) cancels out from both the top and the bottom: = a + b Now, simply add the original numerical values: = 785 + 435 = 1220 Exam Strategy & Shortcut: You should instantly recognize this structure. Whenever you see \frac{Cube + Cube}{Square + Square - Product} , the answer is ALWAYS just the sum of the two base numbers (a + b). You don't need to write down any formulas—just confidently add 785 + 435 in seconds. Common Pitfall: The trap here is attempting to calculate the actual products (e.g., 785 × 785 × 785 ). This will take several minutes, waste massive amounts of scratch paper, and practically guarantee an arithmetic error. Always look for algebraic patterns in massive numbers. Final Answer: Therefore, the correct answer is 1220.

Question 1

A stairway 10 ft. high is such that each step accounts for half a foot upward and one foot forward. What distance will an ant travel if it starts from ground level to reach the top of the stairway?

Accepted Answer

Concept & Logic: A staircase consists of alternating vertical rises and horizontal treads. Reaching the "top" of a staircase logically means ascending the final riser to land on the upper floor level, rendering an additional forward tread unnecessary to satisfy the height completion. Step-by-Step Solution: • Given: • Total stairway height = 10 ft. • Single step vertical rise = 0.5 ft. • Single step horizontal forward (tread) = 1 ft. • Calculation: First, calculate the total number of upward steps (rises) needed to reach 10 ft: Total steps = (10/0.5) = 20 steps The ant must travel upward for every step: Total vertical distance = 20 × 0.5 = 10 ft For n steps reaching a top landing, there are only n-1 intermediate forward treads. The 20th riser puts the ant securely on the top floor. Total horizontal distance = 19 × 1 = 19 ft Sum the vertical and horizontal distances traveled: Total distance = 10 + 19 = 29 ft Exam Strategy & Shortcut: Memorize the geometry of stairs for an entity climbing N steps: Total Distance = (N × rise) + ((N-1) × tread) Here N=20. Distance = 10 + 19 = 29. You can do this visually in your head in seconds. Common Pitfall: Assuming there are equal numbers of vertical rises and horizontal treads (20 of each). This leads to calculating 10 + 20 = 30 ft, tricking students into selecting Option (a). Always remember that the last step up does not require a subsequent tread to be considered "at the top." Final Answer: Therefore, the correct answer is 29 ft.

Question 2

The sum of all proper fractions whose denominators are less than or equal to 100, is

Accepted Answer

Concept & Formula: This problem requires finding the sum of grouped sequences. A proper fraction has a numerator strictly less than its denominator (n < d). By grouping fractions with the same denominator, we can leverage the formula for the sum of the first n natural numbers. Step-by-Step Solution: • Given: Proper fractions (n/d) where 0 < n < d and d \le 100. • Deduction: For a fixed denominator d, the proper fractions are (1/d), (2/d), \dots, (d-1/d). The sum of these fractions for a specific d is: S_d = (1 + 2 + \dots + (d-1)/d) Using the sum of arithmetic progression formula (n(n+1)/2): S_d = (\frac{(d-1)d/2)}{d} = (d-1/2) • Calculation: We need the total sum across all denominators from d=2 to d=100: Total Sum = \sum_{d=2}^{100} (d-1/2) Factor out the (1/2) and expand the summation terms: Total Sum = (1/2) (1 + 2 + 3 + \dots + 99) Apply the sum formula again for n=99: Total Sum = (1/2) × (99 × 100/2) = (9900/4) Total Sum = 2475 Exam Strategy & Shortcut: Recognize the underlying pattern immediately: the sum of all proper fractions for any denominator d is exactly (d-1/2). The total sum for denominators up to N is just half the sum of integers up to N-1. (1/2) × (99 × 100/2) = 2475. This is a purely formula-driven shortcut that takes less than 20 seconds. Common Pitfall: Including n=d (an improper fraction equal to 1) in the sequence or starting the summation at d=1 (which has no valid proper fractions and breaks the math logic). Always strictly adhere to the mathematical definition of a proper fraction. Final Answer: Therefore, the correct answer is 2475.

Question 3

The value of $\frac{1 - x^4}{1 + x} \div \frac{1 + x^2}{x} 	imes \frac{1}{x(1 - x)}$ is

Accepted Answer

Concept & Formula: This problem involves the simplification of rational algebraic expressions. Key techniques include factoring the difference of squares (a^2 - b^2) and converting division by a fraction into multiplication by its reciprocal. Step-by-Step Solution: • Given Expression: (1 - x^4/1 + x) \div (1 + x^2/x) × (1/x(1 - x)) • Step 1: Factorize terms Use the difference of squares formula a^2 - b^2 = (a-b)(a+b) on 1 - x^4: 1 - x^4 = (1 - x^2)(1 + x^2) Factor (1 - x^2) again: 1 - x^4 = (1 - x)(1 + x)(1 + x^2) • Step 2: Convert division to multiplication \div (1 + x^2/x) becomes × (x/1 + x^2) • Step 3: Combine and simplify Substitute everything back into the main expression: ((1 - x)(1 + x)(1 + x^2)/1 + x) × (x/1 + x^2) × (1/x(1 - x)) Cancel (1 + x) from the first fraction: = (1 - x)(1 + x^2) × (x/1 + x^2) × (1/x(1 - x)) Cancel (1 + x^2): = (1 - x) × x × (1/x(1 - x)) Cancel x and (1 - x): = 1 Exam Strategy & Shortcut: Use the value substitution method to bypass algebra completely. Pick a simple, permissible value for x, like x = 2 (avoid x=0, 1, -1 to prevent zero denominators). Substitute x=2: • Term 1: (1 - 16/1 + 2) = (-15/3) = -5 • Term 2: (1 + 4/2) = (5/2). Division by this is × (2/5). • Term 3: (1/2(1-2)) = (1/-2). Combine: -5 × (2/5) × (1/-2) = -2 × (1/-2) = 1. Since it equals a constant 1, it matches Option (a). Common Pitfall: Performing the multiplication before the division due to misinterpreting BODMAS/PEMDAS rules. Multiplication and division have the exact same precedence and must be evaluated strictly left-to-right unless grouped by parentheses. Final Answer: Therefore, the correct answer is 1.

Question 4

What is $$ \frac{\frac{7}{8} 	imes \frac{7}{8} + \frac{5}{6} 	imes \frac{5}{6} + \frac{7}{8} 	imes \frac{5}{3}}{\frac{7}{8} 	imes \frac{7}{8} - \frac{5}{6} 	imes \frac{5}{6}} $$ equal to?

Accepted Answer

Concept & Formula: This problem masks fundamental algebraic identities inside complex-looking fraction multiplication. The two governing formulas to recognize here are: (a + b)^2 = a^2 + b^2 + 2ab a^2 - b^2 = (a - b)(a + b) Step-by-Step Solution: Let's substitute the repeating fractions with variables to spot the pattern more easily: • Let a = (7/8) • Let b = (5/6) Look closely at the numerator: a × a + b × b + a × (5/3) The trick lies in the fraction (5/3) . Notice that (5/3) is exactly double (5/6) . Therefore, (5/3) = 2 × b . Substitute this back into the numerator: Numerator = a^2 + b^2 + a(2b) = a^2 + b^2 + 2ab Which simplifies to (a + b)^2 . Now look at the denominator: Denominator = a × a - b × b = a^2 - b^2 Which expands to (a - b)(a + b) . Combine the simplified numerator and denominator: Expression = ((a + b)^2/(a - b)(a + b)) Cancel out one (a + b) term from top and bottom: Expression = (a + b/a - b) Now, evaluate (a + b) and (a - b) by substituting the fractions back. Find a common denominator for 8 and 6, which is 24. • a = (7/8) = (21/24) • b = (5/6) = (20/24) Calculate the sum and difference: • a + b = (21/24) + (20/24) = (41/24) • a - b = (21/24) - (20/24) = (1/24) Finally, divide the sum by the difference: (41/24/1/24) = 41 × (24/1) = 41 Exam Strategy & Shortcut: Whenever you see a a^2 + b^2 + \dots structure, actively look for how the remaining term represents 2ab. Here, seeing 5/3 alongside 5/6 immediately signals 2b. You can then instantly simplify the whole block of math down to (a+b)/(a-b) , requiring only one basic fraction addition and subtraction. Common Pitfall: Failing to recognize 5/3 as 2 × 5/6 causes students to abandon the algebraic approach entirely and attempt brute-force arithmetic, which under exam conditions almost always results in a miscalculation given the large mixed denominators. Final Answer: Therefore, the correct answer is 41.

Question 5

$ \frac{(469 + 174)^2 - (469 - 174)^2}{469 	imes 174} = x $

Accepted Answer

Concept & Formula: This is another classic aptitude exam pattern testing your knowledge of derived algebraic formulas. In this case, it utilizes the subtraction of two expanded binomial squares. The specific derived formula is: (a + b)^2 - (a - b)^2 = 4ab Step-by-Step Solution: Let's replace the large numbers with variables to expose the underlying structure: • Let a = 469 • Let b = 174 Substitute these variables into the expression given in the question: ((a + b)^2 - (a - b)^2/a × b) Let's prove the numerator's behavior by expanding both terms: Numerator = (a^2 + b^2 + 2ab) - (a^2 + b^2 - 2ab) Carefully distribute the negative sign into the second set of parentheses: = a^2 + b^2 + 2ab - a^2 - b^2 + 2ab The a^2 and b^2 terms cancel out, leaving only the ab terms: = 2ab + 2ab = 4ab Now, place this simplified numerator back over the denominator: = (4ab/ab) The ab terms cancel out completely from the top and the bottom: = 4 As demonstrated, the actual numerical values (469 and 174) play no role in determining the final answer. Exam Strategy & Shortcut: Much like the previous problem, you should memorize this format: ((a+b)^2 - (a-b)^2/ab) . Any equation following this exact mathematical structure will ALWAYS evaluate to 4. Recognizing this instantly saves you minutes of calculation. Common Pitfall: If you mistakenly write the denominator as (a^2 + b^2) rather than ab, you might confuse this identity with the previous one and answer 2. Pay very close attention to whether the operator connecting the terms is a plus or a minus, and what the denominator consists of. Final Answer: Therefore, the correct answer is 4.

Question 6

If $ \frac{x^2 - 1}{x + 1} = 4 $, $ x = $

Accepted Answer

Concept & Formula: This problem tests your knowledge of basic algebraic factorization. Specifically, it uses the difference of two squares identity to simplify a rational expression. The core formula is: a^2 - b^2 = (a - b)(a + b) Step-by-Step Solution: Examine the numerator of the given fraction: x^2 - 1 . Since 1 = 1^2 , this can be written as x^2 - 1^2 . Apply the difference of squares formula to factor the numerator: x^2 - 1 = (x - 1)(x + 1) Substitute this back into the original equation: ((x - 1)(x + 1)/x + 1) = 4 Cancel the common term (x + 1) from the numerator and denominator: x - 1 = 4 Add 1 to both sides to solve for x : x = 4 + 1 x = 5 Exam Strategy & Shortcut: Whenever you see a quadratic term like x^2 - y^2 divided by a linear term like x + y or x - y , immediately factor the quadratic. The test maker intentionally designed the problem so the denominator will cancel out. Bypassing the cross-multiplication step saves time and reduces the risk of errors. Common Pitfall: A common mistake is to immediately cross-multiply, resulting in x^2 - 1 = 4(x + 1) \Rightarrow x^2 - 1 = 4x + 4 \Rightarrow x^2 - 4x - 5 = 0 . While factoring this quadratic equation will eventually give you x = 5 and x = -1 (which is extraneous), it is significantly slower and more prone to calculation mistakes. Final Answer: Therefore, the correct answer is 5.

Question 7

$ \frac{\left(3 \frac{2}{3}\right)^2 - \left(2 \frac{1}{2}\right)^2}{\left(4 \frac{3}{4}\right)^2 - \left(3 \frac{1}{3}\right)^2} \div \frac{3 \frac{2}{3} - 2 \frac{1}{2}}{4 \frac{3}{4} - 3 \frac{1}{3}} = x $

Accepted Answer

Concept & Formula: This complex fraction relies on recognizing that the algebraic structure can be simplified entirely before any arithmetic on the fractions is performed. The primary identity used is the difference of squares: x^2 - y^2 = (x - y)(x + y) Step-by-Step Solution: First, to make the structure clear, assign variables to the mixed fractions: • Let a = 3 (2/3) • Let b = 2 (1/2) • Let c = 4 (3/4) • Let d = 3 (1/3) Substitute these variables into the original equation: (a^2 - b^2/c^2 - d^2) \div (a - b/c - d) Expand the squared terms using the difference of squares formula: ((a - b)(a + b)/(c - d)(c + d)) \div (a - b/c - d) Recall that dividing by a fraction is the same as multiplying by its reciprocal: ((a - b)(a + b)/(c - d)(c + d)) × (c - d/a - b) Cancel out the common (a - b) and (c - d) terms from the numerator and denominator: = (a + b/c + d) Now, evaluate a + b and c + d by converting to improper fractions: a + b = 3 (2/3) + 2 (1/2) = (11/3) + (5/2) = (22 + 15/6) = (37/6) c + d = 4 (3/4) + 3 (1/3) = (19/4) + (10/3) = (57 + 40/12) = (97/12) Divide the sum (a + b) by the sum (c + d) : (37/6/97/12) = (37/6) × (12/97) = (37 × 2/97) = (74/97) Exam Strategy & Shortcut: Whenever you see a structure matching (x^2 - y^2/z^2 - w^2) \div (x - y/z - w) , you should immediately recognize that the negative components will cancel out. You can instantly skip all middle steps and jump to evaluating (x + y/z + w) . Common Pitfall: The most dangerous pitfall is attempting to square the mixed fractions right at the beginning (e.g., trying to calculate (11/3)^2 = 121/9 ). This creates massive, unwieldy fractions that will consume minutes of exam time and almost certainly lead to arithmetic errors. Final Answer: Therefore, the correct answer is (74/97) .

Question 8

The simplified value of  $$ \frac{\left(1 + \frac{1}{1 + \frac{1}{100}}\right)\left(1 + \frac{1}{1 + \frac{1}{100}}\right) - \left(1 - \frac{1}{1 + \frac{1}{100}}\right)\left(1 - \frac{1}{1 + \frac{1}{100}}\right)}{\left(1 + \frac{1}{1 + \frac{1}{10…

Accepted Answer

Concept & Formula: Despite the intimidating nested fractions, this is fundamentally a simple algebraic simplification problem based on the difference of squares identity. The governing formula is: (a^2 - b^2/a + b) = a - b Step-by-Step Solution: First, simplify the visual complexity by abstracting the repeating expressions into variables: • Let a = 1 + (1/1 + \frac{1){100}} • Let b = 1 - (1/1 + \frac{1){100}} Rewrite the given expression using these variables. Notice that the numerator consists of two identical terms multiplied together, minus another two identical terms multiplied together: (a \cdot a - b \cdot b/a + b) = (a^2 - b^2/a + b) Apply the difference of squares formula to expand the numerator: = ((a - b)(a + b)/a + b) The (a + b) terms cancel each other out, leaving only: = a - b Now, substitute the actual expressions back in and evaluate a - b : a - b = ( 1 + (1/1 + \frac{1){100}} ) - ( 1 - (1/1 + \frac{1){100}} ) Distribute the negative sign: = 1 + (1/1 + \frac{1){100}} - 1 + (1/1 + \frac{1){100}} The 1s cancel out, leaving two identical fractions: = 2 × ( (1/1 + \frac{1){100}} ) Simplify the inner denominator: 1 + (1/100) = (101/100) Substitute this back: = 2 × ( (1/\frac{101){100}} ) = 2 × (100/101) = (200/101) Exam Strategy & Shortcut: Don't let nested fractions bait you into immediate calculation. Notice the (x^2 - y^2/x + y) pattern right away. This allows you to deduce that the answer is simply x - y . When calculating x - y , the "1"s cancel out immediately, leaving 2 × the nested fraction . Common Pitfall: Many students attempt to fully evaluate the value of 1 + (1/1 + 1/100) and square it before looking at the rest of the equation. This leads to squaring fractions like 201/101 , which is an enormous waste of time. Always look for algebraic structure before doing arithmetic. Final Answer: Therefore, the correct answer is (200/101) .

Question 9

If $ x - y = 1 $ and $ x^2 + y^2 = 41 $, then the value of $ x + y $ will be

Accepted Answer

Concept & Formula: This problem links the sum and difference of two variables via their squared terms. You need to use the expansion formulas for both (x - y)^2 and (x + y)^2 to bridge the gap using the product xy . The key identities are: (x - y)^2 = x^2 + y^2 - 2xy (x + y)^2 = x^2 + y^2 + 2xy Step-by-Step Solution: Step 1: Find the value of 2xy We are given x - y = 1 and x^2 + y^2 = 41 . Use the squared difference formula: (x - y)^2 = x^2 + y^2 - 2xy Substitute the given values into the formula: (1)^2 = 41 - 2xy 1 = 41 - 2xy Rearrange to solve for 2xy : 2xy = 41 - 1 2xy = 40 Step 2: Find the value of (x + y)^2 Now, use the squared sum formula: (x + y)^2 = x^2 + y^2 + 2xy Substitute the known value of x^2 + y^2 (which is 41) and our calculated value for 2xy (which is 40): (x + y)^2 = 41 + 40 (x + y)^2 = 81 Step 3: Solve for x + y Take the square root of both sides. Remember that squaring either a positive or a negative number yields a positive result. x + y = \pm \sqrt{81} x + y = \pm 9 Exam Strategy & Shortcut: For small integer squares, look for Pythagorean triplets or familiar square sums. We need two numbers whose squares add to 41. Testing small squares: 1, 4, 9, 16, 25, 36 . Notice that 16 + 25 = 41 . These are the squares of 4 and 5. Check the first condition: 5 - 4 = 1 . This matches! So, x = 5 and y = 4 . Their sum is 5 + 4 = 9 . Since variables can also be negative (e.g., -4 and -5 where -4 - (-5) = 1 ), the sum can also be -9 . Thus, \pm 9 . Common Pitfall: The most common trap is calculating (x + y)^2 = 81 and simply selecting 9 as the answer, forgetting that the square root of 81 is algebraically \pm 9 . Exam setters often provide both options to catch this exact oversight. Final Answer: Therefore, the correct answer is \pm 9 .

Question 10

If $ x = a + m $, $ y = b + m $, $ z = c + m $, the value of  $$ \frac{x^2 + y^2 + z^2 - yz - zx - xy}{a^2 + b^2 + c^2 - ab - bc - ca} $$  is

Accepted Answer

Concept & Formula: This problem centers on a special cyclic algebraic identity. It demonstrates that a specific combination of squares and cross-products is invariant under constant addition. The critical algebraic identity is: x^2 + y^2 + z^2 - xy - yz - zx = (1/2)[ (x - y)^2 + (y - z)^2 + (z - x)^2 ] Step-by-Step Solution: We are given the equations: 1. x = a + m 2. y = b + m 3. z = c + m Let's calculate the differences between the variables x, y, and z : x - y = (a + m) - (b + m) = a - b y - z = (b + m) - (c + m) = b - c z - x = (c + m) - (a + m) = c - a Notice that the constant m cancels out entirely. The difference between the new variables is exactly the same as the difference between the original variables. Now, look at the expression we need to evaluate: Numerator = x^2 + y^2 + z^2 - xy - yz - zx Apply the cyclic identity to rewrite the numerator: Numerator = (1/2)[ (x - y)^2 + (y - z)^2 + (z - x)^2 ] Substitute the differences we calculated: Numerator = (1/2)[ (a - b)^2 + (b - c)^2 + (c - a)^2 ] Now look at the denominator of the given expression: Denominator = a^2 + b^2 + c^2 - ab - bc - ca Apply the exact same cyclic identity to rewrite the denominator: Denominator = (1/2)[ (a - b)^2 + (b - c)^2 + (c - a)^2 ] Finally, put the rewritten numerator over the rewritten denominator: (\frac{1/2)[ (a - b)^2 + (b - c)^2 + (c - a)^2 ]}{(1/2)[ (a - b)^2 + (b - c)^2 + (c - a)^2 ]} Because the numerator and denominator are identical, the entire fraction simplifies to 1. Exam Strategy & Shortcut: Use the "Value Putting" method. Since the expression must hold true for *any* values of a, b, c, and m , pick simple numbers. Let a = 1, b = 2, c = 3 . Let m = 1 . Then x = 2, y = 3, z = 4 . Denominator: 1^2 + 2^2 + 3^2 - (2) - (6) - (3) = 14 - 11 = 3 . Numerator: 2^2 + 3^2 + 4^2 - (6) - (12) - (8) = 29 - 26 = 3 . Result: 3 / 3 = 1 . This practical method bypasses complex algebraic proofs entirely. Common Pitfall: The biggest mistake is trying to expand the numerator by substituting (a + m)^2 , (b + m)^2 , etc. This creates a massive polynomial with 18+ terms, which is extremely difficult to simplify without making a sign or grouping error. Final Answer: Therefore, the correct answer is 1.

Question 11

$ \left( \frac{785 	imes 785 	imes 785 + 435 	imes 435 	imes 435}{785 	imes 785 + 435 	imes 435 - 785 	imes 435} ight) $ simplifies to

Accepted Answer

Concept & Formula: This problem is a classic aptitude question designed to test your recognition of the "Sum of Cubes" algebraic identity. The formula is: a^3 + b^3 = (a + b)(a^2 - ab + b^2) When rearranged as a fraction, it becomes a powerful simplification tool: (a^3 + b^3/a^2 - ab + b^2) = a + b Step-by-Step Solution: To make the algebraic structure obvious, let's replace the large numbers with variables: • Let a = 785 • Let b = 435 Now, substitute these variables into the expression given in the problem: Numerator = a \cdot a \cdot a + b \cdot b \cdot b = a^3 + b^3 Denominator = a \cdot a + b \cdot b - a \cdot b = a^2 + b^2 - ab Putting it all together, the expression is exactly: (a^3 + b^3/a^2 - ab + b^2) Using the algebraic identity, we can expand the numerator: = ((a + b)(a^2 - ab + b^2)/a^2 - ab + b^2) The entire trinomial block (a^2 - ab + b^2) cancels out from both the top and the bottom: = a + b Now, simply add the original numerical values: = 785 + 435 = 1220 Exam Strategy & Shortcut: You should instantly recognize this structure. Whenever you see \frac{Cube + Cube}{Square + Square - Product} , the answer is ALWAYS just the sum of the two base numbers (a + b). You don't need to write down any formulas—just confidently add 785 + 435 in seconds. Common Pitfall: The trap here is attempting to calculate the actual products (e.g., 785 × 785 × 785 ). This will take several minutes, waste massive amounts of scratch paper, and practically guarantee an arithmetic error. Always look for algebraic patterns in massive numbers. Final Answer: Therefore, the correct answer is 1220.

Question 12

$ \frac{(13)^3 + 7^3}{(13)^2 + 7^2 - x} = 20 $

Accepted Answer

Concept & Formula: This problem tests your structural understanding of the "Sum of Cubes" formula. Instead of asking for the final result, it provides the result and asks you to identify a missing structural component. The underlying identity is: (a^3 + b^3/a^2 - ab + b^2) = a + b Step-by-Step Solution: Let's define our variables based on the obvious base numbers: • Let a = 13 • Let b = 7 Notice the result given on the right side of the equation is 20. Observe that 20 = 13 + 7 = a + b . So, the equation provided is essentially stating: (a^3 + b^3/a^2 + b^2 - x) = a + b We know from our standard algebraic identity that for this relationship to hold true, the denominator MUST be exactly a^2 - ab + b^2 . Let's rearrange our formula's denominator to match the visual layout of the problem's denominator: a^2 + b^2 - ab Comparing this perfect formula denominator ( a^2 + b^2 - ab ) to the given problem's denominator ( 13^2 + 7^2 - x ), it becomes clear that x is simply the missing ab term. Therefore: x = a × b x = 13 × 7 x = 91 Exam Strategy & Shortcut: When you see a sum of cubes fraction that equals the sum of the two base numbers ( 13^3 + 7^3 leading to 20 ), you instantly know the equation is utilizing the standard formula. The missing piece subtracting from the squares in the denominator is always the product of the two numbers. Multiply 13 × 7 mentally to get 91. Common Pitfall: Students who rely on brute-forcing arithmetic might attempt to cube 13 (2197) and cube 7 (343), add them together (2540), and then solve (2540/218 - x) = 20 . While this eventually yields 91, it takes a massive amount of unnecessary time and invites calculation errors. Use algebra to bypass arithmetic. Final Answer: Therefore, the correct answer is 91.

Question 13

$ \frac{38 	imes 38 	imes 38 + 34 	imes 34 	imes 34 + 28 	imes 28 	imes 28 - 38 	imes 34 	imes 84}{38 	imes 38 + 34 	imes 34 + 28 	imes 28 - 38 	imes 34 - 34 	imes 28 - 38 	imes 28} $ is equal to

Accepted Answer

Concept & Formula: This massive calculation is built entirely around one of the most complex, yet standard, algebraic identities involving the sum of three cubes. The identity is: a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca) When restructured as a fraction, it provides an instant simplification: (a^3 + b^3 + c^3 - 3abc/a^2 + b^2 + c^2 - ab - bc - ca) = a + b + c Step-by-Step Solution: Let's establish our variables: • Let a = 38 • Let b = 34 • Let c = 28 The denominator perfectly matches the formula: a^2 + b^2 + c^2 - ab - bc - ca . The numerator looks like a^3 + b^3 + c^3 , but the final term is - 38 × 34 × 84 . Where is the -3abc ? Let's calculate what 3abc should be: 3abc = 3 × 38 × 34 × 28 Notice that 3 × 28 = 84 . Therefore, 3abc can be rewritten as 38 × 34 × 84 . This means the numerator is indeed exactly a^3 + b^3 + c^3 - 3abc . Substitute the entire structure into our formula fraction: (a^3 + b^3 + c^3 - 3abc/a^2 + b^2 + c^2 - ab - bc - ca) According to the identity, the massive polynomial cancels out, leaving only: = a + b + c Substitute the numbers back in and sum them up: = 38 + 34 + 28 = 100 Exam Strategy & Shortcut: Whenever you see a fraction containing the sum of three cubes ( a^3 + b^3 + c^3 ) in the numerator and the sum of three squares in the denominator, you can confidently assume it is testing this specific identity. Don't even worry about verifying the 3abc term; test makers design these specifically to cancel out. Jump straight to adding the three base numbers ( 38 + 34 + 28 = 100 ). Common Pitfall: The main point of confusion is the 84 at the end of the numerator. Students expect to see a 3 and all three variables explicitly written out. When they see 84 , they assume the formula doesn't apply and either abandon the question or attempt impossible brute-force arithmetic. Always check if constants have been multiplied together to hide the formula. Final Answer: Therefore, the correct answer is 100.

Question 14

The value of  $$ \frac{(x - y)^3 + (y - z)^3 + (z - x)^3}{9 (x - y) (y - z) (z - x)} $$ is equal to

Accepted Answer

Concept & Formula: This problem leverages a highly specific conditional case of the three-variable sum of cubes formula. The core conditional identity is: If A + B + C = 0 , then it is always true that: A^3 + B^3 + C^3 = 3ABC Step-by-Step Solution: Let's define new variables to represent the binomials in the expression: • Let A = (x - y) • Let B = (y - z) • Let C = (z - x) First, let's check the condition A + B + C by adding these terms together: A + B + C = (x - y) + (y - z) + (z - x) = x - y + y - z + z - x Notice that all the variables perfectly cancel each other out: = 0 Since A + B + C = 0 is definitively true, we can apply the conditional identity: A^3 + B^3 + C^3 = 3ABC Substitute the original binomials back into this identity: (x - y)^3 + (y - z)^3 + (z - x)^3 = 3(x - y)(y - z)(z - x) Now, substitute this simplified result back into the numerator of the original problem: Expression = (3(x - y)(y - z)(z - x)/9(x - y)(y - z)(z - x)) The entire block (x - y)(y - z)(z - x) is identical in both the top and bottom. Cancel it out completely: = (3/9) Simplify the remaining fraction: = (1/3) Exam Strategy & Shortcut: Recognize the "cyclic format". Whenever you see expressions looping sequentially like (x-y) , (y-z) , (z-x) , their sum is zero. Therefore, their cubes add up to three times their product. Seeing \frac{3 × Product}{9 × Product} allows you to instantly bypass all writing and evaluate 3/9 = 1/3 in a matter of seconds. Common Pitfall: A major mistake is attempting to individually expand (x - y)^3 , (y - z)^3 , and (z - x)^3 . This generates 12 separate algebraic terms, creating an overwhelming polynomial that is highly susceptible to sign errors and takes entirely too long to manually cancel out. Always look for zero-sum conditions first. Final Answer: Therefore, the correct answer is (1/3) .

Question 15

If $ x = 5 $, $ y = 3 $, the value of $ \frac{x^3 - y^3}{x^2 - y^2} - \frac{3xy}{x + y} $ will be

Accepted Answer

Concept & Formula: This problem tests your ability to simplify rational algebraic expressions before inserting numerical values. The goal is to combine the two fractions into a much simpler form. The necessary factorization formulas are: x^3 - y^3 = (x - y)(x^2 + xy + y^2) x^2 - y^2 = (x - y)(x + y) Step-by-Step Solution: Let's start with the first fraction in the expression: (x^3 - y^3/x^2 - y^2) Factor both the numerator and the denominator using the standard formulas: = ((x - y)(x^2 + xy + y^2)/(x - y)(x + y)) Cancel out the common (x - y) term from the top and bottom: = (x^2 + xy + y^2/x + y) Now, bring back the second part of the original expression and subtract it: (x^2 + xy + y^2/x + y) - (3xy/x + y) Because both fractions now share a common denominator ( x + y ), we can combine their numerators: = (x^2 + xy + y^2 - 3xy/x + y) Combine the like terms ( +xy - 3xy = -2xy ): = (x^2 - 2xy + y^2/x + y) Notice that the numerator is now a perfect square trinomial. Factor it: = ((x - y)^2/x + y) The expression is fully simplified. Now substitute the given values: x = 5 and y = 3 . = ((5 - 3)^2/5 + 3) = ((2)^2/8) = (4/8) = (1/2) Exam Strategy & Shortcut: Instead of algebraic manipulation, you can use direct substitution ("Value Putting") immediately, as the numbers are very small. Fraction 1: (5^3 - 3^3/5^2 - 3^2) = (125 - 27/25 - 9) = (98/16) = (49/8) Fraction 2: (3(5)(3)/5 + 3) = (45/8) Difference: (49/8) - (45/8) = (4/8) = (1/2) . For small integers, basic arithmetic is often faster than performing multi-step polynomial factorization. Common Pitfall: When simplifying algebraically, students sometimes mistakenly cancel (x^2 - y^2) out of the denominator without properly factoring the numerator, or miscalculate the combination +xy - 3xy as +2xy , which leads to an entirely incorrect final value. Final Answer: Therefore, the correct answer is (1/2) .

Question 16

The value of  $$\frac{(x - y)^3 + (y - z)^3 + (z - x)^3}{(x^2 - y^2)^3 + (y^2 - z^2)^3 + (z^2 - x^2)^3}$$ is

Accepted Answer

Concept & Formula: This fraction simplifies using the conditional identity for the sum of cubes: If A + B + C = 0, then A^3 + B^3 + C^3 = 3ABC. Step-by-Step Solution: • Let the numerator components be A = x - y, B = y - z, and C = z - x. • Notice that A + B + C = (x - y) + (y - z) + (z - x) = 0. • Therefore, the numerator becomes 3(x - y)(y - z)(z - x). • Now, let the denominator components be P = x^2 - y^2, Q = y^2 - z^2, and R = z^2 - x^2. • Notice that P + Q + R = (x^2 - y^2) + (y^2 - z^2) + (z^2 - x^2) = 0. • Therefore, the denominator becomes 3(x^2 - y^2)(y^2 - z^2)(z^2 - x^2). • Expand the denominator using the difference of squares a^2 - b^2 = (a-b)(a+b): 3(x - y)(x + y)(y - z)(y + z)(z - x)(z + x) • Divide the simplified numerator by the simplified denominator: (3(x - y)(y - z)(z - x)/3(x - y)(x + y)(y - z)(y + z)(z - x)(z + x)) • Cancel the common terms 3(x - y)(y - z)(z - x) from top and bottom: = (1/(x + y)(y + z)(z + x)) • Rewrite using a negative exponent: = [(x + y)(y + z)(z + x)]^{-1} Exam Strategy & Shortcut: Whenever you see a sum of three cubes in a pattern that looks cyclical (like x-y, y-z, z-x), immediately check if they sum to zero. If they do, instantly replace them with 3 times their product. This visual recognition avoids lengthy expansions. Common Pitfall: A common error is to try expanding (x - y)^3 individually. This creates a massive, unmanageable polynomial that guarantees a loss of time and inevitable calculation errors. Final Answer: Therefore, the correct answer is [(x + y) (y + z) (z + x)]^{-1}.

Question 17

If $a + b + c = 0$, the value of $\frac{a^2}{bc} + \frac{b^2}{ca} + \frac{c^2}{ab}$ is

Accepted Answer

Concept & Formula: This tests the fundamental condition of cubic identities: If a + b + c = 0, then a^3 + b^3 + c^3 = 3abc. Step-by-Step Solution: • Given: a + b + c = 0. • Write out the expression to be evaluated: (a^2/bc) + (b^2/ca) + (c^2/ab) • Find a common denominator for the three fractions, which is abc. • Multiply the numerator and denominator of each fraction to achieve the common denominator: = (a^2 \cdot a/abc) + (b^2 \cdot b/abc) + (c^2 \cdot c/abc) • Combine into a single fraction: = (a^3 + b^3 + c^3/abc) • Apply the conditional identity (since a + b + c = 0): a^3 + b^3 + c^3 = 3abc • Substitute this back into the numerator: = (3abc/abc) • Cancel abc from the top and bottom (assuming a, b, c 
eq 0): = 3 Exam Strategy & Shortcut: You can verify this quickly by picking small integers that sum to zero. For example, let a=1, b=1, c=-2. Substitute: 1/(-2) + 1/(-2) + 4/(1) = -0.5 - 0.5 + 4 = 3. This confirms the result instantly without algebra. Common Pitfall: Forgetting to find a common denominator and attempting to cancel terms prematurely, or mistakenly assuming a^3 + b^3 + c^3 = 0. Final Answer: Therefore, the correct answer is 3.

Question 18

If $a = 29$, $b = 24$, $c = 27$, the value of $a^3 + b^3 + c^3 - 3abc$ is

Accepted Answer

Concept & Formula: When the values of a, b, and c are large but relatively close to each other, computing their cubes directly is inefficient. Instead, use the specialized form of the cubic identity involving squares of differences: a^3 + b^3 + c^3 - 3abc = (1/2)(a + b + c)[(a - b)^2 + (b - c)^2 + (c - a)^2] Step-by-Step Solution: • Given: a = 29, b = 24, c = 27. • First, calculate the sum of the variables: a + b + c = 29 + 24 + 27 = 80 • Next, calculate the differences between each pair of variables: a - b = 29 - 24 = 5 b - c = 24 - 27 = -3 c - a = 27 - 29 = -2 • Square these differences: (a - b)^2 = 5^2 = 25 (b - c)^2 = (-3)^2 = 9 (c - a)^2 = (-2)^2 = 4 • Sum the squares of the differences: 25 + 9 + 4 = 38 • Plug everything into the specialized formula: (1/2) × (80) × (38) • Simplify the calculation: = 40 × 38 = 1520 Exam Strategy & Shortcut: Using the difference-of-squares formula (1/2)(a+b+c)[(a-b)^2 + (b-c)^2 + (c-a)^2] is the ultimate shortcut here. It reduces large multi-digit cube calculations to basic addition and single-digit squaring. Common Pitfall: The most common trap is attempting brute-force calculation (29^3 + 24^3 + 27^3 - \dots). This will consume several minutes and likely result in an arithmetic mistake. Final Answer: Therefore, the correct answer is 1520.

Question 19

$(32)^3 + (79)^3 - (111)^3 + 3 	imes 32 	imes 79 	imes 111$ is equal to

Accepted Answer

Concept & Formula: This expression evaluates to zero based on a clever disguised application of the conditional identity: If a + b + c = 0, then a^3 + b^3 + c^3 - 3abc = 0. Step-by-Step Solution: • Look closely at the structure of the expression: (32)^3 + (79)^3 + (-111)^3 - 3(32)(79)(-111). • Notice that the negative signs have been distributed. The expression exactly matches the form a^3 + b^3 + c^3 - 3abc. • Let a = 32, b = 79, and c = -111. • Note that in the given expression, subtracting (111)^3 is mathematically identical to adding (-111)^3. • Similarly, adding + 3 × 32 × 79 × 111 is identical to subtracting -3 × (32) × (79) × (-111). • Check the sum of the variables: a + b + c = 32 + 79 + (-111) = 111 - 111 = 0 • Because a + b + c = 0, the entire expression a^3 + b^3 + c^3 - 3abc must equal 0. Exam Strategy & Shortcut: Any time an aptitude question asks you to add and subtract massive cubed numbers alongside a product of those same numbers, it is always testing the a+b+c=0 rule. Don't calculate anything; just verify that the base numbers sum to zero. Common Pitfall: Failing to recognize that c is negative (-111) and incorrectly assuming the formula doesn't apply because the signs look slightly different than the standard textbook formula. Final Answer: Therefore, the correct answer is 0.

Question 20

If $(x + y) = 3$, $xy = 2$, then what is the value of $x^3 + y^3$?

Accepted Answer

Concept & Formula: This requires directly calculating the sum of cubes using the sum and product of the variables, circumventing the need to find individual variable values. x^3 + y^3 = (x + y)^3 - 3xy(x + y) Step-by-Step Solution: • Given: The sum (x + y) = 3 and the product xy = 2. • Substitute these known values directly into the derived identity: x^3 + y^3 = (3)^3 - 3(2)(3) • Calculate the cube: 3^3 = 27 • Calculate the product term: 3 × 2 × 3 = 18 • Subtract: 27 - 18 = 9 Exam Strategy & Shortcut: While the formula is fast, for small integers you can quickly guess the roots. What two numbers add to 3 and multiply to 2? The numbers are 2 and 1. Therefore, x = 2 and y = 1 (or vice versa). Calculate 2^3 + 1^3 = 8 + 1 = 9. This is even faster than writing out the formula. Common Pitfall: A frequent mistake is using the formula for (x+y)^3 and forgetting to subtract the middle terms to isolate x^3 + y^3, leading students to incorrectly select 27 if it were an option. Final Answer: Therefore, the correct answer is 9.

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