Question 1

Solve $14 	imes 627 \div \sqrt{1089} = x^3 + 141$

Accepted Answer

Concept & Formula: This problem evaluates operational hierarchy (BODMAS/VBODMAS) along with finding perfect square roots and solving basic cubic equations. • Square root calculation: \sqrt{1089} = 33 Step-by-Step Solution: • Step 1: Find the square root of 1089: \sqrt{1089} = 33 \quad (since 30^2 = 900 and 40^2 = 1600) • Step 2: Substitute back into the equation and apply division first: 14 × (627 \div 33) = x^3 + 141 627 \div 33 = 19 • Step 3: Perform the multiplication: 14 × 19 = 266 • Step 4: Equate and solve for x^3: 266 = x^3 + 141 x^3 = 266 - 141 = 125 • Step 5: Compute the cube root: x = \sqrt[3]{125} = 5 Exam Strategy & Shortcut: Quick Estimation: Knowing basic squares and cubes speeds up this process tremendously. Knowing \sqrt{1089} = 33 simplifies 627/33 to just under 20 (specifically 19). 14 × 19 is roughly 266. Subtracting 141 yields exactly 125. Recognizing 125 as a perfect cube (5^3) immediately reveals x = 5. Common Pitfall: • Wrong Order of Operations: Executing 14 × 627 before dividing by \sqrt{1089} creates massive numbers that are difficult to manage. Always perform division before multiplication according to BODMAS rules. Final Answer: Therefore, the correct answer is 5.

Question 2

If $x + y + z = 0$, then $x^3 + y^3 + z^3 + 3xyz$ is equal to

Accepted Answer

Concept & Formula: This problem is built upon a well-known algebraic identity regarding cubic sums: x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx) If the condition x + y + z = 0 is fulfilled, the entire right side becomes zero, leading to: x^3 + y^3 + z^3 = 3xyz Step-by-Step Solution: • Given: x + y + z = 0 • Step 1: From the standard cubic identity, we know that if x+y+z=0, then: x^3 + y^3 + z^3 = 3xyz • Step 2: The question asks for the value of the expression x^3 + y^3 + z^3 + 3xyz. Substitute 3xyz in place of x^3 + y^3 + z^3: (3xyz) + 3xyz • Step 3: Combine the like terms: 6xyz Exam Strategy & Shortcut: Value Substitution Method: Choose simple values for x, y, z that satisfy the condition x + y + z = 0. Let x = 1, y = -1, z = 0. Substitute these test values into the required expression: 1^3 + (-1)^3 + 0^3 + 3(1)(-1)(0) = 1 - 1 + 0 + 0 = 0 Now check the options using our selected values (6xyz = 6(1)(-1)(0) = 0). While option (a) is also 0, trying a second set like 1, 1, -2 quickly eliminates option (a) because 1 + 1 - 8 + 3(1)(1)(-2) = -6 - 6 = -12, matching 6xyz = 6(1)(1)(-2) = -12. Common Pitfall: • Misreading the Expression: Students frequently mix up x^3 + y^3 + z^3 - 3xyz = 0 with the required expression. They assume the answer is 0 automatically without noticing that the question contains a plus sign (+ 3xyz) instead of a minus sign. Final Answer: Therefore, the correct answer is 6 xyz.

Question 3

Direction: In the question given below, if the given mathematical symbols are changed from '+' to '\div', '-' to '	imes', '\div' to '-' and from '	imes' to '+', then choose your answers form the following options.  Solve $67 	imes 119 + 17 - 27 \…

Accepted Answer

Concept & Formula: This problem requires evaluating an expression after performing character-based operator substitutions. Once the operators are replaced, evaluation must follow the standard BODMAS rule hierarchy: 1. Brackets 2. Orders (Powers/Roots) 3. Division / Multiplication (left-to-right) 4. Addition / Subtraction (left-to-right) Step-by-Step Solution: • Given: The original expression is 67 × 119 + 17 - 27 \div 259. • Step 1: Apply the symbol transformations: • × \rightarrow + • + \rightarrow \div • - \rightarrow × • \div \rightarrow - The new expression becomes: 67 + 119 \div 17 × 27 - 259 • Step 2: Perform division according to BODMAS (119 \div 17 = 7): 67 + 7 × 27 - 259 • Step 3: Perform multiplication (7 × 27 = 189): 67 + 189 - 259 • Step 4: Perform addition (67 + 189 = 256): 256 - 259 • Step 5: Complete the subtraction: -3 Exam Strategy & Shortcut: Rapid Substitution & Mental Division: Write the operational signs directly above or below the original question to save time rewriting the digits. Immediately calculate 119 \div 17 = 7. Next, evaluate 7 × 27 = 189. Finally, use units-digit tracking or quick rounding: 67 + 189 = 256, and 256 - 259 = -3. Common Pitfall: • Ignoring Order of Operations: A frequent mistake is performing the calculations linearly from left to right (e.g., executing 67 + 119 first). Always follow the mandatory BODMAS precedence table. Final Answer: Therefore, the correct answer is -3.

Question 4

Solve $\frac{(0.73)^3 + (0.27)^3}{(0.73)^2 + (0.27)^2 - 0.73 	imes 0.27} = x$

Accepted Answer

Concept & Formula: This expression matches the structure of a standard cubic algebraic identity: a^3 + b^3 = (a + b)(a^2 - ab + b^2) Rearranging this identity to match our rational expression format yields: (a^3 + b^3/a^2 - ab + b^2) = a + b Step-by-Step Solution: • Step 1: Assign variable mappings for the decimal figures: Let a = 0.73 and b = 0.27. • Step 2: Substitute these variables into the given expression structural template: (a^3 + b^3/a^2 + b^2 - ab) • Step 3: Expand the cubic terms in the numerator using the identity: ((a + b)(a^2 - ab + b^2)/(a^2 - ab + b^2)) • Step 4: Cancel the identical algebraic expressions from both the numerator and the denominator: a + b • Step 5: Compute the final numeric sum: 0.73 + 0.27 = 1.00 = 1 Exam Strategy & Shortcut: Algebraic Form Matching: Identify the template (a^3 + b^3/a^2 - ab + b^2) immediately upon looking at the structure. Recognize that this template always simplifies directly down to (a + b). The calculation becomes a simple addition exercise: 0.73 + 0.27 = 1. Common Pitfall: • Long Multiplication of Decimals: Attempting to manually calculate (0.73)^3 or (0.27)^3 leads to multi-digit fractional decimals, wasting critical exam runtime and introducing immediate transcription or precision errors. Final Answer: Therefore, the correct answer is 1.

Question 5

For what value of $x$ the statement $\left( \frac{x}{15} \right) \left( \frac{x}{135} \right) = 1$ is true?

Accepted Answer

Concept & Logic When multiplying fractions involving the same unknown variable, combine the numerators and denominators to form a quadratic equation, then isolate the variable. Step-by-Step Solution • Given: ( (x/15) ) ( (x/135) ) = 1 • Calculation: Multiply the fractions on the left side: (x^2/15 × 135) = 1 • Cross-multiply to isolate x^2: x^2 = 15 × 135 • To avoid large multiplication, factor 135 into smaller, manageable pieces to easily find the square root: x^2 = 15 × (15 × 9) x^2 = 15^2 × 3^2 • Take the square root of both sides: x = 15 × 3 • x = 45 Exam Strategy & Shortcut Instead of multiplying 15 × 135 to get 2025 and then struggling to find its square root, break 135 down immediately into 15 × 9. Taking the square root of (15 × 15 × 9) visually reduces straight to 15 × 3 = 45. Common Pitfall Multiplying the denominators completely (15 × 135 = 2025) and then spending excessive time trying to extract the square root of a large number without a calculator. Final Answer Therefore, the correct answer is 45.

Question 6

$\frac{3}{7}$ of $\frac{5}{4}$ of $3024 = x$ [SBI—Jr. Associates (Pre.) Exam, 2016]

Accepted Answer

Concept & Consecutive Fractions: When finding a fraction "of" a fraction "of" a number, it translates to a continuous multiplication sequence. Value = A × B × C Step-by-Step Solution: • Given expression: (3/7) × (5/4) × 3024 • Combine the fractions first to simplify the multiplication: • (3 × 5/7 × 4) = (15/28) • Now, the expression is: (15/28) × 3024 • Divide 3024 by 28. To do this easier, divide by 4 first, then by 7: • (3024/4) = 756 • (756/7) = 108 • Now, multiply the remaining factors: • 15 × 108 • Calculate: 15 × 100 = 1500, and 15 × 8 = 120. • 1500 + 120 = 1620. Exam Strategy & Shortcut: Look at the unit digits. 15 × 108: The unit digit of 5 × 8 is 0. However, all options end in 0. Let's look at the approximate value: 15 × 100 = 1500. Since 108 > 100, the answer must be greater than 1500. Only one option fits this criteria. Common Pitfall: Trying to multiply 3024 by the numerators (3 and 5) before dividing can lead to very large numbers, increasing the chance of calculation errors. Always divide (cancel out) first. Final Answer: Therefore, the correct answer is 1620.

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