Question 1

What is the remainder when $4^{96}$ is divided by 6?

Accepted Answer

Concept & Formula: This problem deals with the cyclicity and patterns of remainders (modular arithmetic). Specifically, we explore the properties of powers of 4 when divided by 6. 4^n \equiv constant remainder \pmod 6 Step-by-Step Solution: • Step 1: Let's analyze the remainder generated by small powers of 4 when divided by 6: • 4^1 = 4 \rightarrow 4 \div 6 leaves a remainder of 4 • 4^2 = 16 \rightarrow 16 \div 6 = 2 with a remainder of 4 • 4^3 = 64 \rightarrow 64 \div 6 = 10 with a remainder of 4 • Step 2: Observe the clear rule: Any positive integer power of 4 divided by 6 always yields a remainder of 4. • Step 3: Apply this rule to 4^{96}. Since 96 is a positive integer, the remainder remains unchanged. Exam Strategy & Shortcut: Property Pattern: Remember this standard number theory property: 4^n \pmod 6 = 4 \quad for all n \in \mathbb{Z}^+ Since the exponent 96 is a positive integer, you can instantly mark the answer as 4 without doing any complex modular reduction. Common Pitfall: • Over-complicating with Totient/Fermat Theorem: Students sometimes try to apply Euler's totient function or Fermat's Little Theorem directly. However, because 4 and 6 are not co-prime (\gcd(4,6) = 2), standard basic theorems do not apply directly, leading to confusing breakdowns if the base relationship isn't noticed. Final Answer: Therefore, the correct answer is 4.

Question 2

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 3

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 4

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 5

If $0.13 \div p^2 = 13$, then $p$ equals

Accepted Answer

Concept & Formula Rearrange the basic division equation to isolate the squared variable, then apply decimal division rules before extracting the square root. Step-by-Step Solution • Given: (0.13/p^2) = 13 • Calculation: Multiply both sides by p^2 and divide by 13 to isolate p^2: p^2 = (0.13/13) • Perform the division. You can rewrite the decimal as a fraction to make it visually easier: p^2 = (\frac{13/100)}{13} p^2 = (13/13 × 100) p^2 = (1/100) p^2 = 0.01 • Take the square root of both sides to find p: p = \sqrt{0.01} p = 0.1 Exam Strategy & Shortcut Look at the significant digits. 13 divided by something becomes 13, meaning the divisor must just be shifting decimal places. Specifically, 0.13 has to be divided by 0.01 to shift the decimal two places to the right to become 13. Thus, p^2 = 0.01. The square root of 0.01 is 0.1. Common Pitfall Failing to properly account for the decimal places when extracting the root, often mistakenly choosing option (a) 0.01 because they forget that p^2 = 0.01, not p. Final Answer Therefore, the correct answer is 0.1.

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.

Question 7

Directions: Study the table and answer the given questions.  | Months | Gross Revenue | Amount Allocated for Commission | Amount Allocated for discount and offer | Net Revenue | |---|---|---|---|---| | March | ₹ 360000 | ₹ 31200 | — | — | | April | …

Accepted Answer

Concept & Formula: The problem requires calculating two missing values using the provided algebraic relationship and then finding their difference. Gross = Net + Commission + Discount Net = Gross - Commission - Discount Step-by-Step Solution: 1. Calculate April Net Revenue: - Net Revenue = 320000 - 28000 - 16000 - Net Revenue = 276000 2. Calculate June Gross Revenue: - Gross Revenue = 330000 + 42000 + 30000 - Gross Revenue = 402000 3. Find the Difference: - Difference = 402000 - 276000 = 126000 Exam Strategy & Shortcut: Work systematically column by column. Compute April's net (320k - 44k = 276k) and June's gross (330k + 72k = 402k) mentally, then perform the final subtraction. Common Pitfall: Extracting the wrong data row (e.g., May instead of June) or confusing Gross and Net calculations will lead to an incorrect result. Final Answer: Therefore, the correct answer is ₹ 126000.

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