Question 1

Balls are arranged in rows to form an equilateral triangle. The first row consists of one ball, the second row consists of two balls and so on. If 669 more balls are added, then all the balls can be arranged in the shape of a square and each of the …

Accepted Answer

Concept & Formula: This problem is built on the concept of Arithmetic Progressions and Sum of N Natural Numbers. An equilateral triangle arrangement of balls means the total number of balls is the sum of the first n natural numbers. The formula for the sum of the first n natural numbers is: Total = (n(n+1)/2) The problem requires us to equate this initial sum plus the added balls to the area of a square arrangement, leading to a quadratic equation. Step-by-Step Solution: Given: • Let the number of rows (or balls in the side) of the initial triangle be n. • Initial number of balls = (n(n+1)/2). • Balls added = 669. • Side of the new square = n - 8. • Total balls in square = (n - 8)^2. Calculation: Step 1: Set up the core equation. (n(n+1)/2) + 669 = (n - 8)^2 Step 2: Expand and simplify. n^2 + n + 1338 = 2(n^2 - 16n + 64) n^2 + n + 1338 = 2n^2 - 32n + 128 Step 3: Bring all terms to one side to form a standard quadratic equation. n^2 - 33n - 1210 = 0 Step 4: Factorize the quadratic equation. We need two numbers that multiply to -1210 and add to -33. The factors are -55 and 22. (n - 55)(n + 22) = 0 Since the number of balls cannot be negative, n = 55. Step 5: Find the initial number of balls. Initial balls = (55 × (55+1)/2) = (55 × 56/2) = 55 × 28 = 1540. Exam Strategy & Shortcut: Reverse Engineering from Options: Solving large quadratics like n^2 - 33n - 1210 = 0 can eat up time. Instead, use the options to test the final state. If initial balls = 1540 (Option B). New total after adding 669 = 1540 + 669 = 2209. Check if 2209 is a perfect square. \sqrt{2209} = 47. So the square side is 47. If the square side is 47, the triangle side n should be 47 + 8 = 55. Check if n=55 yields 1540 balls in the triangle: (55 × 56/2) = 1540. It matches perfectly! Option B is correct. Common Pitfall: The most frequent error is misinterpreting "each side of the triangle". Students sometimes confuse this with the perimeter. In a triangular stacking arrangement (like bowling pins), the number of rows is exactly equal to the number of balls in the bottom-most side. Final Answer: Therefore, the correct answer is 1540.

Question 2

If $\frac{a}{b} + \frac{b}{a} = 2$, then the value of $(a - b)$ is

Accepted Answer

Concept & Formula: This problem elegantly masks a core Algebraic Identity: (a - b)^2 = a^2 - 2ab + b^2 By finding a common denominator for the given expression, it transforms directly into a perfect square trinomial, revealing the exact relationship between the variables a and b. Step-by-Step Solution: Given: (a/b) + (b/a) = 2 Calculation: Step 1: Find a common denominator (ab) for the left side of the equation. (a × a + b × b/ab) = 2 (a^2 + b^2/ab) = 2 Step 2: Multiply both sides by ab. a^2 + b^2 = 2ab Step 3: Move all terms to one side to form a quadratic expression. a^2 - 2ab + b^2 = 0 Step 4: Recognize the perfect square identity. (a - b)^2 = 0 Step 5: Take the square root of both sides. a - b = 0 Exam Strategy & Shortcut: The Value Assumption Method: In competitive exams, when you see symmetrical algebraic equations that equal 2, test the assumption that the variables are equal to 1. Let a = 1 and b = 1. Check the condition: (1/1) + (1/1) = 1 + 1 = 2. It satisfies the given condition perfectly! Now, simply plug these values into the target expression: (a - b) = 1 - 1 = 0. This takes less than 3 seconds. Common Pitfall: Students often try to isolate one variable in terms of the other (like a = 2b - (b^2/a)) and end up in endless loops of substitution. Never isolate variables in symmetric fraction sums; immediately cross-multiply to look for standard identities. Final Answer: Therefore, the correct answer is 0.

Question 3

$24.96^2 \div (34.11 \div 20.05) + 67.96 - 89.11 = x$

Accepted Answer

Concept & Strategy: This question is a classic test of Approximation. In banking and competitive exams, exact calculation of such convoluted decimals is mathematically punitive and entirely unnecessary. The strategy is to round every single term to its nearest whole number or highly compatible integer *before* performing any operations. Step-by-Step Solution: Calculation: Step 1: Approximate all values to the nearest sensible integer. • 24.96 \approx 25 • 34.11 \approx 34 • 20.05 \approx 20 • 67.96 \approx 68 • 89.11 \approx 89 Step 2: Rewrite the equation with approximated values. 25^2 \div (34 \div 20) + 68 - 89 = x Step 3: Solve using BODMAS. Focus on the division first. 625 \div ((34/20)) + 68 - 89 625 × ((20/34)) + 68 - 89 Step 4: Simplify the fraction block. 625 × ((10/17)) Notice that 625 \div 17 is roughly 36.7 (since 17 × 30 = 510 and 17 × 6 = 102, 510+102 = 612). So, 36.7 × 10 \approx 367. Step 5: Perform the final addition and subtraction. 367 + 68 - 89 435 - 89 = 346 Exam Strategy & Shortcut: Fraction Scaling: If doing 625 × ((10/17)) feels slow, you can quickly round the ratio itself. 34 \div 20 = 1.7. So, 625 \div 1.7. We know 17 × 4 = 68, so 625 \div 1.7 is a little bit less than 680/1.7 = 400. It is around 360-370. 370 + 68 - 89 = 349. Looking at the options: 884, 346, 252, 424, 366. The only option solidly in this ballpark is 346. Common Pitfall: The trap is attempting to calculate exact decimal values. Multiplying 24.96 × 24.96 will drain 3 minutes of your exam time and likely result in an arithmetic error, leading you to guess wildly. Always boldly round numbers in approximation questions. Final Answer: Therefore, the correct answer is 346.

Question 4

If $\frac{x+1}{x-1} = \frac{a}{b}$ and $\frac{1-y}{1+y} = \frac{b}{a}$, then the value of $\frac{x-y}{1+xy}$ is

Accepted Answer

Concept & Formula: The core mathematical tool for this problem is the rule of Componendo and Dividendo, which states that if (p/q) = (r/s), then (p+q/p-q) = (r+s/r-s). This property allows us to instantly untangle variables trapped in (+1) and (-1) fractional relationships. Step-by-Step Solution: Given: 1) (x+1/x-1) = (a/b) 2) (1-y/1+y) = (b/a) Calculation: Step 1: Apply Componendo & Dividendo to the first equation. ((x+1) + (x-1)/(x+1) - (x-1)) = (a+b/a-b) (2x/2) = (a+b/a-b) \implies x = (a+b/a-b) Step 2: Apply Componendo & Dividendo to the second equation. ((1-y) + (1+y)/(1-y) - (1+y)) = (b+a/b-a) (2/-2y) = (a+b/-(a-b)) \implies -(1/y) = -(a+b/a-b) \implies y = (a-b/a+b) Step 3: Analyze the relationship between x and y. Notice that x = (a+b/a-b) and y = (a-b/a+b). They are perfect reciprocals! Therefore, xy = 1. Step 4: Substitute into the target expression (x-y/1+xy). Since xy = 1, the denominator is 1 + 1 = 2. So we just need to find (x-y/2). Step 5: Calculate (x - y). x - y = (a+b/a-b) - (a-b/a+b) = ((a+b)^2 - (a-b)^2/(a-b)(a+b)) Using the algebraic identity (a+b)^2 - (a-b)^2 = 4ab: x - y = (4ab/a^2 - b^2) Step 6: Final substitution. (x-y/1+xy) = (\frac{4ab/a^2 - b^2)}{2} = (2ab/a^2 - b^2) Exam Strategy & Shortcut: The Value Substitution Method: Instead of heavy algebra, substitute simple numbers. Let a=3 and b=1. From (x+1/x-1) = 3 \implies 3x - 3 = x + 1 \implies 2x = 4 \implies x=2. From (1-y/1+y) = (1/3) \implies 1 + y = 3 - 3y \implies 4y = 2 \implies y = 0.5. Target expression: (2 - 0.5/1 + (2 × 0.5)) = (1.5/2) = (3/4). Now, plug a=3, b=1 into Option A: (2(3)(1)/3^2 - 1^2) = (6/8) = (3/4). It matches perfectly, bypassing all the algebraic manipulation! Common Pitfall: When applying Componendo & Dividendo to (1-y/1+y), students frequently mess up the negative signs, ending up with y = (a+b/a-b) instead of its reciprocal. Careless sign errors in the denominator (1-y) - (1+y) = -2y ruin the entire derivation. Final Answer: Therefore, the correct answer is (2ab/a^2 - b^2).

Question 5

$200 \div 25 	imes 4 + 12 - 3 = x$

Accepted Answer

Concept & Rule: This is a direct application of the BODMAS rule (Brackets, Orders, Division, Multiplication, Addition, Subtraction). When mathematical operators appear in a sequence, they must be resolved in this strict hierarchical order to arrive at the correct result, regardless of how they are arranged from left to right. Step-by-Step Solution: Given: 200 \div 25 × 4 + 12 - 3 Calculation: Step 1: Perform Division (D). (200 \div 25) × 4 + 12 - 3 8 × 4 + 12 - 3 Step 2: Perform Multiplication (M). (8 × 4) + 12 - 3 32 + 12 - 3 Step 3: Perform Addition (A). (32 + 12) - 3 44 - 3 Step 4: Perform Subtraction (S). 44 - 3 = 41 Exam Strategy & Shortcut: Chunking: To solve this rapidly under pressure, mentally group the multiplication and division into one "chunk" and the addition/subtraction into another. Visually process 200 \div 25 × 4 as (200/25) × 4 = 8 × 4 = 32. Then append the simple arithmetic: 32 + 9 = 41. You should not need paper for this question. Common Pitfall: The most frequent error is reading left-to-right but accidentally prioritizing multiplication over division based on visual grouping. A student might incorrectly evaluate 25 × 4 = 100 first, and then do 200 \div 100 = 2, resulting in a final wrong answer of 2 + 12 - 3 = 11. Always divide before you multiply if division comes first sequentially in a chunk! Final Answer: Therefore, the correct answer is 41.

Question 6

Evaluate: $(923 - 347) / x = 32$

Accepted Answer

Concept & Logic: This is a straightforward arithmetic equation relying on basic Subtraction and Division. The strategy is to resolve the brackets first as dictated by BODMAS, and then rearrange the resulting simple linear equation to isolate the unknown variable x. Step-by-Step Solution: Given: (923 - 347/x) = 32 Calculation: Step 1: Evaluate the expression inside the parentheses. 923 - 347 = 576 The equation is now: (576/x) = 32 Step 2: Rearrange to solve for x. Multiply both sides by x and divide by 32: x = (576/32) Step 3: Perform the division. You can simplify by breaking it down: 32 × 10 = 320. Remaining: 576 - 320 = 256. 32 × 8 = 256. Therefore, 10 + 8 = 18. x = 18 Exam Strategy & Shortcut: Unit Digit Tracking: Once you have (576/x) = 32, this means 32 × x = 576. Look purely at the unit digits. 2 × (unit digit of x) = (ends in 6). What single digit multiplied by 2 gives a number ending in 6? It must be either 3 (since 2 × 3 = 6) or 8 (since 2 × 8 = 16). Look at the given options: 35, 20, 18, 15, 40. The only option ending in a 3 or an 8 is 18. You can select it immediately without actually dividing 576 by 32! Common Pitfall: A common unforced error is messing up the initial subtraction due to carrying digits poorly. 923 - 347 might be mistakenly calculated as 586 if a student forgets to borrow properly from the tens and hundreds columns, which throws off the entire subsequent division. Final Answer: Therefore, the correct answer is 18.

Question 7

Direction: What approximate value will come in place of the question mark ($x$) in the following question? (You are not expected to calculate the exact value).  $1559.95 - 7.99 	imes 24.96 - x^2 = 1154$

Accepted Answer

Concept & Strategy: This question evaluates your ability to apply Decimal Approximation combined with the BODMAS rule. In competitive exams, dealing with numbers like 1559.95 wastes time. The strategy is to round each decimal to its nearest whole integer before performing any operations, especially before squaring or finding square roots. Step-by-Step Solution: Calculation: Step 1: Approximate all given values to their nearest integers. • 1559.95 \approx 1560 • 7.99 \approx 8 • 24.96 \approx 25 Step 2: Rewrite the equation using these approximated values. 1560 - 8 × 25 - x^2 = 1154 Step 3: Apply BODMAS (Multiplication before Subtraction). 1560 - 200 - x^2 = 1154 Step 4: Perform the subtraction. 1360 - x^2 = 1154 Step 5: Isolate x^2. x^2 = 1360 - 1154 x^2 = 206 Step 6: Find the approximate square root. We know that 14^2 = 196 and 15^2 = 225. 206 is closer to 196. Since the question asks for an approximate value and 14 is one of our options, x \approx 14. Exam Strategy & Shortcut: Mental Anchor Points: Memorize squares up to 30. Once you reach 1360 - x^2 = 1154, you can quickly estimate 1350 - 1150 = 200. What square is near 200? The answer is 14 (196). You can skip the precise subtraction and instantly lock in option (a) without needing to write down the final algebraic steps. Common Pitfall: The most common trap is attempting to multiply 7.99 × 24.96 exactly. This will lead to an unwieldy decimal (199.4304) that takes a minute to calculate and drastically increases the chance of a careless arithmetic error. Always round first! Final Answer: Therefore, the correct answer is 14.

Question 8

Solve $1\frac{4}{5} + 20 - 280 \div 25 = x$

Accepted Answer

Concept & Rule: This problem is a direct test of the BODMAS rule mixed with Fraction and Decimal Conversions. To solve this efficiently, it is often easier to convert simple mixed fractions and division operations into decimals, perform the linear addition/subtraction, and convert back to a fraction if required by the options. Step-by-Step Solution: Calculation: Step 1: Convert the mixed fraction into a decimal. 1(4/5) = 1 + 0.8 = 1.8 Step 2: Process the division component. 280 \div 25 To divide by 25 easily, multiply the number by 4 and divide by 100. 280 × 4 = 1120 \rightarrow (1120/100) = 11.2 Step 3: Substitute the decimal values back into the equation. 1.8 + 20 - 11.2 = x Step 4: Perform addition and subtraction from left to right. 21.8 - 11.2 = x x = 10.6 Step 5: Convert the final decimal back to a mixed fraction. 10.6 = 10 + (6/10) = 10(3/5) Exam Strategy & Shortcut: Fraction Isolation: Keep whole numbers and fractions separate to avoid creating large improper fractions. Equation: 1 + (4/5) + 20 - (280/25) Simplify (280/25) by dividing by 5: (56/5) Convert (56/5) to a mixed fraction: 11(1/5) Now evaluate: (1 + 20 - 11) + ((4/5) - (1/5)) Whole numbers: 21 - 11 = 10 Fractions: (3/5) Combine: 10(3/5). This skips decimals entirely! Common Pitfall: Students often convert everything into a massive single fraction, calculating (9/5) + (100/5) - (56/5). While this works here because the denominators happen to align easily to 5, it forms a bad habit that wastes time on harder questions. Treat whole numbers and fractional parts independently when adding or subtracting. Final Answer: Therefore, the correct answer is 10(3/5).

Question 9

Directions: What approximate value should come in place of question mark($x$) in the following questions? (NOTE: You are not expected to calculate the exact value).  $421 \div 35 	imes 299.97 \div 25.05 = x^2$

Accepted Answer

Concept & Strategy: This is an Approximation problem involving a chain of multiplication and division. Because multiplication and division have the same precedence in BODMAS (evaluated left-to-right), rounding the numbers to highly compatible multiples ensures smooth cancellation of fractions without needing to compute large intermediary decimals. Step-by-Step Solution: Calculation: Step 1: Approximate the values to nearest compatible integers. • 421 \approx 420 (Since it is easily divisible by 35) • 35 remains 35 • 299.97 \approx 300 • 25.05 \approx 25 Step 2: Rewrite the equation with approximations. 420 \div 35 × 300 \div 25 = x^2 Step 3: Convert the chain of operations into a single fraction for easier visual cancellation. ((420/35)) × ((300/25)) = x^2 Step 4: Simplify the individual parts. (420/35) = 12 (300/25) = 12 Step 5: Multiply the simplified results. 12 × 12 = x^2 144 = x^2 Step 6: Solve for x. x = 12 Exam Strategy & Shortcut: Fraction Scaling: If (420/35) isn't instantly obvious, scale it up or down. Divide both by 7 to get (60/5) = 12. Similarly for (300/25), think of 100 having four 25s. Therefore, 300 has 3 × 4 = 12 twenty-fives. Realizing both blocks equal 12 means you are looking at 12 × 12 = x^2, which immediately gives x=12 without even calculating 144. Common Pitfall: A severe mistake here is evaluating strictly left to right without converting to fractions, like trying to do (420 \div 35) = 12, then 12 × 300 = 3600, and then 3600 \div 25. While it gives the same answer, dealing with larger numbers like 3600 increases the mental load and risk of simple errors compared to keeping the numbers small and modular. Final Answer: Therefore, the correct answer is 12.

Question 10

Directions: What approximate value should come in place of question mark($x$) in the following questions? (NOTE: You are not expected to calculate the exact value).  $19.99 	imes 15.98 + 224.98 + 125.02 = x$

Accepted Answer

Concept & Rule: This question tests Decimal Rounding paired with foundational BODMAS application. By rounding each decimal component to its nearest whole number, the complex arithmetic collapses into a standard primary-school level multiplication and addition problem. Step-by-Step Solution: Calculation: Step 1: Round all values to their nearest integers. • 19.99 \approx 20 • 15.98 \approx 16 • 224.98 \approx 225 • 125.02 \approx 125 Step 2: Rewrite the equation using the approximated integers. 20 × 16 + 225 + 125 = x Step 3: Apply BODMAS and perform the multiplication first. 320 + 225 + 125 = x Step 4: Perform the addition. Group friendly numbers first. 225 + 125 = 350 320 + 350 = 670 Exam Strategy & Shortcut: Complementary Addition: When adding up the final string of numbers 320 + 225 + 125, look for "complements" that round out nicely. The 25 from 225 and the 25 from 125 instantly form a 50, and 200+100=300. So the right block is 350. Adding 320 + 350 takes less than a second mentally. Always look for tens or hundreds pairings before doing column addition. Common Pitfall: The danger here is neglecting BODMAS. A student rushing might add 16 + 225 = 241, and then multiply by 20. Multiplication must always happen before addition, even in approximation problems. Final Answer: Therefore, the correct answer is 670.

Question 11

Evaluate : $123 	imes 999 + 123$

Accepted Answer

Concept & Formula: This problem can be efficiently solved using the Distributive Law of Multiplication over addition: a × b + a × c = a × (b + c) Step-by-Step Solution: • Step 1: Rewrite the expression to make the common factor explicit: 123 × 999 + 123 × 1 • Step 2: Factor out the common term 123: 123 × (999 + 1) • Step 3: Simplify inside the parentheses and multiply: 123 × 1000 = 123000 Exam Strategy & Shortcut: Distributive Property: Never multiply 123 × 999 directly. Recognize that 999 is 1 less than 1000. Adding 123 to 123 × 999 simply completes the group to make exactly 123 × 1000, which instantly gives 123000. Common Pitfall: • Brute Force Multiplication: Spending valuable time performing long multiplication for 123 × 999 increases the risk of calculation errors. Always look for algebraic simplifications when numbers like 99, 999, or 9999 appear. Final Answer: Therefore, the correct answer is 123000.

Question 12

Solve 4376 + 3209 - 1784 + 97 = 3125 + x

Accepted Answer

Concept & Formula: To find the value of x, group all constants on one side of the equation using the standard rules of arithmetic operations (BODMAS). Step-by-Step Solution: • Step 1: Simplify the left-hand side (LHS) of the equation: Add positive numbers first: 4376 + 3209 + 97 = 7682 Subtract the negative number: 7682 - 1784 = 5898 • Step 2: Equate the simplified LHS to the right-hand side (RHS): 5898 = 3125 + x • Step 3: Isolate x by subtracting 3125 from 5898: x = 5898 - 3125 x = 2773 Exam Strategy & Shortcut: Unit Digit Trick: Calculate the unit digit of both sides: LHS unit digit: 6 + 9 - 4 + 7 = 18 - 4 = 14 \rightarrow 4 RHS expression: 5 + x \rightarrow must end in 4. Therefore, the unit digit of x must be 9 (since 5 + 9 = 14). Options (b) and (c) end in 3, option (d) ends in 93. Let's check the tens digits: LHS last two digits: 76 + 09 - 84 + 97 = 98 RHS last two digits: 25 + x \rightarrow 98 - 25 = 73. Thus, the answer must end in 73, which leaves 2773 as the correct choice. Common Pitfall: • Premature Transposition Error: When shifting 3125 to the other side, make sure its sign flips to negative. Accidentally adding it instead of subtracting is a common point of failure under exam stress. Final Answer: Therefore, the correct answer is 2773.

Question 13

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 14

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 15

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 16

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 17

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 18

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 19

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 20

$\frac{3}{7}$ 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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