Math & Reasoning One Liner MCQs

"The inequality |x-3| > 10 implies x < -7 or x > 13 as it represents values of x whose distance from 3 is greater than 10. Solving we get x-3 < -10 or x-3 > 10 yielding the solution set. The absolute value of x |x| depends on x’s magnitude. For x = -15 (within x < -7) |x| = 15; for x = 14 (within x > 13) |x| = 14. Among options 8 fits as a possible |x| for certain x values aligning with the inequality’s solution."

"To find the square of the distance between points A(4 -1) and B(3 5) use the distance formula: distance = √[(x₂-x₁)² + (y₂-y₁)²]. Here (x₁ y₁) = (4 -1) and (x₂ y₂) = (3 5). Calculate: (3-4)² + (5-(-1))² = (-1)² + (6)² = 1 + 36 = 37. The square of the distance is 37. This application of the distance formula is essential in geometry navigation and computer graphics for determining spatial relationships and optimizing designs."

"To solve the system of equations x+y=6 and 4x-y=4 add the equations to eliminate y: (x+y) + (4x-y) = 6 + 4 yielding 5x = 10 so x = 2. Substituting x = 2 into x+y=6 gives 2+y=6 so y=4. This method called elimination is a fundamental algebraic technique for solving simultaneous equations. Such systems are critical in mathematics economics and engineering for modeling relationships between variables ensuring accurate solutions to real-world problems."

"To calculate the distance a car travels at 55 km/h in 12 minutes convert time to hours: 12 minutes = 12/60 = 0.2 hours. Using the formula distance = speed × time we get 55 × 0.2 = 11 km. This calculation demonstrates the application of uniform motion principles widely used in physics and engineering to determine travel distances. Understanding such computations is essential for real-world applications like navigation logistics and vehicle performance analysis ensuring accurate planning and efficiency."

"To find the money Akbar has left after spending $8.4 from $12 subtract: 12 - 8.4 = 3.6. Thus Akbar has $3.6 remaining. This basic arithmetic operation is essential in financial literacy budgeting and everyday calculations. Understanding subtraction in monetary contexts ensures accurate financial planning and resource management applicable in personal finance business accounting and economic modeling where precise calculations are critical for decision-making and maintaining fiscal responsibility."

"A circle can have infinite tangents as a tangent line touches the circle at exactly one point and can be drawn from any point on the circle’s circumference. Each point on the circle supports a unique tangent and since a circle has infinitely many points the number of tangents is infinite. This geometric property is fundamental in mathematics physics and engineering particularly in applications like optics and motion analysis where tangent lines model interactions with circular objects or paths."

"To find the simple interest rate use the formula: Simple Interest (SI) = Principal × Rate × Time / 100. Given Principal = 12500 Amount = 15500 Time = 4 years SI = 15500 - 12500 = 3000. Thus 3000 = 12500 × Rate × 4 / 100. Solving Rate = (3000 × 100) / (12500 × 4) = 6%. This calculation is vital in finance for determining loan or investment returns aiding in economic planning and understanding interest-based financial systems."

"The sequence 6 9 18 45 126 follows a pattern where each term is derived by multiplying the previous term by an increasing integer and adding a constant: 6×1+3=9 9×2+0=18 18×2.5+0=45 45×2.8+0≈126 126×2.9286+0≈369. The next number is 369. Sequence analysis is crucial in mathematics and computer science for modeling patterns forecasting trends and solving problems in algorithms finance and data science where recognizing patterns drives predictive accuracy."

"The series 1 1 2 6 24 represents factorials (1! 2! 3! 4! 5!) making the next number 6! = 120. This pattern illustrates the mathematical concept of factorials used in probability and combinatorics. Understanding the number 120 in this context highlights the application of factorials in advanced mathematics a key topic in algebra and statistics."

"Adding 1 to the largest four-digit number 9999 yields 10000 a five-digit number. This simple arithmetic operation demonstrates the transition between numerical place values. Understanding the calculation of 10000 highlights fundamental number theory and place value concepts a critical topic in basic mathematics education."

"Dividing 39 by 26 yields 1.5 which simplifies to the fraction 3/2. This calculation demonstrates basic fraction reduction and division principles. Understanding the value 3/2 highlights the importance of simplifying fractions in mathematical computations a fundamental topic in arithmetic and algebra studies."

"The series 19 26 33 40 47 follows an arithmetic pattern with a common difference of 7 but 53 breaks this pattern. Identifying 53 as the odd number demonstrates pattern recognition skills. Understanding this anomaly highlights the importance of sequences in mathematical analysis a key topic in number theory and algebra."

"A cuboidal tank with a capacity of 50000 liters (50 m³) and dimensions of length 2.5 m and depth 10 m has a breadth of 50 ÷ (2.5 × 10) = 2 m. This calculation applies the volume formula for cuboids. Understanding the breadth of 2 m highlights the practical use of geometry in real-world applications like engineering a key topic in mathematics."

"Solving |n − 2| = 10 yields n = 12 or n = −8 with their sum being 4. This absolute value equation demonstrates linear algebra techniques. Understanding the sum 4 highlights the application of absolute values in solving equations a critical topic in algebra and mathematical problem-solving."

"A book’s discount price is 80% of the retail price (100% − 20%). James buys it at 70% of the discount price (80% × 70% = 56% of retail). This percentage calculation showcases compound discounts. Understanding the 56% value highlights practical applications of percentage calculations in commerce a key topic in financial mathematics."

"If a sum triples in 20 years at simple interest it doubles in half that time 10 years as the interest rate is constant. This demonstrates simple interest principles. Understanding the time of 10 years highlights the application of interest formulas in financial mathematics a critical topic in economics and math studies."

"The sequence 5 25 125 625 follows a geometric pattern where each term is multiplied by 5. The nth term is expressed as Tₙ = 5ⁿ where n is the term position. For example T₁ = 5¹ = 5 and T₂ = 5² = 25. This formula is key in geometric sequences used in mathematics and finance. Understanding the nth term formula enhances problem-solving in pattern analysis and predictive modeling."

"The curved surface area of a hemisphere with radius 7 cm is calculated using the formula 2πr². Substituting r = 7 gives 2 × π × 7² = 2 × π × 49 ≈ 308 cm² (using π ≈ 3.14). This formula accounts for the hemisphere’s outer surface excluding the base. Understanding this calculation is essential in geometry for analyzing three-dimensional shapes applied in engineering and design for precise measurements."

"To find the smallest of three consecutive integers summing to 33 let the integers be x x+1 and x+2. Their sum x + (x+1) + (x+2) = 33 simplifies to 3x + 3 = 33. Solving 3x = 30 gives x = 10. Thus the smallest integer is 10. This algebraic approach is fundamental for solving sequence problems applicable in mathematics and real-world scenarios like scheduling."

"Simplifying (x² - x - 20) / (x² - 4) involves factoring the numerator to (x + 5)(x - 4) and the denominator a difference of squares to (x + 2)(x - 2). The simplified form is (x + 5)(x - 4) / (x + 2)(x - 2). This process is fundamental in algebra aiding in solving rational expressions critical for mathematical problem-solving in various fields."