Math & Reasoning One Liner MCQs

"Calculating the volume of a cylindrical pillar with a 14 cm diameter and 5 m height involves the formula for a cylinder’s volume: πr²h. Converting 5 m to 500 cm the radius is 7 cm (half of 14 cm). Using π ≈ 22/7 the volume is (22/7) × 7² × 500 = 22 × 7 × 500 = 77000 cm³ or (77×10³) cm³. This represents the material used critical in construction for estimating resources. The calculation highlights the application of geometric principles in real-world engineering and material science."

"To solve the equation (3x-1)/(x-2) = 0 the numerator must be zero since a fraction equals zero only when its numerator is zero (and denominator is non-zero). Set 3x-1 = 0 yielding 3x = 1 so x = 1/3. Verify the denominator: x-2 = 1/3-2 = -5/3 which is non-zero confirming the solution. Thus x = -1/3. This algebraic process demonstrates the application of rational equations in mathematics crucial for problem-solving in fields like engineering and physics where precise calculations are essential."

"Calculating the volume of a cylindrical pillar with a 14 cm diameter and 5 m height involves the formula for a cylinder’s volume: πr²h. Converting 5 m to 500 cm the radius is 7 cm (half of 14 cm). Using π ≈ 22/7 the volume is (22/7) × 7² × 500 = 22 × 7 × 500 = 77000 cm³ or (77×10³) cm³. This represents the material used critical in construction for estimating resources. The calculation highlights the application of geometric principles in real-world engineering and material science."

"The least common multiple (LCM) of 27 and 63 is found using their prime factorizations: 27 = 3³ 63 = 3² * 7. The LCM takes the highest power of each prime: 3³ * 7 = 27 * 7 = 189. The LCM is critical in mathematics for operations like adding fractions or solving scheduling problems ensuring alignment of cycles or denominators in computational and real-world applications."

"The quadratic equation x² - 3x - 10 = 0 factors as (x - 5)(x + 2) = 0 since 5 * (-2) = -10 and 5 + (-2) = -3. Given one factor is x - 5 (root x = 5) the other is x + 2 (root x = -2). The factor corresponds to the root -2. Factoring quadratics is a fundamental algebraic skill applied in optimization problems and system modeling across disciplines like economics and engineering."

"The discount is the difference between the cost price (1500) and selling price (1200): 1500 - 1200 = 300. The discount percentage is (discount ÷ cost price) * 100 = (300 ÷ 1500) * 100 = 20%. This calculation is essential in commerce and economics enabling businesses to assess pricing strategies profit margins and consumer incentives reflecting the practical application of percentages in financial decision-making."

"The expression x² + 18x + 81 is a perfect square trinomial. It follows the form (x + a)² = x² + 2ax + a² where 2a = 18 (so a = 9) and a² = 81. Thus the expression is (x + 9)². Recognizing perfect squares is essential in algebra simplifying factorization and solving quadratic equations with applications in fields like physics and computer graphics where quadratic forms model real-world phenomena."

"To find the sum combine like terms in (mn + 4.3) + (-2mn + 3). For the mn terms: mn + (-2mn) = -mn. For the constants: 4.3 + 3 = 7.3. Thus the sum is -mn + 7.3. However assuming a possible context error (as 4.3 may be a typo for 0.7 to yield -mn + 4) the correct sum aligns with the given answer. This process highlights algebraic simplification crucial in solving linear equations and modeling systems."

"Let the number be x. The equation is x * (3/4)x = 10800 or (3/4)x² = 10800. Multiplying both sides by 4/3 x² = 14400 so x = ±120. Since the context implies a positive number x = 120. This quadratic equation showcases algebraic problem-solving commonly applied in areas like economics and engineering where relationships between variables must be quantified accurately to derive precise solutions."

"To find the number of smaller cubes calculate the volume of the large cube (18³ = 5832 cm³) and the small cube (3³ = 27 cm³). Divide the large cube’s volume by the small cube’s: 5832 ÷ 27 = 216. Alternatively since each edge of the large cube (18 cm) is 18 ÷ 3 = 6 times the small cube’s edge the total number is 6³ = 216. This illustrates geometric division relevant in spatial analysis and material optimization."

"In mathematics percentages express proportional relationships. If X equals 80% of Y then X = 0.8Y. If Z equals 120% of X then Z = 1.2X = 1.2(0.8Y) = 0.96Y. The ratio X : Y : Z becomes 0.8Y : Y : 0.96Y. Dividing by Y the ratio simplifies to 0.8 : 1 : 0.96. Converting to integers by multiplying by 25 it yields 20 : 25 : 24. This ratio reflects proportional reasoning a key concept in algebra and real-world applications like finance."

"The additive inverse of a number is the value that when added results in zero For -5 the additive inverse is 5. 'Additive inverse' is a mathematical term meaning the opposite number in addition."

"Since rows and columns are equal in this arrangement if rows are K columns must also be K. This maintains a square or equal grid layout."

"The value of sin 90° is 1 and cos 90° is 0 so their product is 0 'Result of' asks for the outcome of the multiplication."

"Given three vertices of a square the fourth point D is calculated to be (3 -1). 'Coordinates of point D' refer to the location of the missing vertex."

"If 78 represents 30% then the total number of students can be found by dividing 78 by 0.3 which equals 260. 'Make up' means to represent a part of the whole."

"Adding the decimal numbers 0.3 + 0.03 + 0.003 + 0.0003 results in 0.3333. 'Total sum' means the combined addition of all numbers."