Mathematics

The tangent function is positive in the first and third quadrants because tangent is the ratio of sine to cosine, and both sine and cosine have the same sign in these quadrants, making their ratio positive.

The symbol ">" is commonly used as a comparison operator in arithmetic and programming to mean "greater than." It is not primarily a logic function, text formatting option, or bullet point.

The number 1 is the only number whose square (1² = 1) and cube (1³ = 1) are equal. Other numbers have different values for their square and cube.

To find the sales tax, multiply the item cost by the tax rate: $2000 × 3% = $2000 × 0.03 = $60. Therefore, the correct tax amount is $60, which corresponds to option B.

Let the two numbers be 5x and 7x. Their sum is 12x. According to the problem, subtracting 5 from the sum gives 79, so 12x - 5 = 79. Solving for x, we get 12x = 84, so x = 7. Therefore, the numbers are 5*7 = 35 and 7*7 = 49, which corresponds to option D.

The least common multiple (LCM) of 70 and 80 is the smallest number divisible by both. The prime factors of 70 are 2, 5, and 7; for 80, they are 2^4 and 5. Taking the highest powers, LCM = 2^4 × 5 × 7 = 16 × 5 × 7 = 560.

The volume of a sphere is given by the formula V = (4/3)πr³. To find the radius r in terms of the volume V, rearrange the formula: r³ = (3V) / (4π). Taking the cube root of both sides gives r = ³√(3V / 4π). This corresponds exactly to option C. Therefore, the correct answer is C, not B.

The expression 'ab/c/cd/a' is ambiguous as written, but if interpreted as (ab/c) ÷ (cd/a), it simplifies to (ab/c) × (a/cd) = a^2 b / c^2 d, which matches option B. However, the original expression is unclear due to missing parentheses and ambiguous notation, so the safest choice is 'None of these' unless clarified.

Twelve men take 6 hours to finish the work, so total work = 12 × 6 = 72 man-hours. After working for 1 hour, 12 men complete 12 man-hours of work, leaving 60 man-hours remaining. Now, 20 men work to complete the remaining 60 man-hours. Time required = 60 ÷ 20 = 3 hours. Therefore, 20 men will take 3 more hours to finish the remaining work, making the correct answer option A (3 hours).

Let the four consecutive two-digit odd numbers be x, x+2, x+4, and x+6. Their sum is 4x + 12. When this sum is divided by 10, the quotient must be a perfect square. We want (4x + 12)/10 = k, where k is a perfect square integer. Thus, 4x + 12 = 10k. Since x is odd and two-digit, testing perfect squares for k shows that k can be 16 or 36. For k=16, sum = 160, so 4x + 12 = 160 => 4x = 148 => x = 37, giving numbers 37, 39, 41, 43. For k=36, sum = 360, so 4x + 12 = 360 => 4x = 348 => x = 87, giving numbers 87, 89, 91, 93. Among the options given (21, 25, 41, 67), only 41 appears in the valid set (37, 39, 41, 43). Therefore, option C (41) is the correct answer.