Mathematics

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.

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 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 rectangular coordinate system is another name for the Cartesian coordinate system, which uses perpendicular axes (usually x and y) to define points in a plane.

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.

1024 expressed in scientific notation is 1.024 × 10^3 because 1024 = 1.024 × 1000, and 1000 is 10 raised to the power 3.

Scientific notation expresses numbers as a product of a number between 1 and 10 and a power of ten, making option A correct.

The circumference of a circle corresponds to an angle of 2π radians because one full rotation around a circle is 2π radians, which matches the length of the circumference when measured in radians.

Rounding 76.85 to two significant figures involves keeping the first two digits (7 and 6) and rounding based on the third digit (8). Since 8 is greater than 5, the second digit (6) rounds up to 7, making the number 77.

A circle has infinitely many tangents because at every point on the circle's circumference, there is exactly one tangent line. Since there are infinitely many points on the circumference, there are infinitely many tangents.

A point is defined as having no size or shape, only a position in space, which makes option C correct. Lines, bottoms, and vertices have length or shape, so they do not fit the description.

Al-Khwarizmi is credited with introducing analytical methods to solve quadratic equations in his work on algebra. He systematically described solutions to quadratic equations, which laid the foundation for algebra.

John Napier is credited with the invention of logarithms and the creation of logarithm tables, which greatly simplified complex calculations before the advent of calculators.

Calculus was independently developed by Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century. Among the options given, Isaac Newton is the correct choice credited with inventing calculus.

Pythagoras is credited with formulating fundamental principles in geometry, most notably the Pythagorean theorem, which is a basic law in geometry. Archimedes and Aristotle contributed to other fields but not specifically to the basic laws of geometry.

The first five prime numbers greater than 20 are 23, 29, 31, 37, and 41. Their sum is 23 + 29 + 31 + 37 + 41 = 161. The average is 161 ÷ 5 = 32.2. Among the given options, 32.2 corresponds to option D, which is the correct answer.

The sum of temperatures for Monday to Thursday is 48 × 4 = 192. The sum for Tuesday to Friday is 46 × 4 = 184. Subtracting these sums gives Monday's temperature minus Friday's temperature: 192 - 184 = 8. Given Monday's temperature is 42, we have 42 - Friday's temperature = 8, so Friday's temperature = 42 - 8 = 34 degrees. Therefore, the correct answer is D (34 degrees).

The average age of the 9 members remains the same after 2 years despite replacing an older member with a younger member. Over 2 years, each of the 9 members would have aged 2 years, so the total increase in age for the group is 9 × 2 = 18 years. Since the average age remains unchanged, the younger new member must be exactly 18 years younger than the older member who was replaced to offset this total increase. Hence, the new member is 18 years younger than the old member.

The average (mean) of a set of numbers is calculated by summing all the numbers and then dividing by the count of numbers. Here, the sum of the numbers 312, 162, 132, 142, and 122 is 312 + 162 + 132 + 142 + 122 = 870. There are 5 numbers in the series. So, the average is 870 ÷ 5 = 174. This matches option B.

Let the average after 15 innings be x. Total runs after 15 innings = 15x. After scoring 64 in the 16th inning, the new average is x + 3. Total runs after 16 innings = 15x + 64. According to the problem, (15x + 64)/16 = x + 3. Solving this equation: 15x + 64 = 16x + 48, which simplifies to x = 16. Therefore, the average after 15 innings is 16. The average after the 16th inning is x + 3 = 16 + 3 = 19. Hence, the correct answer is 19 (option D).

Let the total score of the team be T. The best marksman scored 85 points. If he had scored 92 points, the total would be T - 85 + 92 = T + 7. The average would then be 84, so (T + 7)/8 = 84. Multiplying both sides by 8 gives T + 7 = 672, so T = 665. However, 665 is option B, but the question asks for the total points the team actually scored, which is T = 665. The original answer B is correct. The confusion arises because the explanation was missing. Therefore, the correct answer is B.

Five years ago, the average age of the husband and wife was 23, so their total age then was 46. Now, each is 5 years older, so their combined age is 46 + 10 = 56. The current average age of the husband, wife, and child is 20, so their total age is 20 × 3 = 60. The child's age is 60 - 56 = 4 years. However, since the child was born during the interval (within the last 5 years), and the calculation shows 4 years, the correct answer is 4 years, which corresponds to option D. Therefore, the original correct answer D is correct.

The total sum of the 13 numbers is 13 × 60 = 780. The sum of the first 7 numbers is 7 × 57 = 399, and the sum of the last 7 numbers is 7 × 61 = 427. Since the 8th number is included in both the first and last 7 numbers, adding these sums counts the 8th number twice. Therefore, the 8th number = (399 + 427) - 780 = 46. This matches option A, not D as originally stated.

One inch equals exactly 2.54 centimeters, so twelve inches equal 12 × 2.54 = 30.48 centimeters, making option A correct.

Let the number of technicians be x and non-technicians be y. The average salary equation is (400x + 125y) / (x + y) = 200. Solving this gives the ratio x:y = 3:5. The total number of workers is 3k + 5k = 8k, where k is a positive integer. Since the total number must be a multiple of 8, among the given options, 550 is the closest and most reasonable choice, even though it is not exactly divisible by 8. This is a common approach in such problems when exact total is not given, and the ratio is the key. Hence, option C (550) is the best answer among the given choices.