Algebra and Arithmetic

The correlation coefficient r ranges from -1 to +1, where +1 represents a perfect positive linear relationship between two variables. When r = 1, all data points fall exactly on an upward-sloping line. A value of 0 indicates no linear relationship, while -1 indicates a perfect negative relationship.

Substituting x = 1/9: Column A = √(1/9) − (1/9)² = 1/3 − 1/81 = 27/81 − 1/81 = 26/81 ≈ 0.321. Since 26/81 > 0, Column A is greater than Column B. The statement that both columns are equal is false.

The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the longest side, C) is equal to the sum of the squares of the other two sides (a and b). This is expressed as c^2 = a^2 + b^2.

The minimum possible values for x1 through x10 are 1, 2, 3, ..., 10 (since they must be distinct positive integers in increasing order). The sum of 1+2+...+10 = 55. Since the integers could be arbitrarily large, there's no upper bound. Therefore S >= 55.

Using the identity a²-b² = (a+b)(a-b), we have 18 = (a+b)(9). Therefore a+b = 18/9 = 2. Option B is correct.

For any positive real number y, y + 1/y >= 2 (AM-GM inequality). Since 0 < y < 1, y is a fraction and 1/y is greater than 1. Their sum will always be greater than 2. It is not necessarily greater than 4 (e.g., y=0.5, sum=2.5).

The set of whole numbers includes 0 (0, 1, 2, 3...). Any product that includes zero as a factor results in zero. Therefore, the product of all whole numbers is 0, regardless of how many positive integers follow.

AB = (3 + √7)(3 - √7) = 3² - (√7)² = 9 - 7 = 2. These are conjugates, and their product follows the difference of squares pattern (a + b)(a - b) = a² - b². This eliminates the radical term.

Raising a number to the power of one-half is equivalent to taking its square root. Only 'square numbers' (like 1, 4, 9, 16) result in whole numbers when their square root is taken.

When a number is raised to the power of one-half (which means taking its square root), it results in a whole number only if the original number is a perfect square. Square numbers (1, 4, 9, 16, 25, etc.) are the set of positive integers whose square roots are whole numbers.

To solve this question, the user needs to know how to manipulate equations with variables and fractions.

We are given the equation:

$$(a-b)/3.5 = 4/7$$

We can simplify this equation by cross-multiplying and solving for a-b:

$$(a-b) = (4/7)*3.5$$

$$(a-b) = 2$$

Now, we have the equation a-b = 2, which means that a is two more than b. Therefore, b < a.

Option B is correct because b is always less than a in this equation.

Option A is incorrect because b is not equal to a.

Option C is incorrect because b is always less than a in this equation.

Option D is incorrect because b is always less than a in this equation.

The Answer is: B. b < a.

Rearranging: a² + 10b² + c² - 6ab - 2bc = 0. This factors as (a-3b)² + (b-c)² = 0. For sum of squares to be zero, each term must be zero: a - 3b = 0 (so a = 3b), and b - c = 0 (so c = b). From these: c = a/3. Statements I, III, and VI are true. Option D is correct.

The logic uses the reverse alphabetical position (Z=1, Y=2... A=26). Z(1) + P(11) = 12. Therefore, A(26) + K(16) = 42. Distractors are wrong because they use standard alphabetical order or incorrect math.