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question:The following are four statements: (1) The negation of the proposition There exists an x_0 in mathbb{R} such that x_0^2 x_0 > 0 is For all x in mathbb{R}, x^2 x leq 0; (2) Given that the proposition p land q is false, then both p and q are false; (3) The contrapositive of the proposition If x^2 = 1, then x = 1 is If x^2 = 1, then x neq 1; (4) x = 1 is a necessary but not sufficient condition for x^2 5x 6 = 0; How many of these statements are correct? A: 1 B: 2 C: 3 D: 4
answer:For statement (1), the negation of the proposition "There exists an x_0 in mathbb{R} such that x_0^2 - x_0 > 0" is correctly given as "For all x in mathbb{R}, x^2 - x leq 0". Therefore, this statement is correct. For statement (2), if the proposition p land q is false, then it means at least one of p or q must be false, not necessarily both. Hence, this statement is incorrect. For statement (3), the contrapositive of "If x^2 = 1, then x = 1" is "If x neq 1, then x^2 neq 1". Therefore, the given statement is incorrect as it confuses the contrapositive with negation. For statement (4), if x = -1, then it makes the equation (-1)^2 - 5(-1) - 6 = 0 true. However, x = -1 is not a necessary condition for the equation to be true, as x = 6 also satisfies it. Therefore, "x = -1" is a sufficient but not necessary condition for "x^2 - 5x - 6 = 0". Consequently, this statement is incorrect. In summary, only one of the given statements is correct. The correct choice is: [boxed{A}]
question:A small publishing company is planning to publish a new book. The production costs will include onetime fixed costs and variable costs. The onetime fixed costs will amount to 35,630 dollars. The variable costs will be 11.50 dollars per book. The publisher will sell the finished product to bookstores at a price of some amount per book. The publisher must print and sell 4072 books so that the production costs will equal the money obtained from sales. What is the selling price per book?
answer:To find the selling price per book, we need to calculate the total production costs and then divide that by the number of books that need to be sold to break even. The total production costs consist of the one-time fixed costs and the variable costs per book multiplied by the number of books. Fixed costs = 35,630 Variable costs per book = 11.50 Number of books to break even = 4072 Total production costs = Fixed costs + (Variable costs per book * Number of books) Total production costs = 35,630 + (11.50 * 4072) Now, let's calculate the total variable costs: Total variable costs = 11.50 * 4072 Total variable costs = 46,828 Now, let's add the fixed costs to get the total production costs: Total production costs = 35,630 + 46,828 Total production costs = 82,458 To break even, the money obtained from sales must equal the total production costs. Therefore, the selling price per book must be such that when multiplied by the number of books (4072), it equals the total production costs. Let's denote the selling price per book as "P." 4072 books * P (selling price per book) = 82,458 Now, we solve for P: P = 82,458 / 4072 P = 20.25 Therefore, the selling price per book must be boxed{20.25} to break even.
question:he number 0.8 is how much greater than 1 / 2 ?
answer:To compare 0.8 to 1/2, we first need to express both numbers with a common denominator or in decimal form. Since 0.8 is already in decimal form, we can convert 1/2 to decimal form: 1 / 2 = 0.5 Now we can subtract 0.5 from 0.8 to find out how much greater 0.8 is: 0.8 - 0.5 = 0.3 So, the number 0.8 is boxed{0.3} greater than 1/2.
question:Randomly select 5 employees from 45 in a unit to participate in a community service activity, determining these 5 employees by using a random number table method. Below is a portion of the random number table excerpted: 16 22 77 94 39 49 54 43 54 82 17 37 93 23 78 87 35 20 96 43 84 42 17 53 31 57 24 55 06 88 77 04 74 47 67 21 76 33 50 25 Starting from the number in the 5th column and the 6th row of the first line of the random number table, select two numbers sequentially from left to right. The number for the 5th employee selected is ( ) A: 23 B: 37 C: 35 D: 17
answer:To determine the 5th employee's number using the random number table, we start from the 5th and 6th digits in the first row and select two-digit numbers consecutively from left to right. Starting at the digits `39` and then `49`, do not select because `49` exceeds the total number of employees (45). Move to the next number: - `54` (also exceeds the total number of employees) - `43` - `54` (duplicate, already considered) - `82` (exceeds the total number of employees) - `17` - `37` Here, we finally have five distinct numbers within the range 1 to 45, which correspond to the employee numbers. The fifth number selected is `23`. Thus, the identification number of the 5th employee is boxed{23}. The "random number table method" is a simple random sampling technique. When using the random number table, one can read the numbers from left to right, right to left, top to bottom, etc. It is important to note that the same data appearing in the reading process should only be taken once and numbers exceeding the sample range must be omitted. This problem primarily tests the method of sampling and the use of the random number table. Candidates should not overlook this. Each number in the random number table has equal probability to appear in each position, which means each number has an equal probability of being selected.