answer:A mathematical expression involving a fraction is not defined when the denominator is equal to zero. We need to determine the values of (x) which make the denominator (x^2 - 21x + 110 = 0). - **Factor or find the roots of the quadratic equation:** We start by factoring the quadratic: [x^2 - 21x + 110 = (x - 10)(x - 11) = 0.] - **Solve for (x):** This equation implies that (x - 10 = 0) or (x - 11 = 0), hence: [x = 10 quad text{or} quad x = 11.] These two values are where the denominator turns zero, therefore making the expression undefined. - **Conclusion:** The expression is undefined for (x = 10) and (x = 11), so we conclude: [boxed{x = 10 text{ or } x = 11}.]

answer:1. We start by analyzing the given sets ( A ), ( B ), and ( C ). 2. The set ( A ) is defined as: [ A = {(x, y) mid a x + y = 1, x, y in mathbb{Z}} ] This set represents the integer lattice points on the line ( a x + y = 1 ). 3. The set ( B ) is defined as: [ B = {(x, y) mid x + a y = 1, x, y in mathbb{Z}} ] This set represents the integer lattice points on the line ( x + a y = 1 ). 4. The set ( C ) is defined as: [ C = left{(x, y) mid x^{2} + y^{2} = 1right} ] The elements of ( C ) are the integer points on the unit circle centered at the origin. These points are: [ (1, 0), (-1, 0), (0, 1), (0, -1) ] 5. We are asked to find the value of ( a ) such that the set ( (A cup B) cap C ) contains exactly four elements. 6. Given: [ (A cup B) cap C = (A cap C) cup (B cap C) ] 7. Since ( A cup B ) must intersect ( C ) to form exactly four elements, and because ( C ) contains only four points, each point in ( C ) must be included within either ( A cap C ) or ( B cap C ). 8. It implies that ( A cap C ) and ( B cap C ) each must have exactly two elements, and together they must cover all four elements of ( C ). 9. Evaluating the intersections: - For ( (1, 0) ) to be in ( A cap C ) or ( B cap C ): [ a cdot 1 + 0 = 1 Rightarrow a = 1 ] [ 1 + a cdot 0 = 1 Rightarrow a = 1 ] - However, for ( C = {(1, 0), (-1, 0), (0, 1), (0, -1)} in (A cup B) cap C ), we need: 10. The lines ( a x + y = 1 ) and ( x + a y = 1 ) should neither overlap nor intersect at integer points other than those in ( C ). 11. Therefore, solving for distinct lines without overlap or multiple intersections: - Consider ( a x + y = 1 ) and ( x + a y = 1 ) intersecting at ( (1,0), (-1,0), (0,1), (0,-1) ) only. - The slopes of the lines must satisfy ( a neq 1 ) and ( -a = -frac{1}{a} Rightarrow a = -1 ) or ( a = 1 ). 12. Since ( a = 1 ) results in overlapping lines, ( a neq 1 ). 13. Therefore, the valid solution for non-overlapping and distinct intersection points is: [ a = -1 ] # Conclusion: [ boxed{-1} ]

answer:If P holds, for example, when x=4 and y=1, then Q does not hold. Conversely, if x=2 and y=3, then the statement x+y=5 is true. Therefore, if x+y neq 5, then either x neq 2 or y neq 3 is true. Thus, P is a sufficient but not necessary condition for Q. Hence, the correct choice is boxed{text{A}}.

answer:# Problem: Find any four distinct natural numbers that have the following property: if you add the product of any two of them to the product of the other two, you get a prime number. We will verify that the numbers 1, 2, 3, and 5 satisfy the given condition by checking the required expressions for primality. 1. Consider the numbers (a = 1), (b = 2), (c = 3), and (d = 5). We need to verify the following expressions: [ ab + cd ] [ ac + bd ] [ ad + bc ] 2. Calculate the first expression (ab + cd): [ ab + cd = (1 cdot 2) + (3 cdot 5) = 2 + 15 = 17 ] Since 17 is a prime number, this expression satisfies the condition. 3. Calculate the second expression (ac + bd): [ ac + bd = (1 cdot 3) + (2 cdot 5) = 3 + 10 = 13 ] Since 13 is a prime number, this expression satisfies the condition. 4. Calculate the third expression (ad + bc): [ ad + bc = (1 cdot 5) + (2 cdot 3) = 5 + 6 = 11 ] Since 11 is a prime number, this expression satisfies the condition. Since all three expressions yield prime numbers, the set ({1, 2, 3, 5}) is a correct answer to the problem. # Conclusion: Thus, four distinct natural numbers that meet the required property are: [ boxed{1, 2, 3, 5} ]