answer:We need to determine the range of values for ( | overrightarrow{O A} | ) given the following conditions: - (overrightarrow{A B_{1}} perp overrightarrow{A B_{2}}) - (|overrightarrow{O B_{1}}| = |overrightarrow{O B_{2}}| = 1) - (overrightarrow{A P} = overrightarrow{A B_{1}} + overrightarrow{A B_{2}}) - (|overrightarrow{O P}| < frac{1}{2}) To solve the problem, follow these detailed steps: 1. **Establish Coordinate System and Definitions:** Let ( P ) be point with coordinates ( (a, b) ), and ( O ) be a point with coordinates ( (x, y) ). 2. **Formulate the Conditions with Coordinate Geometries:** Given the conditions: - (overrightarrow{A B_{1}} perp overrightarrow{A B_{2}}), - (|overrightarrow{O B_{1}}| = |overrightarrow{O B_{2}}| = 1), and - (overrightarrow{A P} = overrightarrow{A B_{1}} + overrightarrow{A B_{2}}). Since ( overrightarrow{A B_{1}} ) and ( overrightarrow{A B_{2}} ) are perpendicular, their sum forms a right triangle. Thus, we can write: [ begin{aligned} & (x - a)^2 + y^2 = 1, quad text{(1)} & x^2 + (y - b)^2 = 1, quad text{(2)} & (x - a)^2 + (y - b)^2 < frac{1}{4}. quad text{(3)} end{aligned} ] 3. **Combine these Equations:** Add (1) and (2), then subtract (3): [ (x - a)^2 + y^2 + x^2 + (y - b)^2 - left( (x - a)^2 + (y - b)^2 right) < 1 + 1 - frac{1}{4}. ] This simplifies to: [ x^2 + y^2 < frac{7}{4}. ] But since we are also taking into account the inequality given by (1) and (2): [ x^2 + y^2 leq 2. ] 4. **Determine the Range of ( | overrightarrow{O A} | ):** The correct range for ( x^2 + y^2 ) is: [ frac{7}{4} < x^2 + y^2 leq 2. ] Therefore, ( | overrightarrow{O A} | = sqrt{x^2 + y^2} ) must be: [ left( frac{sqrt{7}}{2}, sqrt{2} right]. ] # Conclusion: The range of values for ( | overrightarrow{O A} | ) is: [ boxed{left( frac{sqrt{7}}{2}, sqrt{2} right]} ]

answer:For proposition p: To determine whether the function y=x^2-4x+1=(x-2)^2-3 is decreasing on the interval (-infty, a), we examine its derivative. The derivative of y with respect to x is given by y'=2(x-2). For y to be decreasing, y' must be less than or equal to zero, which gives us 2(x-2)leq 0. Therefore, x leq 2. Since this must hold for all x in (-infty, a), we conclude that a leq 2 for proposition p to be true. For proposition q: The function y=log_{(7-2a)}x will be increasing on (0, +infty) if the base of the logarithm, 7-2a, is greater than 1. This gives us the inequality 7-2a > 1, which simplifies to a < 3. Hence, for proposition q to be true, the value of a must satisfy a < 3. Given that p lor q is true and p land q is false, it means that one proposition must be true and the other must be false. There are two cases to consider: 1. Proposition p is true and q is false, which gives us a leq 2 and a geq 3, which is not possible, hence the solution set is emptyset. 2. Proposition q is true and p is false, which gives us a > 2 and a < 3, which satisfactorily produces the interval (2, 3). Therefore, the range of the real number a is boxed{(2, 3)}.

answer:To solve the given system of linear equations with the conditions provided, we will proceed step-by-step: Problem: Given ( n > 2 ), consider the following system of linear equations: [ begin{aligned} & a_{11} x_{1} + a_{12} x_{2} + ldots + a_{1n} x_{n} = 0 & a_{21} x_{1} + a_{22} x_{2} + ldots + a_{2n} x_{n} = 0 & vdots & a_{n1} x_{1} + a_{n2} x_{2} + ldots + a_{nn} x_{n} = 0 end{aligned} ] with the conditions: - (a) Each coefficient ( a_{ij} ) is positive. - (b) The sum of the coefficients in each row and column is 1. - (c) ( a_{11} = a_{22} = ldots = a_{nn} = frac{1}{2} ). We aim to show that the system only has the trivial solution ( x_{i} = 0 ) for all ( i ). Proof: 1. **Assumption and Definitions**: Let's assume that there exists a non-trivial solution. Let ( x_i ) be the variable with the maximum absolute value, say ( |x_i| geq |x_j| ) for all ( j ). Without loss of generality, we can consider ( x_i neq 0 ). 2. **Considering Equation (i)**: From the (i)-th equation of the system, we have: [ a_{ii} x_i + sum_{substack{j=1 j neq i}}^n a_{ij} x_j = 0. ] Taking the absolute value, we get: [ |a_{ii} x_i| = left| sum_{substack{j=1 j neq i}}^n a_{ij} x_j right|. ] 3. **Applying the Triangle Inequality**: Using the triangle inequality, we get: [ |a_{ii} x_i| leq sum_{substack{j=1 j neq i}}^n |a_{ij} x_j|. ] Since ( a_{ij} ) are positive and ( |x_i| geq |x_j| ) for all ( j ): [ sum_{substack{j=1 j neq i}}^n |a_{ij} x_j| leq |x_i| sum_{substack{j=1 j neq i}}^n a_{ij}. ] Therefore: [ |a_{ii} x_i| leq |x_i| sum_{substack{j=1 j neq i}}^n a_{ij}. ] 4. **Substituting the Conditions**: Given that ( a_{ii} = frac{1}{2} ) and the sum of the coefficients in each row is 1: [ sum_{j=1}^n a_{ij} = 1. ] This implies: [ sum_{substack{j=1 j neq i}}^n a_{ij} = 1 - a_{ii} = 1 - frac{1}{2} = frac{1}{2}. ] 5. **Combining Inequalities**: Now, substituting ( a_{ii} = frac{1}{2} ) and ( sum_{substack{j=1 j neq i}}^n a_{ij} = frac{1}{2} ) back into the inequality: [ left|frac{1}{2} x_i right| leq |x_i| cdot frac{1}{2}. ] Hence: [ frac{1}{2} |x_i| leq frac{1}{2} |x_i|. ] 6. **Equality Case**: For this inequality to hold, all terms ( left| a_{ij} x_j right| ) must equal ( left| x_i right| cdot a_{ij} ), implying that ( |x_j| = |x_i| ) for all ( j ). 7. **Analyzing Signs and Magnitudes**: Since the absolute values of all variables are equal, let ( |x_1| = |x_2| = cdots = |x_n| ). Without loss of generality, assume ( x_j = x ) or ( x_j = -x ) for some non-zero ( x ). 8. **Sum of Coefficients Contradiction**: Substituting ( x_1 = x_2 = cdots = x_n = x ) back into the first equation: [ a_{11} x + a_{12} x + ldots + a_{1n} x = x sum_{j=1}^n a_{1j} = x cdot 1 = 0. ] Since ( x neq 0 ): [ sum_{j=1}^n a_{1j}=1, text{ contradiction }. ] Hence, our assumption that there exists a non-trivial solution is false. # Conclusion: The only solution is ( x_i = 0 ) for all ( i ). [ boxed{0} ]

answer:Given that S_{10} = 10a_1 + 45d = 120, it follows that 2a_1 + 9d = 24, Therefore, a_2 + a_9 = (a_1 + d) + (a_1 + 8d) = 2a_1 + 9d = 24. Hence, the correct choice is boxed{B}. By utilizing the formula for the sum of the first n terms of an arithmetic sequence to simplify the given equation, we obtain the value of 2a_1 + 9d. Then, by simplifying the expression for what we are asked to find using the general term formula of an arithmetic sequence and substituting the value of 2a_1 + 9d, we can find the answer. This problem tests the formula for the sum of the first n terms of an arithmetic sequence, the general term formula, and the properties of arithmetic sequences. Mastering these formulas is key to solving this problem.