answer:**Analysis** This problem tests our understanding of the arithmetic rules of complex numbers and the concept of the modulus of a complex number. By applying the arithmetic rules of complex numbers, we can solve this basic problem. **Step-by-step solution** Given (1+2i)z=(1-i), First, we multiply both sides of the equation by the conjugate of (1 + 2i), which is (1 - 2i), to eliminate the imaginary part on the left side: (1 - 2i)(1 + 2i)z = (1 - i)(1 - 2i). This simplifies to: 5z = -1 - 3i. Now, isolate z: z = -dfrac{1 + 3i}{5}. To find the modulus of z, denoted as |z|, we calculate the square root of the sum of the squares of the real and imaginary parts: |z| = sqrt{(-dfrac{1}{5})^2 + (dfrac{3}{5})^2} = boxed{dfrac{sqrt{10}}{5}}.

answer:1. **Using the given equation for a triangle:** The problem states that in a triangle, using standard notations for sides and angles, we have a cdot operatorname{tg} alpha + b cdot operatorname{tg} beta = (a + b) operatorname{tg} frac{alpha + beta}{2}. 2. **Convert the sides to sines:** We know from trigonometry that each side of the triangle can be expressed in terms of the circumradius R and the sine of the angle opposite to it: a = 2R sin alpha quad text{and} quad b = 2R sin beta. Using this, we substitute into the original equation: 2R sin alpha cdot operatorname{tg} alpha + 2R sin beta cdot operatorname{tg} beta = (2R sin alpha + 2R sin beta) operatorname{tg} frac{alpha + beta}{2}. 3. **Divide through by 2R:** Simplify the equation by dividing both sides by 2R: sin alpha cdot operatorname{tg} alpha + sin beta cdot operatorname{tg} beta = (sin alpha + sin beta) operatorname{tg} frac{alpha + beta}{2}. 4. **Re-arrange the equation:** Bring terms involving operatorname{tg} frac{alpha + beta}{2} to the left-hand side: sin alpha cdot (operatorname{tg} alpha - operatorname{tg} frac{alpha + beta}{2}) + sin beta cdot (operatorname{tg} beta - operatorname{tg} frac{alpha + beta}{2}) = 0. 5. **Use the tangent subtraction identity:** Apply the identity: operatorname{tg} x - operatorname{tg} y = frac{sin (x-y)}{cos x cdot cos y} to both terms: sin alpha cdot frac{sin frac{alpha - beta}{2}}{cos alpha cdot cos frac{alpha + beta}{2}} + sin beta cdot frac{sin frac{alpha - beta}{2}}{cos beta cdot cos frac{alpha + beta}{2}} = 0. 6. **Factor out common terms:** Notice that sin frac{alpha - beta}{2} appears in both terms, so factor it out: sin frac{alpha - beta}{2} left( frac{sin alpha}{cos alpha cdot cos frac{alpha + beta}{2}} - frac{sin beta}{cos beta cdot cos frac{alpha + beta}{2}} right) = 0. 7. **Multiply through by cos frac{alpha + beta}{2}:** Simplifying further, we get: frac{sin alpha}{cos alpha} cdot sin frac{alpha - beta}{2} - frac{sin beta}{cos beta} cdot sin frac{alpha - beta}{2} = 0. This simplifies to: sin frac{alpha - beta}{2} (operatorname{tg} alpha - operatorname{tg} beta) = 0. 8. **Determine conditions for equality:** The product being zero implies that at least one of the factors must be zero: sin frac{alpha - beta}{2} = 0 quad text{or} quad operatorname{tg} alpha - operatorname{tg} beta = 0. 9. **Evaluate the conditions:** - If sin frac{alpha - beta}{2} = 0, then frac{alpha - beta}{2} = k pi, where k is an integer. Since alpha and beta are angles in a triangle, this simplifies to alpha = beta. - If operatorname{tg} alpha = operatorname{tg} beta, then alpha = beta mod pi. Given alpha, beta are angles in a triangle where each angle is less than 180^circ, this implies alpha = beta. 10. **Conclusion:** Since both conditions lead to alpha = beta, we conclude that the triangle is isosceles, with sides opposite these angles being equal. [ boxed{The , triangle , is , isosceles.} ]

answer:**Analysis** This question examines the probability of equally likely events. It can be solved by first calculating the probability of the complementary event, which is a basic problem. **Solution** When a uniform coin is tossed three times, there are a total of 2^{3}=8 outcomes. There is 1 outcome where all three coins show tails, and 1 outcome where all three coins show heads; therefore P(0 < X < 3)=1-dfrac{2}{8} =0.75. Therefore, the correct answer is boxed{C}.

answer:Given that -x^{m+3}y^{6} and 3x^{5}y^{2n} are like terms, for the terms to be considered like terms, their variables and corresponding exponents must match. This leads us to two equations based on the exponents of x and y: 1. For the exponents of x to match, we have: [m + 3 = 5] Solving for m gives us: [m = 5 - 3] [m = 2] 2. For the exponents of y to match, we have: [6 = 2n] Solving for n gives us: [n = frac{6}{2}] [n = 3] Now, we are asked to find the value of m^{n}. Substituting the values of m and n we found: [m^{n} = 2^{3}] [m^{n} = 8] Therefore, the value of m^{n} is boxed{8}.