answer:Let w = a + bi, where a and b are real numbers. Given |w| = 2, we have sqrt{a^2 + b^2} = 2, so a^2 + b^2 = 4. Now, calculate frac{1}{2 - w}: [ frac{1}{2 - w} = frac{1}{2 - a - bi} = frac{2 - a + bi}{(2 - a - bi)(2 - a + bi)} = frac{2 - a + bi}{(2-a)^2 + b^2} ] [ = frac{2 - a + bi}{4 - 4a + a^2 + b^2} = frac{2 - a + bi}{4 - 4a + 4} = frac{2 - a + bi}{8 - 4a} ] The real part of this complex number is: [ frac{2 - a}{8 - 4a} = frac{1 - frac{a}{2}}{4 - 2a}. ] Simplifying further, the real part is frac{1 - frac{a}{2}}{4(1 - frac{a}{2})} = boxed{frac{1}{4}}.

answer:Given the equation ( x^4 + p^2 x + q = 0 ), we need to determine the positive prime numbers ( p ) and ( q ) such that the equation has a multiple root. 1. Let the multiple root be ( a ). This implies that ( a ) is a root of both the polynomial and its derivative. Hence, we can factor the polynomial in the form: [ x^4 + p^2 x + q = (x + a)^2 (x^2 + bx + c). ] 2. Expanding the right-hand side, we get: [ (x + a)^2 (x^2 + bx + c) = (x^2 + 2ax + a^2)(x^2 + bx + c). ] 3. Further expand this product: [ (x^2 + 2ax + a^2)(x^2 + bx + c) = x^4 + (2a + b)x^3 + (2ab + a^2 + c)x^2 + (a b + 2ac)x + a^2c. ] 4. Comparing coefficients with the equation ( x^4 + p^2 x + q ): (i) From the ( x^3 )-coefficient: [ 2a + b = 0 implies b = -2a. ] (ii) From the ( x^2 )-coefficient: [ 2ab + a^2 + c = 0 implies 2a(-2a) + a^2 + c = 0 implies -4a^2 + a^2 + c = 0 implies c = 3a^2. ] (iii) From the ( x )-coefficient: [ a(b + 2c) = p^2 implies a(-2a + 2 cdot 3a^2) = p^2 implies 4a^3 = p^2. ] (iv) From the constant term: [ a^2c = q implies a^2 cdot 3a^2 = q implies 3a^4 = q. ] 5. Solving for ( a ) and substituting in terms of ( p ) and ( q ): - From ( 4a^3 = p^2 ): [ a^3 = frac{p^2}{4} implies a = sqrt[3]{frac{p^2}{4}}. ] - From ( 3a^4 = q ): [ a^4 = frac{q}{3}. ] 6. We equate the expression for ( a^3 ) and ( a^4 ): [ left(sqrt[3]{frac{p^2}{4}}right)^4 = frac{q}{3} implies left(frac{p^2}{4}right)^{4/3} = frac{q}{3}. ] - Simplifying, we find: [ left(frac{p^2}{4}right)^{4/3} = frac{q}{3} implies frac{p^{8/3}}{4^{4/3}} = frac{q}{3} implies p^{8/3} = 3 cdot 4^{4/3} cdot q. ] 7. Given that ( p ) and ( q ) are both prime numbers, substitute plausible prime numbers and solve. After testing the smallest prime numbers, we arrive at: - Let ( p = 2 ), then ( a = 1 ), making: [ p^2 = 4 implies a = 1, , c = 3 ] - Compute ( q ): [ 3a^4 = q implies 3 cdot 1^4 = q implies q = 3. ] 8. Verify the factorization: [ x^4 + 4x + 3 = (x + 1)^2 (x^2 - 2x + 3). ] 9. Conclusion: The values of the primes ( p ) and ( q ) are: [ boxed{(p, q) = (2, 3)} ]

answer:The rectangle PQJK has width PQ = 2 times 3sqrt{3} = 6sqrt{3}. The distance PJ is equal to the tangent distance from R to the circles, which is a leg of a right triangle with hypotenuse PR = 3sqrt{3} and one leg (radius) 3. Using the Pythagorean theorem: [ RG = sqrt{(3sqrt{3})^2 - 3^2} = sqrt{27 - 9} = sqrt{18} = 3sqrt{2}. ] Hence, the side PJ (height of the rectangle PQJK) is 3sqrt{2}. The area of rectangle PQJK is: [ 6sqrt{3} times 3sqrt{2} = 18sqrt{6}. ] Calculate the area of the right triangles RPG and RQH: [ text{Area} = frac{1}{2} cdot 3 cdot 3sqrt{2} = 4.5sqrt{2} ] each. Calculate the area of sectors PGR and QHR each covering 45^circ: [ text{Area} = frac{45^circ}{360^circ} cdot pi cdot 3^2 = frac{1}{8} cdot 9pi = frac{9pi}{8} ] each. The total area of triangles and sectors to subtract from the rectangle is: [ 2(4.5sqrt{2} + frac{9pi}{8}) = 9sqrt{2} + frac{9pi}{4}. ] Thus, the area of the shaded region RGJHK is: [ 18sqrt{6} - left(9sqrt{2} + frac{9pi}{4}right) = 18sqrt{6} - 9sqrt{2} - frac{9pi}{4} = boxed{18sqrt{6} - 9sqrt{2} - frac{9pi}{4}}. ]

answer:To calculate the time it takes for the bullet train to pass the man, we need to determine the relative speed between the two. The bullet train is moving at 50 kmph, and the man is running in the opposite direction at 4 kmph. Since they are moving in opposite directions, we add their speeds to find the relative speed. Relative speed = Speed of bullet train + Speed of man Relative speed = 50 kmph + 4 kmph Relative speed = 54 kmph Now, we need to convert the relative speed from kmph to m/s to match the length of the train, which is given in meters. To convert kmph to m/s, we use the conversion factor: 1 kmph = 1000 m / 3600 s Relative speed in m/s = 54 kmph * (1000 m / 3600 s) Relative speed in m/s = 54 * (1000 / 3600) Relative speed in m/s = 54 * (5 / 18) Relative speed in m/s = 15 m/s Now that we have the relative speed in m/s, we can calculate the time it takes for the bullet train to pass the man. Time = Distance / Speed The distance to be covered is the length of the bullet train, which is 120 m. Time = 120 m / 15 m/s Time = 8 seconds Therefore, it will take the bullet train boxed{8} seconds to pass the man.