answer:To find the imaginary part of the complex number z=frac{2+mi}{1+i}, given that z is purely imaginary, we proceed as follows: First, we simplify z by multiplying both the numerator and the denominator by the complex conjugate of the denominator: z = frac{2+mi}{1+i} = frac{(2+mi)(1-i)}{(1+i)(1-i)}. Expanding both the numerator and the denominator, we get: z = frac{2(1-i) + mi(1-i)}{1^2 - i^2} = frac{2 - 2i + mi - mi^2}{1 + 1}. Since i^2 = -1, we can further simplify this to: z = frac{2 - 2i + mi + m}{2} = frac{2+m}{2} + frac{m-2}{2}i. For z to be purely imaginary, the real part must be equal to 0. Therefore, we set the real part equal to 0 and solve for m: frac{2+m}{2} = 0 implies 2+m = 0 implies m = -2. Substituting m = -2 into our expression for z, we find: z = frac{2-2}{2} + frac{-2-2}{2}i = 0 - 2i = -2i. Thus, the imaginary part of the complex number z is -2. Therefore, the answer is: boxed{-2}.

answer:Let the first term of the geometric sequence be a, and the common ratio be r. The terms of the sequence can be represented as follows: - The fourth term: ar^3 = 6! - The seventh term: ar^6 = 7! We can find r by dividing the seventh term by the fourth term: [ r^3 = frac{ar^6}{ar^3} = frac{7!}{6!} = 7 ] Thus, r = sqrt[3]{7}. Now, solve for a using the expression for the fourth term: [ a = frac{6!}{r^3} = frac{720}{7} ] [ a = frac{720}{7} ] Therefore, the first term of the sequence is boxed{frac{720}{7}}.

answer:# Solution: Part 1: Estimating the Probabilities **For Female University Students:** Given that there are 200 university students in total and 120 are boys, we can calculate the number of girls as follows: - Total students = Boys + Girls - 200 = 120 + Girls - Girls = 200 - 120 = 80 Now, knowing that 30 girls like running, we can find the probability of a female university student liking running: - Probability = frac{text{Number of girls who like running}}{text{Total number of girls}} = frac{30}{80} = frac{3}{8} So, the probability of female university students liking running is boxed{frac{3}{8}}. **For Male University Students:** To find the number of boys who like running, we use the total number of boys and subtract those who do not like running: - Boys who do not like running = 50 - Total boys = 120 - Boys who like running = Total boys - Boys who do not like running = 120 - 50 = 70 The probability of a male university student liking running is: - Probability = frac{text{Number of boys who like running}}{text{Total number of boys}} = frac{70}{120} = frac{7}{12} Thus, the probability of male university students liking running is boxed{frac{7}{12}}. Part 2: Relation to Gender To determine if liking running is related to gender among university students, we use the given chi^2 formula: - {chi^2}=frac{{n{{({ad-bc})}^2}}}{{({a+b})({c+d})({a+c})({b+d})}} Substituting the values: - a = 70 (Boys who like running) - b = 50 (Boys who do not like running) - c = 30 (Girls who like running) - d = 50 (Girls who do not like running, calculated as 80 - 30) - n = 200 We calculate chi^2 as follows: - {chi^2}=frac{200times({70times50-50times30})^2}{120times80times100times100}=frac{200times(3500-1500)^2}{120times80times100times100}=frac{200times2000000}{96000000}=frac{25}{3}approx8.333 Consulting the reference table, we find P(chi^2geq 6.635)=0.010. Since our calculated chi^2 value of 8.333 is greater than 6.635, we can conclude with 99% certainty that the preference for running among university students is related to gender. Therefore, we can be boxed{99%} certain that whether university students in this location like running is related to gender.

answer:Let's analyze the problem step by step, following the given solution closely: 1. **Given Information**: Curve C is defined by the property that for any point P(x,y) on it, the product of the distances from P to two fixed points F_{1}(-1,0) and F_{2}(1,0) is equal to 4. This can be mathematically represented as |PF_{1}|cdot |PF_{2}|=4. 2. **Equation of Curve C**: To find the equation of curve C, we use the distances |PF_{1}| and |PF_{2}|: [ |PF_{1}| = sqrt{(x+1)^{2}+y^{2}}, quad |PF_{2}| = sqrt{(x-1)^{2}+y^{2}} ] Substituting these into the given condition |PF_{1}|cdot |PF_{2}|=4 gives: [ sqrt{(x+1)^{2}+y^{2}}cdot sqrt{(x-1)^{2}+y^{2}} = 4 ] Squaring both sides and simplifying leads to: [ [(x+1)^{2}+y^{2}][(x-1)^{2}+y^{2}] = 16 ] Expanding and rearranging gives the equation of curve C. 3. **Checking Statement ①**: Substituting x=0 and y=0 into the equation of curve C to check if the origin lies on C: [ [(0+1)^{2}+0^{2}][(0-1)^{2}+0^{2}] - 16 = 1cdot 1 - 16 = -15 neq 0 ] Since this does not hold true, the curve C does not pass through the origin, making statement ① incorrect. 4. **Checking Statement ②**: To check if curve C is symmetric with respect to the origin, substitute (-x,-y) into the equation of curve C. This substitution does not alter the equation, indicating that curve C is indeed symmetric with respect to the origin, making statement ② correct. 5. **Checking Statement ③**: The area of triangle F_{1}PF_{2} can be calculated as: [ text{Area} = frac{1}{2}|PF_{1}|cdot |PF_{2}|cdot sin angle F_{1}PF_{2} = frac{1}{2}times 4times sin angle F_{1}PF_{2} leqslant 2 ] This is because the maximum value of sin angle F_{1}PF_{2} is 1, making statement ③ correct. 6. **Checking Statement ④**: For the curve frac{x^2}{4}+frac{y^2}{3}=1, solving for y^{2} gives: [ y^{2} = 3left(1-frac{x^{2}}{4}right) ] Substituting this into the equation of curve C and solving for x gives x^{2}(x^{2}-32)=0, which simplifies to x^{2}=32 (discarded) and x=0. Therefore, y=±sqrt{3}, indicating that curve C intersects the given curve at exactly two points (0, pm sqrt{3}), making statement ④ correct. Therefore, the correct statements are ②, ③, and ④. Hence, the answer is: boxed{D}.