answer:Solution: (1) When m=1, the equation of curve C is: (x-1)^{2}+(y-2)^{2}=4, which represents a circle with center at (1,2) and radius of 2, because line l passes through point P and has only one common point with curve C, therefore line l is tangent to the circle. bullet When the slope of line l does not exist, the equation of line l is: x=3. bullet When the slope of line l exists, suppose the equation of line l is: y=k(x-3)-1. That is kx-y-3k-1=0. left( dfrac {|k-2-3k-1|}{ sqrt {k^{2}+1}}=2right) Rightarrow k=-dfrac {5}{12}, the equation of line l is: 5x+12y-3=0. In summary, the equations of the sought line l are: x=3, 5x+12y-3=0. (2) Completing the square for the equation of curve C gives: (x-1)^{2}+(y-2)^{2}=5-m. If the equation represents a circle, then 5-m > 0 Rightarrow m < 5. The distance d from the center of the circle to the line x+2y+5=0 is d= dfrac {10}{ sqrt {5}}=2sqrt{5}, According to the formula for the length of a chord of a circle 2sqrt{r^{2}-d^{2}}=2sqrt{5}, Rightarrow 2sqrt{5-m-20}=2sqrt{5}, Rightarrow m=-20. Therefore, the final answers are: (1) The equations of line l are boxed{x=3} and boxed{5x+12y-3=0}. (2) The value of the real number m is boxed{-20}.

answer:We are given 100 products, with precisely 3 being defective. We need to find the number of ways to draw 4 products such that exactly 2 are defective. This task involves drawing 2 defective products from the 3 available and simultaneously drawing 2 non-defective products from the remaining 97. The number of ways to choose 2 defective products from 3 is given by the combination formula C(3,2). Similarly, the number of ways to choose 2 non-defective products from 97 is given by C(97,2). Using the formula for combinations, which is C(n,k) = frac{n!}{k!(n-k)!}, we can calculate the required number of ways. The total number of ways to draw such a sample is the product of these two combinations: C(3,2) times C(97,2) = frac{3!}{2!(3-2)!} times frac{97!}{2!(97-2)!}. We then simplify the expressions for each combination: C(3,2) = frac{3!}{2!(3-2)!} = frac{3 times 2}{2 times 1} = 3, and C(97,2) = frac{97!}{2!(97-2)!} = frac{97 times 96}{2 times 1} = 4656. Multiplying these together, we get the total number of drawing methods: 3 times 4656 = 13968. Therefore, the number of ways to randomly draw 4 products from 100, where exactly 2 are defective, is boxed{13968}.

answer:To find the minimum value of the expression |x - 4| + |x + 7| + |x - 5|, we need to consider the points where the expression changes its slope, which are the points where each of the absolute value terms equals zero. These points are x = 4, x = -7, and x = 5. We will examine the expression in the intervals determined by these points: 1. For x < -7, all three terms are negative, so the expression becomes -(x - 4) - (x + 7) - (x - 5) = -3x + 6. 2. For -7 ≤ x < 4, the term |x + 7| becomes positive, so the expression becomes -(x - 4) + (x + 7) - (x - 5) = -x + 8. 3. For 4 ≤ x < 5, the terms |x - 4| and |x + 7| are positive, so the expression becomes (x - 4) + (x + 7) - (x - 5) = x + 8. 4. For x ≥ 5, all three terms are positive, so the expression becomes (x - 4) + (x + 7) + (x - 5) = 3x - 2. Now, we need to find the minimum value in each interval: 1. For x < -7, the expression -3x + 6 decreases as x decreases, so there is no minimum in this interval. 2. For -7 ≤ x < 4, the expression -x + 8 decreases as x increases, so the minimum value in this interval is at x = 4, which gives us -4 + 8 = 4. 3. For 4 ≤ x < 5, the expression x + 8 increases as x increases, so the minimum value in this interval is at x = 4, which gives us 4 + 8 = 12. 4. For x ≥ 5, the expression 3x - 2 increases as x increases, so the minimum value in this interval is at x = 5, which gives us 3(5) - 2 = 13. Comparing the minimum values at the transition points, we see that the overall minimum value of the expression is 4, which occurs at x = boxed{4} .

answer:1. To find x and y, subtract the first equation from the second: [ 3x + y - (x + y) = 18 - 12 Rightarrow 2x = 6 Rightarrow x = 3 ] 2. Substitute x = 3 into the first equation to find y: [ 3 + y = 12 Rightarrow y = 9 ] 3. Calculate x^2 - y^2 using the formula (x-y)(x+y): [ x^2 - y^2 = (x-y)(x+y) = (3-9)(3+9) = (-6)(12) = -72 ] Thus, x^2 - y^2 = boxed{-72}.