Appearance
question:The equation of the line connecting the centers of the circles x^{2}+y^{2}-4x+6y=0 and x^{2}+y^{2}-6x=0 is __________.
answer:To find the equation of the line connecting the centers of the two circles, we first need to determine the centers of each circle. For the first circle, x^{2}+y^{2}-4x+6y=0, we can rewrite it as (x-2)^{2}+(y+3)^{2}=2^{2}+3^{2}=13, which means the center is at (2, -3).For the second circle, x^{2}+y^{2}-6x=0, we can rewrite it as (x-3)^{2}+y^{2}=3^{2}, which means the center is at (3, 0).The equation of the line connecting these two centers (2, -3) and (3, 0) can be found using the two-point form of a line equation. The slope of the line is frac{0-(-3)}{3-2}=3. Therefore, the equation of the line in slope-intercept form is y-(-3)=3(x-2), which simplifies to 3x-y-9=0.Thus, the equation of the line connecting the centers of the two circles is boxed{3x-y-9=0}.
question:Example 2. Given: f(x)=2 x+3,|x| leqslant 2, and x in Z. Find the range of the function.
answer:begin{array}{c}text { Solve } because|x| leqslant 2, text { and } x in Z, therefore x text { can only be }-2,-1,0,1,2 . text { Also } because f(-2)=-1, f(-1)=1, f(0) =3, f(1)=5, f(2)=7,end{array}therefore the range of the function is {-1,1,3,5,7}.
question:Calculate the sum of the cubes of the first eight positive integers: 1^3 + 2^3 + 3^3 + 4^3 + ldots + 8^3.
answer:To solve 1^3 + 2^3 + 3^3 + 4^3 + ldots + 8^3, we calculate each term individually:= 1 + 8 + 27 + 64 + 125 + 216 + 343 + 512,= 1296.Therefore, the answer is boxed{1296}.
question:Let a, b, c, and d be positive integers, and a^2 + b^2 = c^2 + d^2. Prove that a + b + c + d must be a composite number.
answer:Answer: The judgment can be made based on the parity of a and a^2 being the same.Proof: Since a^2 + b^2 has the same parity as a + b, and c^2 + d^2 has the same parity as c + d, and a^2 + b^2 = c^2 + d^2,Therefore, a^2 + b^2 and c^2 + d^2 have the same parity, which means a + b and c + d have the same parity.Therefore, a + b + c + d is an even number, and a + b + c + d geq 4,Therefore, a + b + c + d must be a composite number. boxed{text{Composite}}
question:13. In triangle A B C, angle B=frac{pi}{4}, angle C=frac{5 pi}{12}, A C =2 sqrt{6}, A C's midpoint is D. If a line segment P Q of length 3 (point P to the left of point Q) slides on line B C, then the minimum value of A P+D Q is qquad.
answer:13. frac{sqrt{30}+3 sqrt{10}}{2}.Given that B C=6.Draw a line D E / / B C through point D, intersecting A B at point E.Then D E=frac{1}{2} B C=3.Thus, quadrilateral P Q D E is a parallelogram, i.e., D Q=E P.Therefore, the problem is reduced to: finding a point on line B C such that A P+E P is minimized.The minimum value of A P+E P is calculated to be frac{sqrt{30}+3 sqrt{10}}{2}.
question:In a 2times2 contingency table, the value of k^{2}=13.097 is calculated from its data. What is the possibility of a relationship between the two variables?A: 99%B: 95%C: 90%D: No relationship
answer:Given that in a 2times2 contingency table, the value of k^{2}=13.097 is calculated from its data.We know that,P(k^{2} > 13.097) 1 - 0.001 = 0.999Therefore, we can say with 99% confidence that there is a relationship between the two variables.Hence, the answer is: boxed{A}.The solution involves comparing the given observed value with the critical values in the chi-square distribution table to determine the confidence level. This problem tests the application of the independence test, and the key to solving it is being able to read and interpret the critical value table.