Appearance
question:Given that the fifth term of the expansion of {(axfrac{1}{x})}^{n} is a constant term, find the value of n.
answer:**Analysis** This problem tests the properties of binomial coefficients and focuses on the general term formula of binomial expansion, making it a fundamental question. **Solution** The general term of the expansion of {(ax-frac{1}{x})}^{n} is given by {T}_{r+1}=C_{n}^{r}(ax{)}^{n-r}(- frac{1}{x}{)}^{r}=C_{n}^{r}{a}^{n-r}(-1{)}^{r}{x}^{n-2r}. Since the fifth term of the expansion of {(ax-frac{1}{x})}^{n} is a constant term, we have: When r=4, n-2r=0, Thus, n=8. **Conclusion** The value of n is boxed{8}.
question:f set A={1,0,1,2}, B={x|x+1 > 0}, then Acap B= ______.
answer:Given that set A={-1,0,1,2}, B={x|x+1 > 0}={x|x > -1}, therefore, Acap B={0,1,2}. Hence, the answer is boxed{{0,1,2}}. First, we find the sets A and B separately, and then use the definition of intersection to find Acap B. This question tests the method of finding intersections, which is a basic problem. When solving it, one should carefully read the problem and properly apply the properties of intersections.
question:Let ABC be a triangle. There exists a positive real number k, such that if the altitudes of triangle ABC are extended past A, B, and C, to A', B', and C', as shown, such that AA' = kBC, BB' = kAC, and CC' = kAB, then triangle A'B'C' is equilateral. [asy] unitsize(0.6 cm); pair[] A, B, C; pair D, E, F; A[0] = (2,4); B[0] = (0,1); C[0] = (5,0); D = (A[0] + reflect(B[0],C[0])*(A[0]))/2; E = (B[0] + reflect(C[0],A[0])*(B[0]))/2; F = (C[0] + reflect(A[0],B[0])*(C[0]))/2; A[1] = A[0] + (1/sqrt(3))*(rotate(90)*(C[0] B[0])); B[1] = B[0] + (1/sqrt(3))*(rotate(90)*(A[0] C[0])); C[1] = C[0] + (1/sqrt(3))*(rotate(90)*(B[0] A[0])); draw(A[0]B[0]C[0]cycle); draw(A[1]D); draw(B[1]E); draw(C[1]F); label(A, A[0], NW); dot(A', A[1], N); label(B, B[0], S); dot(B', B[1], SW); label(C, C[0], S); dot(C', C[1], SE); [/asy] Find k.
answer:To solve for k, we first express the vertices of the triangle and their extended points in the complex plane. Let's denote the vertices A, B, and C by complex numbers a, b, and c, respectively, and their extended points A', B', and C' by a', b', and c', respectively. Given that AA' = kBC, BB' = kAC, and CC' = kAB, and considering the rotation by 90^circ to get from A to A' (and similarly for B and C), we can write the positions of A', B', and C' as follows: begin{align*} a' &= a + ki(c - b), b' &= b + ki(a - c), c' &= c + ki(b - a). end{align*} For triangle A'B'C' to be equilateral, the complex numbers must satisfy the relation: [c' - a' = e^{pi i/3} (b' - a').] Substituting the expressions for a', b', and c', and using the fact that e^{pi i/3} = frac{1}{2} + frac{sqrt{3}}{2} i, we obtain: [c + ki(b - a) - a - ki(c - b) = left( frac{1}{2} + frac{sqrt{3}}{2} i right) [b + ki(a - c) - a - ki(c - b)].] Expanding and simplifying both sides, we get: begin{align*} &(-1 - ki) a + 2ki b + (1 - ki) c &= frac{-k sqrt{3} - 1 + ki - i sqrt{3}}{2} cdot a + frac{- k sqrt{3} + 1 + ki + i sqrt{3}}{2} cdot b + (k sqrt{3} - ki) c. end{align*} To ensure the triangle A'B'C' is equilateral, the coefficients of a, b, and c on both sides of the equation must be equal. By equating the coefficients of c, we find: [1 - ki = k sqrt{3} - ki,] which simplifies to k = 1/sqrt{3}. For this value of k, the coefficients of a and b also match as required. Therefore, the value of k that makes triangle A'B'C' equilateral is k = boxed{frac{1}{sqrt{3}}}.
question:Uncle Jude baked 256 cookies. He gave 15 cookies to Tim, 23 cookies to Mike, kept 188 cookies in the fridge, and gave some cookies to Anna. What is the ratio of the number of cookies given to Anna to the number of cookies given to Tim?
answer:First, let's calculate the total number of cookies Uncle Jude gave away to Tim, Mike, and the number he kept in the fridge: Cookies given to Tim: 15 Cookies given to Mike: 23 Cookies kept in the fridge: 188 Total cookies accounted for: 15 + 23 + 188 = 226 Now, let's find out how many cookies Uncle Jude gave to Anna by subtracting the total cookies accounted for from the total number of cookies baked: Total cookies baked: 256 Total cookies accounted for: 226 Cookies given to Anna: 256 - 226 = 30 Now we can find the ratio of the number of cookies given to Anna to the number of cookies given to Tim: Cookies given to Anna: 30 Cookies given to Tim: 15 Ratio (Anna : Tim) = 30 : 15 To simplify the ratio, we divide both numbers by the greatest common divisor, which is 15: 30 ÷ 15 : 15 ÷ 15 = 2 : 1 So, the ratio of the number of cookies given to Anna to the number of cookies given to Tim is boxed{2:1} .