Flux

🔍

question:Given p: |x4| leq 6, q: x^2 2x + 1 m^2 leq 0, if neg p is a sufficient but not necessary condition for neg q, find the range of the real number m.

answer:Since p: |x-4| leq 6 implies -2 leq x leq 10; For proposition q, we have x^2 - 2x + 1 - m^2 leq 0, which gives 1-|m| leq x leq 1+|m|. Because neg p is a sufficient but not necessary condition for neg q, it means q is a sufficient but not necessary condition for p, thus, we have begin{cases} 1+|m| leq 10 1-|m| geq -2 end{cases}, which gives -3 leq m leq 3. Therefore, the range of m is boxed{-3 leq m leq 3}.

question:Real numbers x and y satisfy the equations 2^x = 16^{y+1} and 27^y = 3^{x2}. What is xy?

answer:The given equations can be rewritten using their base exponent forms: 1. 2^x = (2^4)^{y+1} = 2^{4(y+1)} 2. 27^y = (3^3)^y = 3^{3y} = 3^{x-2} From the first equation, since the bases are the same, their exponents must be equal: [ x = 4(y + 1) ] From the second equation: [ 3^{3y} = 3^{x-2} ] Since the bases are the same, their exponents must be equal: [ 3y = x - 2 ] Now, we have the system of equations: [ x = 4y + 4 ] [ 3y = x - 2 ] Substitute the expression of x from the first equation into the second equation: [ 3y = (4y + 4) - 2 ] [ 3y = 4y + 2 ] [ y = -2 ] Substitute back into the expression for x: [ x = 4(-2) + 4 = -8 + 4 = -4 ] Thus, the product xy is: [ xy = -4 times -2 = boxed{8} ]

question:My twodigit code is 04. Reckha cannot choose a code that matches exactly one of my digits in its position, transposes two digits of my code, such as 40, or is exactly my code 04. Reckha can otherwise choose any twodigit code where each digit is from the set {0, 1, 2, 3, 4}. How many codes are available for Reckha?

answer:Each digit in a two-digit code can be one of 5 possible digits (0, 1, 2, 3, 4). Therefore, without any restrictions, there would be: [ 5^2 = 25 text{ possible codes}.] Finding forbidden codes: 1. Codes that match exactly one digit with 04 (keeping one digit fixed and changing the other one): - Fix the first digit as 0, change the second: (01, 02, 03), since '04' itself is forbidden. - Fix the second digit as 4, change the first: (14, 24, 34), since '04' itself is forbidden. This results in 3 + 3 = 6 forbidden codes. 2. Codes transposing the digits of 04 are: - 40, since only one permutation is possible as '40'. 3. The code '04' itself is forbidden. Total forbidden codes = 6 + 1 + 1 = 8. Thus, Reckha has: [ 25 - 8 = boxed{17} text { available codes.} ]

question:Walmart sells a multitool with a screwdriver, 3 knives, and two other tools. Target sells a multitool with a screwdriver, a certain number of knives, three files and a pair of scissors. The Target multitool has 5 more tools compared to Walmart. What is the ratio of the number of knives in the Target multitool to the number of knives in the Walmart multitool?

answer:Let's denote the number of knives in the Target multitool as ( k ). The Walmart multitool has a total of ( 1 ) (screwdriver) + ( 3 ) (knives) + ( 2 ) (other tools) = ( 6 ) tools. The Target multitool has a total of ( 1 ) (screwdriver) + ( k ) (knives) + ( 3 ) (files) + ( 1 ) (pair of scissors) = ( k + 5 ) tools. We know that the Target multitool has ( 5 ) more tools than the Walmart multitool, so: ( k + 5 = 6 + 5 ) ( k + 5 = 11 ) ( k = 11 - 5 ) ( k = 6 ) So, the Target multitool has ( 6 ) knives. The ratio of the number of knives in the Target multitool to the number of knives in the Walmart multitool is: ( frac{6}{3} = 2 ) Therefore, the ratio is ( boxed{2:1} ).