Multiple choice

If the mean of three variables x, y, and z is N, and the sum of the products xy, yz, and zx is zero, determine the value of K such that the mean of x2, y2, and z2 is KN2.

  1. 1

  2. 2

  3. 3

  4. 4

  5. a
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C Correct answer
Explanation

Mean of x, y, z is N, so x+y+z = 3N. We know (x+y+z)^2 = x^2+y^2+z^2 + 2(xy+yz+zx). Given xy+yz+zx = 0, we have (3N)^2 = x^2+y^2+z^2 + 0. So x^2+y^2+z^2 = 9N^2. The mean of x^2, y^2, z^2 is (x^2+y^2+z^2)/3 = 9N^2/3 = 3N^2. Thus K=3.

AI explanation

The mean of x, y, and z is N, so x plus y plus z equals 3N, making the square of their sum 9N squared. The sum of the products xy, yz, and zx is zero, so the identity x squared plus y squared plus z squared equals the square of the sum minus 2 times the sum of the products gives 9N squared. The mean of x squared, y squared, and z squared is 9N squared divided by 3, which equals 3N squared, so K equals 3.

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