Problem Solving

All of the stocks on the over the counter market are designated by either a 4 letter or a 5 letter code that is created by using the 26 letters of the alphabet. Which of the following gives the maximum number of different stocks that can be designated with these codes?

Maximum number of stock codes = possible number of 4 letter codes + possible number of 5 letter codes
A 4 letter code can be _ _ _ _, where 1st position can be any one of the 26 letters, similarly 2nd position can be one of the 26 letters, and the same applied for 3rd and 4th positions.
So, possible no. of 4 letter codes = 26 * 26 * 26 * 26 = 26^4
In the same way, possible no. of 5 letter codes = 26 * 26 * 26 * 26 * 26 = 26^5

Therefore, maximum number of stock codes = 26^4 + 26^5 = 26^4(1 + 26) = 27(26^4)

The correct answer is C.

For which of the following functions is f(a + b) = f(a) + f(b) for all positive numbers a and b?

(A) f(x) = x^2
(B) f(x) = x+1
(C) f(x) = √x
(D) f(x) = 2/x
(E) f(x) = -3x

Let's analyze each of the options individually:

1. (a + b)² ≠ (a² + b²) => f(a + b) ≠ f(a) + f(b)
2. (a + b + 1) ≠ (a + 1) + (b + 1) = (a + b + 2) => f(a + b) ≠ f(a) + f(b)
3. √(a + b) ≠ (√a + √b) => f(a + b) ≠ f(a) + f(b)
4. 2/(a + b) ≠ (2/a) + (2/b) = 2(a + b)/ab => f(a + b) ≠ f(a) + f(b)
5. (-3(a + b)) = (-3a) + (-3b) => f(a + b) = f(a) + f(b)

The correct answer is E.