Postby YKM » Mon Aug 29, 2011 5:23 am
#25
Let x and y be positive integers such that 3x + 7y is divisible by 11. Which of the following is true?
a. 4x + 6y
b. x + y + 5
c. 9x + 4y
d. 4x - 9y
e. x + y -1
The answer is d. 4x - 9y
My logic:
Since x and y are positive integers and 3 and 7 are relative primes, then 11 | 3x + 7y would imply that 11 | x and 11 | y. Thus, any linear combination of x and y should be divisible by 11.
I know my logic is wrong, can anyone tell me what's wrong?
#30
The improper integral, from b to a, f(x)*f'(x) dx is
The answer is a. necessarily zero.
My logic:
If I let f(x) = x^2, then f'(x) = 2x, meaning f(x)f'(x) = 2x^3. The integral, from b to a, 2x^3 dx is 1/2 * (b^4 - a^4), which is not zero.
I know my logic is wrong, can anyone tell me what's wrong?
#50
In a game two players take turns tossing a fair coin, the winner is the first one to toss a head. The probability that the player who makes the first toss wins the game is?
The answer is D: 2/3.
My logic:
Let the two player be called A and B. Suppose A go first, and toss a head, the game is done and the probability of this event is 1/2. Suppose A go first and does not toss a head, then B's turn, and toss a head, the game is done, and the probability of this event is 1/2 * 1/2. Since the name A and B are generic, the sum of the probability of these two disjoint events should be the answer, which is 3/4.
I know my logic is wrong, can anyone tell me what's wrong?