1)The function f is defined by the equation f(x)+ x-x^2. Which of the following represents a quadratic with no real zeros?
A)f(x) +1/2
B)f(x)-1/2
C)f(x/2)
D)f(x-1/2)
2) If I^(2k) = 1, and i = radical -1, which of the following must be true about k?
A) k is a multiple of 4
B) k is a positive integer
C)when 2k is divided by 4, the remainder is 1
D) k/2 is an integer
^I got B
3)
For all the numbers x and y, let z be defined by the equation z=I 2^2 - x^2 - y^2I +2^2. What is the smallest possible value of z?
^I got 0
1) f(x) = x - x^2
When the determinant (b^2 - 4ac) of a quadratic is negative, it will have non-real zeros. This is true for f(x) - (1/2)
2) No
In order to get i^x = 1, x must be four, or some multiple of four (could also be negative because 1/1 is equal to 1)
So, if 2k is a multiple of 4, then k is a multiple of 2 hence k/2 is an integer.
(To show that B is untrue, take k = -2 => 2k = -4. i^-4 = 1/1 = 1)
I think number 3 is C?
1) To find a quadratic with no real zeros from the given function f(x) = x - x^2, we need to change the equation to eliminate the possibility of real solutions. This can be achieved by modifying the constant term of the quadratic equation. Let's analyze the answer choices:
A) f(x) + 1/2: This changes the constant term but does not guarantee no real zeros.
B) f(x) - 1/2: This changes the constant term but does not guarantee no real zeros.
C) f(x/2): This modifies the coefficients but does not change the constant term, so it does not guarantee no real zeros.
D) f(x - 1/2): This shifts the entire quadratic to the right by 1/2 unit, effectively changing the constant term. This choice guarantees a quadratic with no real zeros.
Therefore, the answer is D) f(x - 1/2).
2) Given that i^2k = 1 and i = sqrt(-1), we can simplify the equation to:
(i^2)^k = 1
(-1)^k = 1
To satisfy this equation, any even exponent of -1 will yield a result of 1. Thus, k must be a multiple of 2.
Therefore, the answer is that k is a multiple of 4. Choice A is correct.
3) To find the smallest possible value of z in the equation z = |2^2 - x^2 - y^2| + 2^2, we need to minimize the absolute value term. The absolute value of any value is always non-negative, meaning it can never be less than zero.
In this case, the minimum value of |2^2 - x^2 - y^2| is when x^2 + y^2 is maximized. To achieve the maximum value of x^2 + y^2, both x and y should be zero. Therefore, the smallest possible value of z is:
z = |2^2 - 0^2 - 0^2| + 2^2
z = |4 - 0 - 0| + 4
z = |4| + 4
z = 4 + 4
z = 8
Therefore, the smallest possible value of z is 8.
1) To find a quadratic with no real zeros using the function f(x) = x - x^2, we need to find a constant or variable to add or subtract that will not change the nature of the zeros. Let's go through the options:
A) f(x) + 1/2: Adding a constant of 1/2 does not change the nature of the zeros, as it only shifts the graph of the quadratic vertically. Therefore, it is not the correct answer.
B) f(x) - 1/2: Similarly, subtracting a constant of 1/2 does not change the nature of the zeros. This is also not the correct answer.
C) f(x/2): This option involves plugging x/2 into the function f(x). Substituting x/2 for x, we get f(x/2) = x/2 - (x/2)^2 = x/2 - x^2/4. This function is still a quadratic, but the coefficient in front of the x^2 term is different. This means the nature of the zeros might change. Therefore, C is not the correct answer.
D) f(x - 1/2): This option involves substituting x - 1/2 into the function f(x). Substituting x - 1/2 for x, we get f(x - 1/2) = (x - 1/2) - (x - 1/2)^2 = x - 1/2 - x^2 + x - 1/2 + 1/4 = -x^2 + 2x - 3/4. This is a quadratic with a different leading coefficient and constant term compared to the original function. Therefore, it is possible for this quadratic to have no real zeros. Thus, the correct answer is D.
2) Given that i^(2k) = 1, where i is the square root of -1, and we want to determine which statement must be true about k.
We know that i^2 = -1. If we raise both sides to the power of k, we get (i^2)^k = (-1)^k = 1. Since any power of 1 is still 1, we can conclude that k must be an even integer. This means that k is a multiple of 2.
However, the question asks us to choose from the options provided. Let's analyze each option:
A) k is a multiple of 4: Since k is a multiple of 2, it is also a multiple of 4. This statement is true based on our previous analysis.
B) k is a positive integer: We have determined that k must be an even integer, but it does not necessarily have to be positive. Therefore, this statement is not necessarily true.
C) When 2k is divided by 4, the remainder is 1: Since k is a multiple of 2 (an even integer), when we multiply it by 2 (resulting in 2k), the remainder when divided by 4 will always be 0, not 1. Therefore, this statement is not true.
D) k/2 is an integer: As we have determined that k is an even integer, dividing it by 2 will always result in an integer. Therefore, this statement is true.
Based on the analysis above, the correct answer is D - k/2 is an integer.
3) To find the smallest possible value of z given the equation z = |2^2 - x^2 - y^2| + 2^2, we need to determine the minimum value of the expression inside the absolute value and add the squared value of 2.
Since squaring any number results in a non-negative value, the term inside the absolute value should be minimum or equal to zero to achieve the smallest possible value for z.
For the term 2^2 - x^2 - y^2, we want x and y to take values that make the term equal to zero. This implies that x^2 + y^2 should be equal to 2^2. One such possible option is x = 2 and y = 0.
With x = 2 and y = 0, the term 2^2 - x^2 - y^2 = 4 - 4 - 0 = 0. Plugging this into the expression for z, we get z = |0| + 2^2 = 0 + 4 = 4.
Therefore, the smallest possible value of z is 4.