1. Let f be a function such that f(x) = f(-x) for all real numbers x. If the point (-2, 4) lies on the graph of y = f(x) in the xy-plane, which of the following points must also lie on the graph of y = f(x)?
A. (-2, -4)
B. (0, 0)
C. (0, 4)
D. (2, -4)
E. (2, 4) Correct
Explanation
The function f(x) = f(-x) indicates an even function, which is symmetric about the y-axis. Given the point (-2, 4), we have f(-2) = 4. By the property of even functions, f(2) = f(-2) = 4, so the point (2, 4) must lie on the graph. Option B, (0, 0), is not required since f(0) = 0 is not given. Option C, (0, 4), implies f(0) = 4, which is not necessarily true. Options A and D, with y-coordinate -4, change the function value, which is not supported by the symmetry property.
2. If z = a + bi = (3+i)(2-3i), then z is equal to which of the following?
A. (3-7i)
B. (3+5i)
C. (6-6i)
D. (9-7i) Correct
E. (9+5i)
Explanation
To compute z = (3+i)(2-3i), use the distributive property (FOIL): (3*2) + (3*-3i) + (i*2) + (i*-3i) = 6 - 9i + 2i - 3i^2. Since i^2 = -1, this becomes 6 - 7i + 3 = 9 - 7i, matching option D.
3. The sum shown is consistent with which of the following?
A. 1/n
B. n/(n+1) Correct
C. (n+1)/(n+2)
D. 1/(n+1)
Explanation
The general term of the series is n/(n+1), with n from 1 to 99, matching option B. Option A represents a harmonic series, 1/n. Option C generates terms from 2/3 to 100/101. Option D represents reciprocals from 1/2 to 1/100, which does not match the given series.
4. The model for the frog population was given by the function F(m)=m+6, and the model for the turtle population was given by the function T(m)=m^2+3/m+1. Which of the following expressions defines the function F(m)-T(m), the difference in size between the two populations m months after the initial count?
A. 7m+9/m+1
B. -7m-3/m+1
C. 7m+3/m+1 Correct
D. #NAME?
E. #NAME?
Explanation
To find F(m) - T(m), subtract T(m) = m^2 + 3/(m+1) from F(m) = m + 6. Rewrite m + 6 with denominator m + 1 as (m + 6)(m + 1)/(m + 1) = (m^2 + 7m + 6)/(m + 1). Subtract: (m^2 + 7m + 6)/(m + 1) - (m^2 + 3)/(m + 1) = (m^2 + 7m + 6 - m^2 - 3)/(m + 1) = (7m + 3)/(m + 1), matching option C.
5. The equation (x-1)(x^2+x+1)(x^2-3x-4)=0 has exactly how many distinct real roots?
A. One
B. Two
C. Three Correct
D. Four
E. Five
Explanation
Set each factor to zero. For x - 1 = 0, x = 1. For x^2 + x + 1 = 0, the discriminant is 1 - 4 = -3, so no real roots. For x^2 - 3x - 4 = 0, factor as (x - 4)(x + 1) = 0, giving x = 4, x = -1. The distinct real roots are x = 1, 4, -1, totaling three, matching option C.