# Multiple choice practice test

 1. (35 p.) A function $$f$$ is defined by $$f(z)=(4+i)z^2+\alpha z+ \gamma$$ for all complex numbers $$z$$, where $$\alpha$$ and $$\gamma$$ are complex numbers. Given that $$f(1)$$ and $$f(i)$$ are both real, find the smallest possible value for $$|\alpha|+|\gamma|$$.    A    $$1$$    B    $$\sqrt2$$    C    $$2$$    D    $$2\sqrt2$$    E    $$4$$    N

 2. (42 p.) Suppose that the sum of base-10 logarithms of the divisors of $$10^n$$ is 792. Determine $$n$$.    A    11    B    12    C    13    D    14    E    15    N

 3. (7 p.) Points $$C$$ and $$D$$ are on the same side of diameter $$AB$$ of circle $$k$$. Assume that $$\angle AOC=30^{\circ}$$ and $$\angle DOB=45^{\circ}$$. Let $$\alpha_1$$ denote the area of the smaller sector $$COD$$ of the circle, and let $$\alpha$$ denote the area of the entire circle. Calculate the ratio $$\frac{\alpha_1}{\alpha}$$.    A    $$\frac29$$    B    $$\frac14$$    C    $$\frac5{18}$$    D    $$\frac7{24}$$    E    $$\frac3{10}$$    N

 4. (1 p.) A basketball player made five successful shots during a game. Each shot was worth either 2 or 3 points. How many different numbers could represent the total points scored by the player?    A    2    B    3    C    4    D    5    E    6    N

 5. (12 p.) The operation $$\circ$$ is defined on the set of real number as $$a\circ b=(a-b)^2$$. What is $$(x-y)^2\circ(y-x)^2$$?    A    0    B    $$x^2+y^2$$    C    $$2x^2$$    D    $$2y^2$$    E    $$4xy$$    N

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