In the following figure, the circle centered at n has a radius of 4. What is the area of the shaded region?

Solution
Use the formula for the area of a circle:
A = πr^{2} = π(4)^{2} = 16π
The right angle accounts for 90° of the 360° circle, which is ^{1}⁄_{4} of it.
So the shaded region of the circle is ^{3}⁄_{4} the area of the circle:
^{3}⁄_{4}(16π) = 12π
Noreen recently took a job helping people register to vote. The job has a mandatory 10day period of probation during which her success rate is strictly monitored. On her first day, she registered 30 people. Then, for each of the next 9 days, she registered 4 more people than she did on the previous day. How many people did she register altogether during her probationary period?

Solution
Jot down how many people Noreen registered on each day:
30 34 38 42 46 50 54 58 62 66
To save time adding all these numbers, notice that the total of the first and 10th numbers is 30 + 66 = 96.
This total is the same for the 2nd and 9th, the 3rd and 8th, the 4th and 7th, and the 5th and 6th.
Therefore, you have five pairings of days on which Noreen registered 96 people.
You can simply multiply to find the total:
96 × 5 = 480
If x = 6 and y = –2, what is the value of 3xy + 2x^{2} – y^{3}?

Solution
First, plug in 6 for x and –2 for y throughout the expression:
3xy + 2x^{2} – y^{3} = 3(6)( –2) + 2(6)^{2} – (–2)^{3}
Now simplify using the order of operations:
= 3(6)( –2) + 2(6)(6) – (–2)(–2)(–2) = –36 + 72 + 8 = 44
What is the missing number in the sequence 1, 5, 10, 16, 23, 31, __?

Solution
Each number in the sequence is a bit higher than the one before it, so see how much needs to be added to each number to produce the rest.
As you can see, if you add 4, then 5, then 6, and so forth, the numbers add up correctly:
1 (+ 4) = 5 (+ 5) = 10 (+ 6) = 16 (+ 7) = 23 (+ 8) = 31 (+ 9) = 40
Jackson worked 25 hours and received $225. At the same rate of pay, how much would he make if he worked 40 hours?

Solution
Jackson worked 25 hours for $225, so divide to find out how much he earned per hour:
225 ÷ 25 = 9
Now multiply $9 per hour by 40 hours:
40 × 9 = 360
Therefore, he would earn $360 in 40 hours.
The equation y = ax^{b} + c produces the following (x, y) coordinate pairs: (0, 2), (1, 7), and (2, 42). What is the value of abc?

Solution
To begin, plug x = 0 and y = 2 into the equation y = ax^{b} + c.
Note that the first term drops out:
2 = a(0)^{b} + c
2 = c
Now you can substitute 2 for c in the original equation, giving you y = ax^{b} + 2.
Next, plug in x = 1 and y = 7.
Notice that everything drops out except for the coefficient of the first term:
7 = a(1)^{b} + 2
5 = a
You can now substitute 5 for a in the equation, giving you y = 5x^{b} + 2.
Now plug in x = 2 and y = 42:
42 = 5(2)^{b} + 2
40 = 5(2)^{b}
8 = 2^{b}
2^{3} = 2^{b}
3 = b
Finally, you can see that abc = (5)(3)(2) = 30.
If the least common multiple of 9, 10, 12, and v is 540, which of the following could be v?

Solution
To begin, notice that 540 isn’t a multiple of 24, so you can rule out Choice (B).
Now find the least common multiple (LCM) of 9, 10, and 12.
The LCM of 9 and 10 is 90, so the LCM of 9, 10, and 12 must be a multiple of 90.
Here are the first six multiples of 90:
90, 180, 270, 360, 450, 540
The number 180 is a multiple of 12 as well, so the LCM of 9, 10, and 12 is 180.
However, 180 also is a multiple of 18, 36, and 45.
So if any of these numbers were v,the LCM of 9, 10, 12, and v would be 180.
As a result, you can rule out Choices (A), (D), and (E), leaving Choice (C) as your only answer.
When she chooses a password, Eloise always uses exactly ten different characters: five letters (A, B, C, D, and E) and five numbers (2, 3, 4, 5, and 6). Additionally, she always ensures that no pair of letters is consecutive and that no pair of numbers is consecutive. How many different passwords conform to these rules?

Solution
The first character of the password can be any letter or number, so Eloise has ten options.
Her second choice must be from the set (letter or number) not yet used, so she has five options.
Choice 3 is from the same set as Choice 1, and she has four options left.
Choice 4 is the second item from the same set as Choice 2, so she has four options.
Choices 5 and 6 are from different sets, each with three options; Choices 7 and 8 are from different sets, each with two options; Choices 9 and 10 are from different sets, each with only one option remaining.
You can see this information in the following chart:
1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th
10 5 4 4 3 3 2 2 1 1
To find the total number of possible passwords, multiply these numbers together:
10 × 5 × 4 × 4 × 3 × 3 × 2 × 2 × 1 × 1 = 28,800
If the equation x^{2} + mx + n = 0 has two solutions, x = k and x = 2k, what is the value of mn in terms of k?

Solution
Given that the equation x^{2} + mx + n = 0 has two solutions, x = k and x = 2k, you can work backward to build the original equation. Here’s how:
x = k; x = 2k
x – k = 0 ; x – 2k = 0
Now take the two equations and combine them:
(x – k)(x – 2k) = 0
x^{2} – 2kx – kx + 2k^{2} = 0
x^{2} – 3kx + 2k^{2} = 0
As you can see, m = –3k and n = 2k^{2}, so mn = –6k^{3}.
The following figure shows the graph of y = 5sin(^{x}⁄_{2}). Which of the answer choices are the correct amplitude and period of this function?

Solution
The amplitude is the vertical distance from the midpoint to the crest, so in this case it’s 5.
The period is the horizontal distance between two adjacent crests, so it’s 4π.