(C) Since division by zero is forbidden, x cannot equal 2.

(E) For each value of x there is only one value
for y in each case. Therefore, f, g, and h are all functions.

(D) Complete the square in both x and y to get the standard equation:

(C) Use your calculator to evaluate

(E) The multiplication rule applies. There are (26)(26)(10)5 = 67,600,000 possible codes.

(C) You can either include or exclude each of the seven ingredients in your salad, which means there are 2 choices for each ingredient. According to the multiplication rule, there are 27 = 128 ways of making these yes-no choices.

The best way to solve inverse-function questions on the Math Level 2 is to Plug In numbers. Be sure to pick a number that will work out neatly. For example, for the function f(x) = - 1, a good value for x would be 4. Then, f(4) = - 1 = 2 - 1 = 1. You can see that f(x) turns 4 into 1. So f-1(1) = 4. To find the correct answer, plug 1 into all of the answer choices. The correct answer will produce a value of 4. Only (A) comes out to 4, since (1 + 1)2 = 4, and so (A) is the correct answer.

(A) Average . Therefore, x = 85. [4.1]

D Move the equation around so that it's in y = mx + b formula: y = x - 8. So an equation perpendicular would have a slope of -. The only one is (D).

Sketch a diagram:

(D) Use the quadratic formula program on your graphing calculator to get both zeros and choose the positive one. [1.2]

Drawing this one is helpful. You'll find that the radii of the two circles have to add up to the distance between the circles' centers. You can find that distance using the distance formula, d = You'll find that the two centers are separated by a distance of 8.6023. Since one circle has a radius of 4, the other must have a radius of 4.6023.

D No ordinary calculator can work with exponents this big, and there's no way to spot the biggest values here by looking at them; you've got to get tricky. The important fact about this question is that it's not necessary to find the exact value of any expression merely to compare them. The best way to compare these expressions is to get them into similar forms. To start with, rearrange as many answer choices as possible so that they have exponents of 100. (C) can be expressed as (35)100, or 243100; (D) can be expressed as (44)100, or 256100; and (E) is already there-250100. Suddenly it's easy to see that (D) is the biggest of the three, and eliminate (C) and (E).
Next, take a look at (A). The exponent 999 is approximately 1,000. The expression is therefore worth a little less than (1.7310)100, or (240.14)100. That's definitely smaller than (D), so you can eliminate (A) as well.
Finally, take a look at (B). The expression 2799 is almost equal to 4400. How can you tell? Well, 4400 can be written as (22)400, or 2800. That makes it clear that (D) is bigger than (A). Answer choice (D) reigns supreme.

(D)f(2) = logb 2 = 0.231. Therefore, b0.231 = 2, and so b = 21/0.231 , which (using your calculator) is approximately 20.1. [1.4]

It helps to put parentheses around x + y. Then use FOIL.

Type this into your calculator, being careful to use parentheses. You should get either 3025000000 or something like 3.025e9. To get 3025000000 into scientific notation, you need to move the decimal point 9 places to the left, which means your power of 10 will be 9; pick (E).

This is a visual perception question, and there's no hard and fast technique to follow to solve it. The best plan is to experiment with sketches in your test booklet and use common sense. Remember that this is an EXCEPT question, so you're looking for a shape that can't be made. Any time you find a way to make one of the shapes in the answer choices, that choice can be eliminated.
The intersection of a cube and a plane can be a triangle:

(C) From the Venn diagram below you get the following equations:
a + b + c + d = 30 (1)
b + c = 20 (2)

(E)

Sketch the diagram:

(A) The plane cuts the x-axis at 6, the y-axis at 5, and the z-axis at 3. The base is a right triangle with area

(D) Divide both sides of the equation by 200 to write the equation in standard form . The length of the major axis is [2.1]

A good way to tackle this one is by trying to disprove each of the answer choices. If you start with (A), you're lucky. There's no way to divide 5 by another quantity and get zero; it's the right answer. Even if you weren't sure, the other answer choices are pretty easy to disprove. Just set equal to a quantity prohibited by each answer choice, and solve for x. If there's a real value of x that solves the equation, then the value is in the range after all, and the answer choice is incorrect. Another method is to graph the function on your calculator and see what y-values seem impossible.

(C) Since f(x) = an integer by definition, the answer is Choice C.

E Plug -3 into the function: (-32) - 3(-3) = 18.

D In a 3-digit number containing no zeros, there are nine possibilities for the first digit (1-9); nine possibilities for the second digit (1-9); and nine possibilities for the third digit (1-9). That makes a total of 9 × 9 × 9 possible 3-digit numbers, or 729.

An arithmetic sequence is one that increases by adding a constant amount again and again. The most important information to have when you're working with an arithmetic sequence is the size of the interval between any two consecutive terms in the sequence. Since the 20th term in the sequence is 20 and the 50th term is 100, you know that 30 steps in the sequence produce an increase of 80. That makes each step worth or . To find the first term in the sequence, you can use the formula for the nth term in an arithmetic sequence, an = a1 + (n - 1)d, where n is the number of the term and d is the interval between any two consecutive terms. In this case, you know that a20 = 20, and that d = , or . You can then fill those values into the formula, so

Expand both sides and simplify:

Re-express the equation in standard form:

The question is asking, in other words, for how many positive integer values of x does y turn out to be a positive integer?

(A) The formula for the sum of a geometric series is , where a1 is the first term and r is the common ratio. In this problem a1 = 6 and

C PITA, starting with (C). Use your calculator to see which value makes the equation true. The first one you try, (C), is correct.
To solve the problem algebraically, isolate x one step at a time. First, multiply through by 3 on the right: x - 3 = 3 - 3x. Next, add 3x to each side, and then add 3 to each side, to get 4x = 6. Divide each side by 4 to get x = , or 1.5.

Don't panic because you've never seen a term like "x mod y" before. This isn't something you slept through in math class. It's just one of those terms ETS throws at you sometimes. Some will be little-known math terms, and others will be made up. Either way, it doesn't matter whether you've seen it before, because ETS defines it for you. To find the value of any "x mod y," just take the number in the x position and divide it by the number in the y position. The remainder is the value of "x mod y" for those numbers. The value of 61 mod 7 is 5, because the remainder when 61 is divided by 7 is 5. The value of 5 mod 5 is 0, because the remainder when 5 is divided by 5 is zero. The expression (61 mod 7) - (5 mod 5) can be rewritten as 5 - 0, which equals 5. The correct answer is (D).

This is a combinations question, since rearranging the order of the dishes doesn't change the dinner. Since the dinner's being ordered in two parts (the part from column A and the part from column B), you should calculate the number of combinations in two parts. Start with the five dishes from column A. The number of permutations for five items selected from a group of ten is given by 10 × 9 × 8 × 7 × 6 = 30,240. To find the number of combinations, divide that number by 5! = 252. There are 252 possible combinations of five dishes from column A. You're also selecting five dishes from column B, which has twenty selections. The number of combinations for column B will therefore be given by = 15,504 So there are 15,504 combinations of 5 dishes from column B. To find the total number of possible combinations, multiply these figures together, 15,504 × 252 = 3,907,008. The correct answer is (D).

(C)

(A) Add the two equations: log2 m + log2 n = x + y, which becomes log2 mn = x+y (basic property of logs). 2x + y = mn. [1.4]

A To find the probability, first figure out the total number of possibilities, and then figure out how many meet the condition you want. Since there are 6 possible rolls on a fair cube, the total number of possibilities for two rolls is 6 × 6 = 36. Now you need to figure out all the ways to get a product greater than 18. If you roll a 1, 2, or 3 on the first cube, you're out of luck, since the most you could roll would be 3 × 6 = 18, but you want more than 18. The rolls that will work are 4 × 5, 4 × 6, 5 × 4, 5 × 5, 5 × 6, 6 × 4, 6 × 5, and 6 × 6. That's 8 rolls out of 36, which is a probability of =0.222.

(C) To get from f(x) to f(g(x)), x2 must become 4x2. Therefore, the answer must contain 2x since (2x)2 = 4x2.

First, determine how many numbers are available with just the restriction on the first digit. For the first digit, there are 8 choices, and then for the remaining 6 digits, there are 10 choices. So the pool of available numbers is 8 × 10 × 10 × 10 × 10 × 10 × 10, or 8,000,000. If phone numbers cannot start with the digit pattern 911, then this eliminates 1 × 1 × 1 × 10 × 10 × 10 × 10 numbers, or 10,000. There are another 10,000 numbers starting with 411. So the actual number of available phone numbers is 8,000,000 - 20,000 = 7,980,000.