(A) Either (3,2) or (3,1), which is not an answer choice, must be removed so that 3 will be paired with only one number.

You're looking for sets of three positive integers with a product of 32. The easiest way to do that is just to list them systematically. If the smallest dimension is 1, then the product of the other two dimensions is 32, so there are these possibilities that include an edge of 1:

(D) Converting the log expressions to exponential expressions gives and . Therefore, mn = .

If a varies directly as b, then a is equal to some constant times b:

(E) Rational roots have the form , where p is a factor of 12 and q is a factor of 2.

(D). Since k is in the denominator, it cannot equal 0. [1.1]

First, eliminate choice (E), since it is a linear equation, not a quadratic equation. Then you could attempt to factor each equation to determine the roots, then check the sum and the product of these roots. However, you may recall that in a quadratic equation in the form ax2 + bx + c, the sum of the roots is and the product of the roots is In choice (A), and

To find the average of four expressions, add them up and divide by 4. The sum is 8x + 4. When you divide that by 4, you get 2x + 1.

(C) There are ways of choosing 2 boys out of 5 and ways of choosing 2 girls out of 6. Therefore, there are 10 × 15 = 150 ways of choosing the homecoming court.

(E) Law of sines: .

(C) Since no three points are collinear, every pair of points determines a distinct line. There are such lines.

(C) The word MINIMUM contains 7 letters, which can be permuted 7! ways. The 3 M's can be permuted 3! ways, and the 2 I's can be permuted in 2! ways, so only permutations look different from each other. Therefore, there are distinguishable ways the letters can be arranged. [3.1]

Use substitution.

To find the increase in f(x) as x goes from 2 to 3, calculate f(2) and f(3) by plugging those numbers into the definition of the function. You'll find that f(2) = 0.0589 and f(3) = 1.4308. The increase in f(x) is the difference between these two numbers, 1.3719.

Since g(x) = f(5x), g(2) = f(5(2)) = f(10).

When 5xy = 2, (5xy)5xy = 22 = 4.

(B) The equation of a vertical parabola with its vertex at the origin has the form y = ax2 . Substitute (7,7) for x and y to find . When y = 6, x2 = 42. Therefore, , and the segment = .

It's algebra time. Whenever two fractions are equal, you can cross-multiply.

Draw the parallelogram with the height shown.

(D) If a = b, then f(a) = f(b). Since f(a) f(b), it follows that a b. [1.1]

Use FOIL:

The equations in the beginning of this question can be rearranged into the Law of Sines. A little algebraic manipulation gets you = and = . These equations can be combined into = = which is the Law of Sines. This tells you that the lengths of the triangle's sides are in a ratio of 7:10:4. So, you can call the sides 7x, 10x, and 4x. They add up to 16, so 21x = 16, and x = 0.7619; side a has length 7x, which equals 5.3333. The answer is (C).

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.

You could certainly try out all five answer choices, and solve five sets of simultaneous equations, but it's easier to think of the graphs of these simultaneous equations. Three things can happen with simultaneous equations. First, the two given equations may be identical (or one is simply a multiple of the other), which means they describe the same line. That results in infinite solutions, since there are infinite points on a line. Second, the lines are parallel, because one equation has the same slope as the other, but a different y-intercept. That results in zero solutions, since the lines never intersect. Third, the lines have different slopes, which means they will always intersect in one point. This results in one solution. In this problem, all the answer choices fit the third situation, except (E), which fits the second situation. If you look at the two equations that result

On the Math Level 2, most trigonometry questions like this one are solved by using trigonometric identities to change the form of equations. Most often, the most successful strategy is to start by getting everything in terms of sine and cosine. The ratio you're given can be written in fractional form, like . The secant and cosecant can also be expressed in terms of sine and cosine, = . This fraction simplifies to = , and since , this can be written as tan x = . The cotangent is the reciprocal of the tangent, so cot x = 4. The ratio of tan x to cot x is therefore equal to , or . The correct answer is (A).

The expression 6 + (x - 2)2 is at its minimum when (x - 2)2 is at its minimum value. Since the expression is squared, its minimum is zero. (x - 2)2 = 0 when x = 2.

C You can just factor this one, and then cancel.

Determine the equation of the line through the points (1,5) and (3,9). The slope is which is 2. Find the y-intercept, b: 5 = 2(1) + b, so b = 3. The equation of the line is y = 2x + 3. The line perpendicular to this line has a slope that is the negative reciprocal, namely and the y-intercept is -2, because the given point is (0,-2). Now, find the intersection of these lines.

(D) Even functions are symmetric about the y-axis. Graph each answer choice to see that Choice D is not symmetric about the y-axis.

You could do this one by picking numbers, but for any set of numbers you might pick, more than one of the answer choices will be true. You will therefore have to pick at least two sets of numbers to find the correct answer. The statement that must be true is the one that holds for all a, b, c, and d that fit the given conditions. (A) is tempting: You might think that the product of the larger a and c will be greater than the product of the smaller b and d, and it is if all you pick are positive integers. (A) doesn't hold when you consider negatives, however. But (C) holds for all possible values of a, b, c, and d. The sum of two larger quantities is greater than the sum of two smaller quantities.

This is a complex number, because it has both real and imaginary components. To find its absolute value, visualize the term 6 - 3i on the complex plane, where one axis represents real values and the other represents imaginary values.

(D) The measure of the central angle is superfluous. Areas of similar figures are proportional to the squares of linear measures associated with those figures. Since the ratio of the areas is 1 : 2, the ratio of the radii is , or approximately 0.71 : 1. [1.3]

The point where the horizontal line and the parabola intersect must be the vertex of the parabola. In order to find the point of intersection, set the equations equal to each other.

Since Cassie's stuck in the driver's seat, the number of permutations of the people in the car is simply determined by the possible arrangements of passengers in the 3 passengers' seats. The number of permutations of 3 items in 3 spaces is given by 3!, which equals 3 × 2 × 1, or 6.

(C) Arithmetic mean Geometric mean The difference is 8.

To satisfy the Triangle Inequality Theorem, adding the two shorter sides must give you more than the longest side. In (B), (D), and (E), the two shorter sides add up to less than the longest side, and in (A), they add up to the same as the longest side. Only in (C) do they add up to more.

Always sketch situations that are described but not shown. Here's the situation described in the question:

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).

(D) Substitute in A = Pe0.03t to get 600 = 300e0.03t. Simplify to get 2 = e0.03t . Then take ln of both sides to get ln 2 = 0.03t and t = . Use your calculator to find that t is approximately 23.