(B) Points 6 in. from m form a cylinder, with m as axis, which is tangent to plane X. Points 1 in. from X are two planes parallel to X, one above and one below X. The cylinder intersects only one of the planes in two lines parallel to m. [2.2]

(D) The square of the distance between P and Q is 9, so

(D) The length of the longest segment is

Rewrite the equation to be x3 = 125. Take the cube root of both sides to get x = 5.

Sketch a diagram:

This question tests your understanding of exponent rules. You're told that 4x + 2 = 48, but how do you solve for 4x- Remember that when you multiply exponential terms that have a common base, you add the exponents. In the same way, you can express addition in an exponent as multiplication.

This is the number of permutations of 9 things taken 9 at a time.

(B) There are 49 data values altogether, so the median is the 25th largest. Adding the frequencies up to 25 puts the 25th number at 3.

(C) The line 3x + 2y = 7 has x -intercept and y -intercept . The part of this line that lies in the first quadrant forms a triangle with the coordinate axes. Rotating this triangle about the x -axis produces a cone with radius and height . The volume of this cone is .

(A) A Venn diagram will help you solve this problem.

(B) The inverse is {(2,1),(3,2),(4,3),(1,4),(2,5)}, which is not a function because of (2,1) and (2,5). Therefore, the domain of the original function must lose either 1 or 5.

(A) Since the degree of the polynomial is an even number, both ends of the graph go off in the same direction. Since P(x) increases without bound as x increases, P(x) also increases without bound as x decreases.

The second equation expresses y2 in terms of x, so you can substitute that expression for y2 in the first equation and solve for x:

(B) 1 yen equals 0.0076 euros, and 1 euro equals dollars. Therefore, 1 yen equals 0.0076 × 1.40 = 0.011 dollars. [algebra]

PITA. Plug each answer choice into the equation for t, to see which one makes d = 10. (B) works.

Sketch a diagram:

E All the sides of a square are the same. So each side must be 15. Since the area of a square is (side)2, the area must be 225.

The graph crosses the x-axis at the point where f(x) = 0. So

This simple function question just requires you to plug 6 into g(x). You can start by eliminating (A) and (B), because the entire function is contained within an absolute value sign, so it can't produce negative values. To find the exact value of g(6), Plug In the number. You get |5(6)2 - (6)3|, or |-36|, which equals 36.

PITA. Plug each answer choice into the function for x, to see which one spits out -4. (D) works, because f (1) = (1)2 - 5(1) = 1 - 5 = -4.

The first digit can only be the number 4. Therefore, there is only one option for this digit. There are 4 digits left to choose from for the second digit and 3 digits to choose for the third. Multiply to get 12, which is the number of 3-digit codes that can be formed using each digit only once.

You're dealing with imaginary numbers here, so your calculator won't be much use. To solve this problem, just plug (i - 1) into the expression in place of x and do the math.

If is the midpoint of and C(x, y), then 6 is the average of and x, and is the average of 4 and y :

First isolate one of the variables:

(B) Regardless of what is substituted for x, f(x) still equals 2
Alternative Solution: f(x + 2) causes the graph of f(x) to be shifted 2 units to the left. Since f(x) = 2 for all x, f(x + 2) will also equal 2 for all x. [1.1]

The company charges a flat fee of $42.50, plus x dollars times the number of hours. The problem asks for the cost of a 90-minute call, which is 1.5 hours. The correct expression is therefore 42.50 + 1.5x.

PITA. Starting with (C), Plug In the values from the answer choices for a in the original expression. If the value makes the equation true, you've got the right answer. If not, then pick another answer choice and try again.

After spending 30% on rent, Jackie has 70% of her earnings left. Half of that 70% is 35%, which goes for food and transportation. So the ratio of rent to food and transportation is 30:35, or 6:7. Now you can set up a proportion and solve for the rent:

First, raise both sides of the equation to the third power to eliminate the cube root on the left-hand side. This results in or Cross multiply to get 27(2x + 3) = 40, or 54x + 81 = 40. Subtract 81 from both sides and divide both sides by 54 to get

A Work from the inside out. g(2) = 22 + 7 = 11. Now put in 11 for x: f(11) = 5 - 2(11) = -17.

(D) (g of)(x) = g(f(x)) = . Therefore, x 7. [1.1]

E If you have a graphing calculator, press the Y= key and enter the function. If you check the values of the TABLE, you can find that f(-3.5) and f(2.5) both equal 39. If you don't have a graphing calculator, you can PITA. It may take a while, but you'll get it.

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.

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.

(C) Relative to the statement, answer choice A is the converse, B is the inverse, C is the contrapositive, and D is another form of the inverse. Of these, the contrapositive is the logical equivalent of the original statement. [logic]

(C) The infinite product can be approximated by using your calculator and a "large" value of n. The exponents form a geometric sequence with a first term of and constant ratio of . Enter prod to approximate the product of the first 50 terms as 2.08. Evaluating the product of 75 terms yields the same approximation to 9 decimals, so choose C.

(E) Mean Therefore, x = 8.9. [4.1].

E Since there are variables in the answer choices, you should Plug In. Try x = 3. We are trying to find f(3 + 4) = f(7) = 2(7)2 + 2, which is 100, our target number. Now plug 3 in for x in the answer choices, to see which answer choice hits the target. Only (E) works.

PITA. Plug each answer choice in for x to see which one makes f(x) = 0. (B) works. Alternately, you could graph y = x2 + 6x - 12 on your calculator and see where it crosses the x-axis.

Distance = rate × time.