<ol>
<li>The diameter of the sphere is the long diagonal of the cube. We use the Generalized Pythagorean Theorem:</li>
</ol>
<p>d^2=a^2+b^2+c^2=2^2+2^2+2^2 = 4+4+4=12</p>
<p>So d=sqrt(12)=2sqrt(3), choice (D)</p>
<p>Notes:</p>
<p>(1) Since the volume of the cube is 8, the length of a side of the cube is cuberoot(8) = 2.
(2) a,b,c are the length width and height of the rectangular solid. In this case each is 2.
(3) You don’t need to simplify the square root at the end. You can just put sqrt(12) in your calculator.</p>
<p>Well, for number 1, the volume is 8, so the length (l), to the third power has to equal 8 because l^3=V. So you plug it in and find that a length of the cube is 2. From there you do a bunch of Pythagorean theorem. First you find the diagonal of one of the faces of the cubes. So a^2 + b^2 = c^2. 2^2+2^2=c^2. so c= 2rad(2). Ten you use the diagonal length and one of the edge lengths to find the diagonal from opposite corners of the cube. So 2rad(2)^2 + 2^2 = c^2. so c=rad(12), or 2rad(3). This diagonal is equal to the diameter, so the answe is D.</p>
<p>Drsteve you beat me to it >:0</p>
<ol>
<li>The remainder must be 3 when the number is divided by 4. Let’s start with choice (C). Note that 56 is divisible by 4. Subtracting 1 gives 55, but that’s not an answer choice. So subtract another 4, and we get 51. So the answer is choice (A).</li>
</ol>
<p>Notes:</p>
<p>(1) Remainders are cyclic. So when dividing by 4 the remainders follow the pattern 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0…</p>
<p>So, since 56 is divisible by4, 55 has remainder 3 when dividied by 4, 54 has remainder 2 when divided by 4, etc.</p>
<p>(2) Dividing in your calculator does NOT give you a remainder. You MUST do division by hand to find a remainder!</p>
<p>For number 9, 2f(x) just means to multiply all y values by 2, and the one that does that is D, which is the right answer.</p>
<ol>
<li>If you round 1.783 to the nearest whole, you get 2. If you round it to the nearest tenth, you get 1.8. The difference between them is .2.</li>
</ol>
<p>Haha - sorry giatns. I’ll give you a chance to do some. So I’ll jump to the last one:</p>
<ol>
<li>Just plug in points:</li>
</ol>
<p>From the original graph, f(0)=1 (ie, the point (0,1) is on the graph).</p>
<p>So, 2f(0) = 2*1=2 (so (0,2) is on the new graph).</p>
<p>So the answer is (A) or (E).</p>
<p>But multiplying a function by a nonzero number does NOT change the x-intercepts. So the answer is choice (D).</p>
<ol>
<li>The outside is what’s been added on from the last square (we know it’s a square because it has the same length and width. You have to know that the difference between two consecutive squares is an odd. The only odd there is B, so B is the right answer.</li>
</ol>
<ol>
<li>Draw a picture. If it were a 3,4,5 triangle, then the angle would be 90 degrees (since a 3,4,5 triangle satisfies the Pythagorean Theorem. Since 6 is bigger than 5, the angle opposite 6 must be bigger than 90, choice (E).</li>
</ol>
<p>Remark: In triangles size relationships of sides are the same as the corresponding size relationships of the angles opposite these sides.</p>
<ol>
<li>Just plug in numbers and see what works. So 2^3 + 2^4 = k = 24. 2^5 = 32. so plug 24 in until you get 32, and look at that, 4k/3 works, so that’s your answer.</li>
</ol>
<ol>
<li>Just kinda look at and see that the y tends to decrease as the x increases, therefore the best fit line is negative.</li>
</ol>
<p>Here’s another way to do 5:</p>
<p>We will systematically try values for n, and draw a picture of the situation to determine the corresponding value for k.</p>
<p>Here is the picture for n = 3.</p>
<hr>
<hr>
<hr>
<p>Note that k = 9 – 1 = 8.</p>
<p>Here is the picture for n = 4.</p>
<hr>
<hr>
<hr>
<hr>
<p>Note that k = 16 – 4 = 12.</p>
<p>Here is the picture for n = 5.</p>
<hr>
<hr>
<hr>
<hr>
<hr>
<p>Note that k = 25 – 9 = 16.</p>
<p>So the pattern appears to be 8, 12, 16, 20, 24, 28,…</p>
<p>Make sure that you keep drawing pictures until this is clear to you.</p>
<p>So we see that the answer must be divisible by 4.</p>
<p>So choice (E) is the answer.</p>
<p>Advanced solution: Let’s prove that for each n, the corresponding k is divisible by 4.</p>
<p>For fixed n, the total number of trees is n^2, and the number of trees not on the boundary is (n-2)^2 = n^2 – 4n + 4. Thus, the number of trees on the boundary is k = n^2 – (n^2 – 4n + 4) = n^2 – n^2 + 4n – 4 = 4n – 4 = 4(n – 1) which is divisible by 4.</p>
<ol>
<li>The error is in C. This is because the pronoun ‘he’ is too ambiguous, and it is unclear whether he refers to Rivers or Peters. Therefore, we should replace with the name itself: ‘Rivers’ so that it can read: so Rivers held him in contempt.</li>
</ol>
<p>Another but similar way to come to the conclusion for #9 is to consider x = 2 for f(x), where it is obviously going to be a negative number. A negative number * 2 is basically -f(x) * 2, and so the ‘y’ of the equation will be negative. Choice (A) does not have its ‘y’ remain consistent with being negative around x=2, but Choice (E) does.</p>
<p>As for the first one, I had a different approach. I visualized the cube in the sphere and knew that the diameter is the length from one vertex to the absolute vertical and horizontal opposite vertex. It’s difficult to describe without showing a picture of it and so I’ll leave it be, but just know that finding the answer this way is plausible.</p>
