<p>1) C ; but this has nothing to do with the word “number” in the sentence. Implicit in the sentence is “(The idea/notion/theory) that there exist…”, so the writer is employing the verb with respect to that singular subject. The sentence could be reduced to a simpler form, “The idea [was].”</p>
<p>2) D ; I will elaborate on the “(to contort)” by saying that when ability is used in the context of ability to do, an infinitive is always used. If you are a native speaker of English, “ability of contorting” should jump out at you as unnatural.</p>
<p>3) E ; Think of the quadratic coefficient this way: by increasing the value of (a), you’re increasing the output value for any given input value, which means the “arms” of the parabola will extend higher with less extreme values of (x).</p>
<p>You might also think of it like an acceleration curve, if you are increasing the rate of acceleration by a factor, you will get to a higher velocity, f(x), in a shorter time, (x), making the curve steeper/closer to the y-axis/however it helps to think about it.</p>
<p>Hopefully one of those two explanations makes good sense to you.</p>
<p>4) B ; I think this has already been clearly resolved, but I’m writing anyway.</p>
<p>(s) is an integer, and it is less than 10000. This is the larger variable, so to find the greatest difference, it must be as large as possible. The largest possible integer that is less than 10000 is 9999.</p>
<p>Similarly, (r) is an integer, and (in addition to being greater than 1000) must be at least 9000. To find the greatest difference, (r) must be as small as possible, and the smallest number allowed is 9000. </p>
<p>9999 - 9000 = 999</p>
<p>5) C ; Again, this has already been said, but I like completeness and I have nothing better to do/am procrastinating on something I should be doing.</p>
<p>The equations for these lines will be simply (y = mx), as both cross the origin. So find dy/dx for the line (l), which is 2/3, then take the negative reciprocal of that to determine the slope of (k), as it is perpendicular to (l). The answer is the point which creates a slope equal to that of line (k) when connected to the origin. And stuff.</p>