<ol>
<li>What is the range of the function defined by f(x) = x^(1/3) with x > 2 and = 2x - 1 with x <= 2?
My answer is 2^(1/3) < y < 3 but official guide answer is all real numbers.</li>
<li>If n distinct planes intersect in a line, and another line L intersects one of these planes in a single point, what is the LEAST number of these n planes that L could intersect?
Answer: n - 1</li>
<li>An indirect proof of the statement “If x = 2, then square root of x is NOT a rational number” could begin with the assumption that
A. X = square root of 2
B. X^2= 2
C. Square root of x is rational
D. X is nonnegative
Answer: C</li>
<li>In how many ways can 10 people be divided into two groups, one with 7 people and other with 3 people</li>
</ol>
<p>(1). x^(1/3), x>2: Range is all real numbers greater than 2^(1/3). So when x>2, f(x) is any number larger than ~1.26.</p>
<p>2x-1, x=<2: Range is any number less than 2*2 - 1. So here, x is any number less than 3.</p>
<p>So the range of the function is any number greater than 1.26 and any number less than 3. Any real number satisfies one or both of those, so the range is any real number.</p>
<p>(2). An infinite line must intersect a plane at some point UNLESS it is parallel to that plane. The planes intersect, so they aren’t parallel to each other. So the 2 possibilities are that the line intersects all n planes OR it’s parallel to 1 plane, in which case it intersects all the others (n-1). n-1 is less than n so that’s the answer.</p>
<p>(3). To find an indirect proof, you assume the opposite of the original statement and show that it results in something absurd. If you wanted to indirectly prove Sqrt(2) is irrational, you would assume Sqrt(2) is RATIONAL, think of something that must be true based on that assumption and show that it’s absurd. So the indirect proof starts with “Sqrt(2) is rational”.</p>
<p>(4). If you pick 7 people, the remaining 3 form the other group. Likewise, if you pick 3, the remaining 7 form the other group. Just find the number of possible combinations of 7 people out of 10 OR 3 people out of 10. So either 10C7 or 10C3, both of which give the same answer: 120. [Remember, nCr = nC(n-r) ].</p>