I have a few more questions from GR 0568 that I would greatly appreciate if anyone could answer:

(I highlighted the correct answer in green.)

**#8.**What is the greatest possible area of a triangular region with one vertex at the center of a circle of radius 1 and the other two vertices on the circle?

**(A) 1/2**(B) 1 (C) sqrt(2), (D) pi, (E) (1 + sqrt(2))/4

My guess: Since I would get an isosceles triangle, maybe there is some kind of identity that I can use?

**#27**Consider the two planes $$x + 3y - 2z = 7$$ and $$2x + y - 3z = 0$$ in $$\mathbb{R}^3$$. Which of the following sets is the intersection of these planes?

(A) Empty set

(B) $$\{(0, 3, 1)\}$$

(C) $$\{(x, y, z): x = t, y = 3t, z = 7 - 2t, t \in \mathbb{R}^3\}$$

**(D) $$\{(x, y, z): x = 7t, y = 3 + t, z = 1 + 5t, t \in \mathbb{R}^3\}$$**

(E) $$\{(x, y, z): x - 2y - z = -7\}$$

My guess: Since there should be an infinitely many solutions and those planes are not parallel, the options (A), (B), and (E) are eliminated. Somehow, though, I'm stuck on finding a line that would work for both equations.

Also, why do I not get a right answer by setting those two equations equal? (It seems like I'm forgetting something basic from vector calculus, but I haven't really found a satisfactory answer to this question yet...)

**EDIT:**Oh wait, this one was actually pretty easy. All I need to do was find one point (which was given in the answer for (B)), and then take cross product of those two normal vectors for those planes (i.e. (1, 3, -2) x (2, 1, -3)), and that should give me the equations of the line given in D!

**#55**For how many positive integers k does the ordinary decimal representation of the integer k! end in exactly 99 zeros?

(A) None (B) One (C) Four

**(D) Five**(E) Twenty-four

My guess: Not much clue on this one.

**#61**Which of the following sets has the greatest cardinality?

(A) $$\mathbb{R}$$

(B) The set of all functions from $$\mathbb{Z}$$ to $$\mathbb{Z}.$$

**(C) The set of all functions from $$\mathbb{R}$$ to $$\{0, 1\}.$$**

(D) The set of all finite subsets of $$\mathbb{R}$$.

(E) The set of all polynomials with coefficients in $$\mathbb{R}$$.

My guess: Not so sure about this one either. Except that I know I can eliminate (B) because I think (B) is countable. How do I even find the cardinality of functions, anyway?

-----

That's it for now. If anyone can even give me a hint on just one of these questions above, I would be happy to hear it from you.

Thanks!

PP