Vito Prosciutto: Teaching community college math on the road to a PhD.

Wednesday, October 20, 2004

Math GRE prep: precalculus 

Whoa, that was demoralizing. At the end of chapter 1, I scored 14/25. Where did I go wrong?
  1. Let f be a function such that f(n+1)=1-[f(n)]2 for all nonnegative integers n. Which of the following correctly expresses f(n+2) in terms of f(n)?
    Stupid algebra error. I forgot to keep careful track of exponents
  2. Which of the following best describes the graph of the equation x2+y2=2x+4y+5=0 in the x-y plane?
    Laziness. I looked at this, saw that I was going to have something that simplified to (x-u)2+(y-v)2=r2 and didn't bother to check that r2 was positive.
  3. One of the foci of the hyperbola y2=(x/a)2+1 is the point (0,sqrt(2)). Find a.
    I need to make sure that I memorize all the little details about conic sections, asymptotes and foci
  4. Which one of the following polynomials p(x) has the property that sqrt3-sqrt2 is a root of the equation p(x)=0?
    Tried to be too clever. I figured that it had to be fourth degree and tried to find the product of the roots rather than doing a substitution to turn the product into a quadratic. This is a recurring theme on some of these questions.
  5. When the polynomial p(x) is divided by x-1 it leaves a remainder of 1, and when p(x) is divided by x+1 it leaves a remainder of -1. Find the remainder when p(x) is divided by x2-1.
    Embarrassingly, I wasn't even sure where to start on this one.
  6. Given that p(x) is a real polynomial of degree <= 4 such that one can find five distinct solutions to the equation p(x)=4, what is the value of p(5).
    Another one I didn't know where to start
  7. The hyperbolic sine function, denoted sinh is defined by the equation sinh x=(ex-e-x)/2. Find a formula for sinh-1 x
    It didn't occur to me to, once I had an equation with an e2x term, an ex term and a term without e to rewrite as a quadratic using u=ex.
  8. The hyperbolic cosine function... hyperbolic tangent... find a formula for tanh-1x Having missed the previous one, there was no hope for progress here.
  9. Which one of the following is in the domain of the function f(x)=log(sin x)? You may use the fact that 11111 is just slightly greater than 353,64xπ
    A 11 B 111 C 1111 D 11,111 E None of these
    Perhaps stupid arithmetic, perhaps worse
  10. Determine the exact value of the sum Arctan 1 + Arctan 2 + Arctan 3
    I got killed by not remembering the tan a+b formula. I spent a little bit of time deriving it algebraicly before starting this post, so now I can get it if I need it.
So why post all this here, other than public self-flagellation? I'd like some more practice problems. Feel free to post them in the comments and I'll offer up solutions in future posts. Apologies for those who came here looking for teaching stuff. Not a lot of that likely in the near future.

For the searchers who show up here, I'll continue doing any problems from the comments until (at least) December 10th 2004.

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