Hints: GRDB67
(Note: This is not a full-length practice test. Rather, these questions are from the “Mathematics Test Descriptive Booklet” published by the ETS in 1997.)
Click on a question to uncover the hint.
Question 1
Test points and note that the curve is a parabola.
Question 2
Sketch the bounding semicircle, and remember that the polygon inscribed in it can be very close in shape to the semicircle.
Question 3
\(2^{30}\) is not half of \(2^{60}\).
Question 4
Use expansion by minors to calculate the determinant.
Question 5
Remember that \(|x|=\begin{cases}x&\text{if }x\ge 0\text{,}\\-x&\text{if }x<0\text{.}\end{cases}\) Use this to sketch \(f\).
Question 6
Take the first few derivatives and look for a pattern.
Question 7
Think about even and odd functions.
Question 8
You’re given the area; how do you find the original function? Think basic calculus.
Question 9
Check the definitions of commutativity, associativity, and identity directly.
Question 10
A line with slope \(m\) is perpendicular to a line of slope \(-\dfrac{1}{m}\).
Question 11
Try the boundaries of the feasible region, and try letting \(x\) or \(y\) approach \(\pm\infty\) if possible.
Question 12
Use the Ratio Test.
Question 13
Make a substitution to evaluate the integral, assuming the antiderivative of \(f\) is \(F\).
Question 14
Every multiple of \(9x+5y\) will also be divisible by \(11\). Be on the lookout for alternate ways to represent coefficients mod \(11\) (for example, \(10\equiv -1\)).
Question 15
Represent \((2,3,4)\) as a linear combination of \((1,0,0)\), \((0,1,0)\), and \((0,0,1)\), and use linearity.
Question 16
Power through this one with L’Hôpital’s Rule.
Question 17
Consider the two possible cases: (1) the coin comes up heads, or (2) the coin comes up tails, the die comes up less than \(5\), and the game resets to the original state. This should give you a recursive expression for the probability \(p\).
Question 18
Use logarithms and L’Hôpital’s Rule.
Question 19
Consider a “small” group, and try to come up with subsets that aren’t subgroups while satisfying one of the properties. (It doesn’t HAVE to be multiplicative.)
Question 20
One of these is actually the definition of continuous functions in topology.
Question 21
Don’t let the crazy wording stress you. It may help to think of this in two dimensions before you think of it in three. (Sketch a hill in 2D.)
Question 22
Run through the algorithm a few times to get a feel for what numbers get printed.
Question 23
Again, don’t let the crazy wording stress you. It may help to write down actual numbers (small, manageable ones) in the table, and working through the problem with those.
Question 24
Think about the shape of the graphs — which one will have the most “vibrations” as \(x\) gets large?
Question 25
Think about what \(\dfrac{|x|}{x}\) equals when \(x>0\) and \(x<0\) to sketch the graph.
Question 26
Try sketching the graph, and think about what happens as the rays approach the line where the planes intersect.
Question 27
Remember that just because \(f'(x)=0\) does not mean \(f\) has a local minimum or maximum. (Think of \(f(x)=x^3\).) What about a global maximum or minimum?
Question 28
Similar matrices have the same eigenvalues (and therefore the same traces and determinants).
Question 29
Sketch \(\dfrac{1}{x}\) and \(-\dfrac{1}{x}\), and use those to sketch the graph of \(\dfrac{1}{x}\sin\dfrac{1}{x}\).
Question 30
Write \(\sigma\) as a product of disjoint cycles.
Question 31
The graph of \(f^{-1}\) is a reflection of the graph of \(f\) across the line \(y=x\).
Question 32
Don’t try to divide through! Notice that \(-6\equiv 18\pmod{24}\). Use this to find as many solutions as possible.
Question 33
Notice that the winding number of \(Γ\) is \(2\).
Question 34
Use the Squeeze Theorem for the top and bottom halves of the circle.
Question 35
Use the Cayley-Hamilton Theorem: Any matrix must satisfy its own characteristic equation.