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.

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