Gre Math Subject Worksheet
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6GRE Math Subject Test #2 Solutions.
′
1. C (Calculus) The Fundamental Theorem of Calculus directly gives F
(x) = log x.
2. D (Discrete Math) Repeated use of the recursion yields
1
2
3
n 1
n 1
F (n) = F (n 1)+
= F (n 2)+
= F (n 3)+
= ... = F (n (n 1))+
= 2+
.
2
2
2
2
2
Now, let n = 101.
3. C (Linear Algebra) Apply the formula for the inverse of a 2 × 2 matrix.
4. B (Calculus) First, solve for b from evaluating the given integrals. Finding that
∫
3/2
1
3
2
b =
, the area equals
(x
x) dx =
.
2
6
1
5. E (Calculus) Remember that the derivative of a function at a point is the slope of
′
the tangent line at the point. Accordingly, if f
(x) > 0 (above the x-axis), then f (x)
′
is increasing, and if f
(x) < 0 (below the x-axis), then f (x) is decreasing. To narrow
the remaining choice or two, measure some actual slopes of tangent lines from a
potential plot of f .
6. C (Discrete Math) On each iteration of the loop, k = 999 stays constant, while i
doubles and p increases by 1. After 10 iterations of steps 2-4, we are at k = 999,
10
i = 2
= 1024, and p = 10. Since i > k at last, we print 10.
7. B (Analytic Geometry) Consider the endpoints and orientation; this one is not bad.
√
1
1
2
8. E (Calculus) By u-substitution, this equals
ln(x
+ 1)
= ln
2.
2
0
9. A (Discrete Math) Since f is 1-1 and mapping a finite set to itself, it is
automatically a bijection. Therefore, f essentially permutes the elements of S; and so
there are n! possible functions f .
10. B (Real Analysis) This is similar to Dirichlet’s function. Due to density of the
rationals and irrationals in R, g is automatically discontinuous at any nonzero x.
(The quickest way to establish this is to construct sequences of rationals and
irrationals which converge to x, and note that applying f to each sequence gives
different results.) However, g is continuous at x = 0 by applying the Squeeze
Theorem. For x > 0, we have 1 ≤ g(x) ≤ e
x
, and thus lim
g(x) = 1 = g(0).
+
x→0
Similarly, we have an analogous statement for x → 0 .
11. A (Algebra) Consider the cases x ≥ y and x < y separately.
12. C (Real Analysis) Consider the definitions of supremum and limit point.
13. A (Probability) The probability of choosing the same color can be done in three
1
pairwise disjoint cases (and add the results together): P (blue) =
,
C(8,2)
C(4,2)
1
P (red) =
, and P (yellow) =
C(8,2)
C(8,2)
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