2006 ALABAMA MATHEMATICS CONTEST
6:08 AM
Muhammad Yusuf
, Posted in
1. If sin
α = −5/13 and
α is in Quadrant III, what is the value of cos ( ½ α) ?
(A) −6/13 (B) 2√3/13 (C) −√5/12 (D) 4√3/12 (E) −√26/26
2. The
degree of a polynomial p(x) is
no larger than 3. We know that p(0) = 3, p(-x) = p(x), and p(1/x) = p(x)/x2. Find p(1).
(A) 2 (B) 4 (C) 6 (D) 3 (E) 5
3. M is a perfect square of an
integer. The next larger perfect square is M + 461. What is M ?
(A) 52, 900 (B) 57, 600 (C) 48, 400 (D) 62, 500 (E) 44, 100
4. If x is positive, find x.
(A) 15 (B) 14 (C) 13 (D) 12 (E) 11
5. Find x .
(A) −1 (B) No
solution (C) −3 (D) −2 (E) 0
6. Suppose
that m people can do a task in d days. Assuming that they all work at the same rate,
how many days will it take for m + r people
to do the task ?
7. What
is the smallest value assumed by the polynomial p(x) = 4x2 − 3x + 8 ?
(A) 119/16 (B) 3/8 (C) 8 (D) 9 (E) 15/2
8. If r and s are the (complex) roots of
the equation x2 − √27x + 13 =
0, what is the value of r2 + s2 ?
(A) √27 (B) 1 (C) 13 (D) 53 (E) 27
9. If f(x) = (x – 1) : (x – 2) , what is f−1(4) (here f−1 is the inverse function of f) ?
(A) 3 (B) −3 (C) 3/2 (D) 2/3 (E) 7/3
10. Suppose c ≠ 0 and the roots of 3x2 + 7x + c = 0
are x1 and x2. In simplest form in terms of c, what is the value of 1/x1 + 1/x2?
(A) 3/c (B) −3/c (C) 7/c (D) −7/c (E) 7/3c
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