Download Test - FloridaMAO

March Regional
Pre-Calculus Individual
The acronym NOTA denotes that “none of these answers” are correct. DNE stands for Does Not Exist.
The domain and range of functions are assumed
√ to be the real numbers or the appropriate subset of the
real numbers, unless otherwise implied. i = −1. Good luck, have fun, and may the force be with you!
On the last page are some common values of the sine function to save tedious calculation.
1. Find the sum of the distinct zeros to the polynomial x3 − 7x2 + 15x − 9.
A. -7
B. 4
C. 7
D. 9
E. NOTA
2. Convert the polar coordinates 30, π3 to rectangular
√
coordinates.
√ √
√
π 3 π
π π 3
A. (15 3, 15)
B. (15, 15 3)
C.
,
D.
,
6
6
6
6
E. NOTA
∞
∞
1
1
X
X
k
1
x
x
.
Hint:
for
x
>
1,
=
=
and
2k
xk
xk
(1 − x1 )2
1 − x1
k=1
k=1
k=1
B. 6
C. 8
D. Divergent
E. NOTA
3. Evaluate:
A. 4
∞
X
k2
4. A regular hexagon is inscribed in a circle, which is then inscribed in a square. Let A be the area of the
region that is in the square but not the circle, and let B be the area of the hexagon. Also define the
function f such that f (a + bπ) = a + b when a and b are rational. If the hexagon has side length 2, find
f (A) + B.
√
√
√
√
A. 6 + 6 3
B. 12 + 6 3
C. 6 + 12 3
D. 12 + 12 3
E. NOTA
5. Given that ln(cis(θ)) = iθ for −π < θ ≤ π (where cis(θ) = cos(θ) + i sin(θ)), find
10
X
ln(cis kπ
2 ).
k=1
A.
5π
2 i
B.
7π
2 i
C.
11π
2 i
D.
55π
2 i
E. NOTA
6. The parametric equations x = sin(2t), y = 2 cos2 (t) form what? (Hint: cos(2t) = ...)
A. Line
B. Ellipse
C. Circle
D. Trig Curve
E. NOTA

x2 2 6
7.Let det(M) denote the determinant of a matrix M. Let A =  4x 1 3 .
3 x 3x
2
Find the sum of the values of x for which x + 3x + 1 = det(A).
A. -3
B. -1
C. 0
D. No such values

E. NOTA
8. An ellipse is inscribed within a rectangle 6 units long and 8 units wide. What is the area of the ellipse?
A. 12π
B. 24π
C. 36π
D. 48π
E. NOTA
9. Suppose k > 0, |ln(k)| > 1, and ln(|ln(k)|) > 2. Find the greatest integer less than the smallest possible
2
value of k. (Hint: e ≈ 2.7, e2 ≈ 7.4, ee ≈ 15.2, ee ≈ 1618.2)
A. 2
B. 7
C. 15
D. 1618
E. NOTA
1
March Regional
Pre-Calculus Individual
10. Let g(n) be a function that converts a number n to base-ten. Find
10
X
g(111k ).
k=2
A. 441
11. Let f (x) =
A. x > 4
B. 444
√
C. 447
D. 450
E. NOTA
√
x + 1, g(x) = ln( x − 2), h(x) = ln(x), k(x) = ex . Find the domain of f (k(h(g(x)))).
2
2
B. x > 2 + 1e
C. x ≥ 2 + 1e
D. x > 9
E. NOTA
12. Find the sum of all x ∈ [0, 2π] such that sin2 (x) cos(x) = 0.5 sin(x).
A. π4
B. 3π
C. 2π
D. 5π
2
2
E. NOTA
13. The 8 × 4 rectangle Y ALE is circumscribed about two congruent tangent circles with centers P and
Q. Let |AB| denote the distance between points A and B. Find (|P Y |)(|P A|)(|P L|)(|P E|). Draw a
picture to help.
A. 80
B. 160
C. 320
D. 640
E. NOTA
14. How many unattainable (integral) scores between -30 and 120 are there on a 30-question FAMAT
individual test? (4 points per correct answer, -1 per wrong answer, 0 for answers left blank)
A. 5
B. 6
C. 7
D. 8
E. NOTA
Use the following information for questions 15-17:
The shoelace theorem gives a way to find the area of a convex polygon in the Cartesian plane given its
vertices. If its vertices are (x
1 , y1 ), (x2 , y2 ), ...(xn , y n ) when cycling around
the
polygon,
then the area of
xn−1 yn−1 xn yn 1 x1 y1 x2 y2 +
.
this polygon is given by +
+ ... + xn
yn x1 y1 2 x2 y2 x3 y3 NOTE: A convex polygon is a polygon in which all angles are less than 180◦ .
15. Find the area of the polygon with vertices (0, 0), (4,2), (3,4), (-1, 3), (-4,1).
A. 9
B. 17
C. 18
D. 34
E. NOTA
16. Let k be an arbitrary real number greater than -1. Consider a polygon with vertices (k, 0),
(k + 1, 2k), (0, 1), (2, 0), and (k, 3k + 1). Let S be the set of all distinct value(s) of k which will cause this
polygon to have only four sides, let A be the sum of the elements of S, and let R be the number of
elements in S. Find A + R.
C. 4
D. 14
E. NOTA
A. 3
B. 11
3
3
17. Consider a polygon with vertices (j, 0), (j + 1, 2j), (0, 1), (2, 0), and (j, 3j + 1). If j < 0, find the
minimum
value of j for the
√
√shoelace theorem to√still work.
B. − 2
C. − 2 − 1
D. No minimum
E. NOTA
A. − 2 + 1
√
18. How many times do the graphs of y = x and y = sin(x) intersect for x > 0?
A. 0
B. 1
C. 2
D. 3
2
E. NOTA
March Regional
Pre-Calculus Individual
19. Find the sum of the values of x that satisfy 2x
A. 1
B. 3
C. 4
2 +4
= 16x .
D. 5
E. NOTA
For questions 20 and 21, let fn (x) denote f ◦ f ◦ ◦ ◦ f (x) (that is, f applied n times–for example,
f2 (x) = f (f (x)), f3 (x) = f (f (f (x))), and so on).
20. Suppose f (x) =
x+1
an x + bn
, where an , bn , cn , and dn are functions of n, find
. If fn (x) =
x
cn x + dn
a8 + b9 + c10 + d11 .
A. 110
B. 178
C. 199
D. 246
E. NOTA
21. Let f (x) be defined as in question 20. lim f10 (x) can be expressed in the form m
n , where m and n
x→∞
are relatively prime positive integers. Find m − n.
A. 21
B. 34
C. 55
D. 89
E. NOTA
22. The sine of a certain third-quadrant angle α has the same magnitude as the cosine of a
second-quadrant angle β, which is three times the magnitude of the cosine of α. Find sin(α + β).
C. 45
D. 1
E. NOTA
A. 0
B. 12
23. Mariya the immortal hermit decides that she will go for a run every day. Assume that when she stops
running, she stays exactly where she is until she starts running again the next day. As time goes on, she
becomes lazier–exponentially. On the first day, she runs 24 miles due northeast. On the second day she
runs 12 miles due northwest, on the third day she runs 6 miles due northeast, on the fourth day she runs
3 miles
√ due northwest, and√so on. What is her total
√ displacement in the
√ long run?
A. 8 5
B. 16 2
C. 16 5
D. 24 2
E. NOTA
√
24. Find the sum of the real values of x that satisfy 2x2 − 2x 3 − 3x2 = 1. (Hint: what substitution can
you√make?)
√
√
A. 22
B. 23
C. 1
D. 26
E. NOTA
25. Consider a rigged six-sided die with the numbers 1-6. Let P (k) denote the probability of rolling a
P (2)
P (3)
P (4)
P (5)
number k. Suppose that this die is rigged such that P (1)
6 = 5 = 4 = 3 = 2 = P (6). What is
the probability that, when two dice are rolled, both will show prime numbers?
4
A. 49
B. 19
C. 16
D. 289
E. NOTA
49
441
26. Simplify (where defined): 4 csc(2x) cot(2x).
A. cot2 (x) − tan2 (x)
B. tan2 (x) − cot2 (x)
D. sec2 (x) − csc2 (x)
E. NOTA
27. Let n be a positive integer greater than 100. Evaluate
A. n − 5 + i
B. n − 4
C. n − 4 − i
3
C. csc2 (x) − sec2 (x)
n
X
k=1
ik! .
D. n − 1 + i
E. NOTA
March Regional
Pre-Calculus Individual
28. Suppose f (x + 1337) = f (x − 1337). What is the maximum possible period of f (x)?
A. 668.5
B. 1337
C. 2674
D. 5348
E. NOTA
29. Evaluate:
(i +
√
A. 1024( 3 + i)
√
3)11 .
√
B. 1024( 3 − i)
√
C. 2048( 3 + i)
√
D. 2048( 3 − i)
E. NOTA
30. To verify an if-and-only-if statement, you need to prove that for one condition to be true, the others
must be true; that is, for a statement such as “A if and only if B”, you need to write two proofs–one to
show that A implies B, and one to show that B implies A. Suppose you have the statement “A if and
only if B if and only if C”. What’s the minimum number of proofs you could write to verify this
statement? (Note that you can’t prove both B and C in the same proof assuming A to be true, for
example–if you assume A to be true, in one proof you can show either B or C to be true, but not both)
A. 3
B. 4
C. 5
D. 6
E. NOTA
Common trigonometric
values:
√ √
π
π
sin(0) = 0
sin 12
= 6−4 2
sin 10
=
sin 5π
12 =
√
√
6+ 2
4
√
5−1
4
sin π6 =
sin π2 = 1
4
1
2
sin π4 =
√
2
2
sin π3 =
√
3
2