February Regional
Precalculus Individual Test
REMEMBER—NOTA means None of the Above
1.
The 4th term of a sequence is 4 and the 6th term is 6. Every term after the 2nd is the sum of the
2 preceding terms. What is the 8th term of this sequence?
a) 8
b) 10
c) 12
d) 14
e) NOTA
2. What is the simplified numerical value of the quotient:
sin10 cos10 tan10 cot10 sec10 csc10
?
sin 20 cos 20 tan 20 cot 20 sec 20 csc 20
a) 0
b) ½
c) 1
d) 2
e) NOTA
3. In the series 202  192  182  172  ...  22  12 , the signs alternate between squares of consecutive
numbers. What is the sum of this series?
a) 190
b) 200
c) 210
d) 220
e) NOTA
144
1
1
, what is the numerical value of

?
25
tan x cot x
144
288
b) 1
c)
d)
25
50
4. If tan x  cot x 
a)
25
144
e) NOTA
5. A certain parabola passes through the points (5, 1) and (13, -7) and has the y-axis as its
directrix. What is the sum of all the ordinates of the points at which the vertex of this parabola
could be located?
a) 2
b) 3
c) 4.5
d) 9
e) NOTA
6. Which of the following describes the graph of the equation  x  y   x 2  y 2 ?
2
a) The empty set
b) 1 point
c) 2 lines
7. Which of the following is equivalent to
a) –x
b) x
c) 1
d) a circle
e) NOTA
x
when x  0?
x 1
1
x
d)
x
2
e) NOTA
February Regional
Precalculus Individual Test p2
8. The parabola y  ax 2  bx  c has vertex ( p, p ) and y-intercept (0,  p) , where p  0. What
number does the letter b represent?
a) 0
b) 1
c) 2
d) 4
9. Let a1 , a2 ,... be a sequence for which a1  2, a2  3 and an 
e) NOTA
an 1
for each positive integer n  3.
an  2
What is a2010 ?
a)
1
2
b)
2
3
c)
3
2
d) 2
e) NOTA
10. Alex and Zach each have a bag that contains one ball of each of the colors blue, green, orange,
red and purple. Zach randomly selects one ball from his bag and puts it in Alex’s bag. Alex
then randomly selects one ball from his bag and puts it into Zach’s bag. What is the probability
that after this process the contents of the 2 bags are the same?
1
1
1
1
a)
b)
c)
d)
e) NOTA
10
6
5
3
11. For how many real distinct values of x is 120  x an integer?
a) 3
b) 6
c) 9
d) 10
e) NOTA
12. The arithmetic mean of a set of 50 numbers is 32. The arithmetic mean of a second set of 70
numbers is -53. Find the arithmetic mean of the numbers if the sets are completely combined.
7
1
1
1
a) -17
b) -15
c) -11
d) -10
e) NOTA
12
2
6
2
13. From a rectangular piece of cardboard 12 x 14, an isosceles trapezoid and a square (of side s )
are removed. What must be the value of s so that their combined area is a maximum?
a) 1
b) 5
c) 10
d) 12
e) NOTA
14. Find the sum (in terms of n) of the n terms of the arithmetic series whose first term is the sum
of the first n natural numbers and whose common difference is n.
a) n
b) n 2
c) n3
d) n 4
e) NOTA
February Regional
15. Express F 
a)
3
Precalculus Individual Test p3
3
2
with a rational denominator.
1 2  3 4
3
4  3 12  1
2
b)
3
432
1
c)
3
6  3 12  1
1
d)
3
6  3 12  3 2
2
e) NOTA
16. Suppose that a boy remembers all but the last digit of his friend’s phone number. But, trying to
call his friend anyway, he chooses a number at random for that last digit. If he only has 2
quarters in his pocket and it costs 25 cents a call, what is the probability that he dials the right
number before running out of money?
a) .10
b) .20
c) .25
d) .30
e) NOTA
17. Decompose
a) 6
7 x3  x 2  x  1
into partial fractions and then find the sum of their numerators.
x3 ( x  1)
b) 8
c) 10
d) 12
e) NOTA
18. If f ( x)  1  f ( x  1), express f ( x  1) in terms of f ( x  1).
a) f ( x  1)
b) 2 f ( x  1)
c) ½ f ( x  1)
d) 4 f ( x  1)
e) NOTA
19. If there are 10,000 bicycles, each of which has a license number from 1 to 10,000 inclusive. No
two bicycles have the same number. Find the probability that the number on the 1st bicycle you
pick at random will not have any 8’s among its digits.
a) 4(.9)
b) 5(.9)
c) (.9)4
d) (.9)5
e) NOTA
20. Find the value of sin 2 10  sin 2 20  sin 2 30  ...  sin 2 90 .
a) 1
b) 2
c) 4
d) 5
21. Find the sum of all real solutions of xlog x 
a) 3
b) 10
c) 30
22. Find x if Tan 1 x  Tan 1 4  Tan 1 6.
10
10
a) -10
b)
c)
23
23
e) NOTA
x3
.
100
d) 100
e) NOTA
d) 10
e) NOTA
February Regional
Precalculus Individual Test p4
23. A parabola y  ax 2  bx  c has vertex (4, 2). If (2, 0) is on the curve, find the product abc.
a) -24
b) -12
c) 12
d) 24
e) NOTA
24. If g ( x)  2 x  8 and f ( x) 
a) -2
b)
5
2
1
, find g f 1 ( 2).
x2
5
c)
2
d) 3
e) NOTA
1 4 c 
25. For what value of c will there be no inverse for the matrix  2 1 7  ?


 3 2 11
a) -1
b) 0
c) 1
d) 2
3 6
26. In the harmonic sequence, 6, 3, 2, , ,…,what is the 8th term?
2 5
1
3
4
a)
b)
c)
d) 2
2
4
3
27. Solve for x  0 : e
a) ln 2
b)
xe
x
xe
ln 2
e) NOTA
e) NOTA
 2.
c) ln 4
d)
ln 4
e) NOTA
28. Find the probability that the ace of spades lies next to the jack of diamonds in an ordinary deck
of playing cards.
1
1
1
1
a)
b)
c)
d)
e) NOTA
26
13
4
52
29. If ln x 4  (ln x)3 , then find the product of all real values of ln x.
a) -4
b) -2
c) 0
d) 1
e) NOTA
30. What is the product of the first 10 terms of a geometric series whose 1st term is 1 and whose
10th term is 2?
a) 1
b) 16
c) 32
d) 34
e) NOTA