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January Regional
Pre-Calculus
Note: For the entirety of this test, please assume that all solutions are complex and that , unless stated
otherwise, is equal to the square root of – 1. In addition, all of the figures are not drawn to scale. You can,
however, draw your own diagrams if you wish to, and hopefully to scale as well.
1. Given that ln 5 = a. What is the value of ln (-5)?
a) a b) a + c) -a d) - a
e) NOTA
2. Given the function y = 2 cos(2πx) + 2, what is the sum of the period and the amplitude of the function?
a) 1/2 b) 1
c) 2
d) 3
e) NOTA
3. Which of the answer choices is not a co-terminal angle of π/4?
a) 5π/4
b) 9π/4
c) - 7π/4
d) - 15π/4
e) NOTA
4. Given that x, y, and z can take on values of positive integers. How many different combinations of x, y,
and z exist such that x + y + z = 15?
a) 15
b) 91
c) 182
d) 5040
e) NOTA
5. Given that a = 9π/17 and b = tan2(a) – sec2(a). What is the value of √ ?
a) b) d) c) e) NOTA
6. Mr. Wiggins and Mr. Frazer are both standing exactly at the point (0,0) on the Cartesian plane. Mr.
Wiggins starts walking at a speed of 5 meters per second for 10 seconds at an angle of π/6 to the positive
x axis. At the same time, Mr. Frazer starts walking at a speed of 4 meters per second for 10 seconds at an
angle of –π/6 to the positive x axis. What is the distance between Mr. Wiggins and Mr. Frazer after 10
seconds?
b) √210
a) 90
c) 10√21
d) 21√10
e) NOTA
7. Given function f is a real function and that f(x) = cos(x), with the domain of (0, ). What is the domain
of f -1 (x) ?
a) All reals
b) (0,1]
c) (0,1)
d) [0,1)
e) NOTA
8. Wenda started to do his math homework but soon realized the first question was way over his head.
)
Help Wenda simplify the expression of ).
a) cos(x)
b) sin(x)
c) 1
d) tan(x)
e) NOTA
d) 1
e) NOTA
9. Given that =∑ ! − 1), what is the value of " ?
a) b) - c) -1
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January Regional
Pre-Calculus
10. What is the area of a triangle with sides 5, 6, and 7?
a) 6
b) 12
c) 3√6
d) 6√6
e) NOTA
11. Jerome has drawn a right triangle on the argand diagram with leg lengths of 3 and . What is the
length of the hypotenuse of Jerome’s right triangle?
a) √10
c) 4
b) 2√2
d) 2
e) NOTA
12. What is the coefficient of the first degree term of the completed expansion of % + )' ?
a) 1
b) 2
c) 5
d) 10
e) NOTA
13. James graphed the following equation on the Argand diagram: (r sin(() + rcos(())2 - 3r2sin2(() =
2)*+,() + 1. What conic shape did James just graph?
a) circle
b) ellipse
c) parabola
d) hyperbola
e) NOTA
14. Given that 1 + ) = , what is the magnitude of ?
a) 64
b) 128
c) 256
d) 1024
e) NOTA
15. How many values satisfy the equation: tan(ax) = 1, with 0<x< , given that a = the number of books
in Euclid’s elements. Hint: a also equals to the cube root of 2197.
a) 5
b) 6
c) 7
d) 8
e) NOTA
16. Hansol and Kathy are bored in Mr. Frazer’s class so Hansol takes out her analog watch and shows it
to Kathy. The time on the watch reads (correctly) 12:05 . The hour hand of Mr. Frazer’s clock also has
interesting habit of only moving when the hour changes. Hansol then asks Kathy to state the acute angle
measured by the hands of the analog watch. Kathy, being not so good at math, accidently states (, the
complementary angle of the correct answer, in radians. What is the value of sin ( ?
a) b)
√
c)
√/
d) 1/4
e) NOTA
17. How many distinguishable permutations can be made from using all the letters in the word
January, given that the u must be between the two a’s ?
a) 24
b) 60
c) 120
d) 240
e) NOTA
18. Given csc ( = 0 for 0 < ( < , which of the following expressions represent tan ( ?
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January Regional
Pre-Calculus
a) 21 − 0 b)
2
3 c)20 − 1
3 d)
23 3 e) NOTA
19. What is the amplitude of the function f(x) = 2sin(x) + 2cos(x) ?
b) 2√2
a)2
c)2√3
d) 4
e) NOTA
20. What is the volume of the parallelepiped determined by the points (0,0,1), (1,1,1), and (1,2,1) ?
a) b) /
c) d) 1
e) NOTA
21. What is the range of the function f(y) = 3 tan(2y) + 5?
a) 2, 5)
b) 2, 8)
c) [2, 5]
d) [2, 8]
e) NOTA
'
22. If sec(x) = - and tan(x) > 0, what is the value of sin(x) ?
a) '
b) − '
c)
√
'
d) -
√
'
e) NOTA
23. Given the statement: If he has a ring, then he is a king. What is the contrapositive of the converse of
the statement?
a) If he has a ring, then he is a king.
b) If he doesn’t have a ring, then he is not a king.
c) If he isn’t a king, then he doesn’t have a ring.
d) If he doesn’t have a ring, then he is a king.
e) NOTA
24. How many palindromes are contained by the first 9999 whole numbers?
a) 99
b) 189
c) 198
d) 199
e) NOTA
25. Given that tan(x) = y, for 0 < x < , which of the following expressions can be written as tan(2x)?
3
3 b) 3
3
3
e) NOTA
a)
3
d)
3
c) 3
26. What is the value of 81a? Where a is log / 5
a)15
b) 125
c) 243
d) 625
3
e) NOTA
January Regional
Pre-Calculus
<
'
27. Given that cos-1 + sin-1
=
= tan-1 x, and all angles are located in the first quadrant. What is the value
of 17x ?
a) 100
b) 121
c) 144
d) 169
e) NOTA
28. How many of the following statements is/are true about the function y = 0 ?
The function is an even function.
The function is not an odd function.
The domain of the function is all reals except for 0.
The range of the function is a subset of the set of natural numbers.
o
o
o
o
a) 1
b) 2
c) 3
d) 4
e) NOTA
29. When the point (1, 1) on the Cartesian plane is converted into polar coordinates, one of the possible
polar coordinates is:
a) (1, )
<
b) (1, )
c) (√2,
<
)
d) (-√2,
/
<
)
e) NOTA
30. Which of the answers choices is not equivalent to sin(4π)?
a) cos( )
b) tan(4π)
/
d) cot( )
c) 0!
4
e) NOTA