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FAMAT January Pre-Calculus Regional
NOTA means “None of the Above.” Luck!
1. Mrs. Sowers has always said: “The calculus in calculus is not hard. The algebra in calculus
is hard.” Often while simplifying a particularly nasty derivative you will obtain something like
1
2
1


the following: 2  1  3x  3  2x   1  3x  3  3   . Prove you are ready to take the training
3

wheels off, and factor this expression as neatly as possible.
a) 2  1  3x 

b) 2  1  3x 
2
3
 1  3x 
2
3
 1  4x 

1
c) 2  1  3x  3  1  3x 
1
d) 2  1  3x  3  1  4x 
e) NOTA
2. Dr. Evil, Dr. Doom, Dr. Frankenstein, and Dr. Morris are each comparing their sinusoidal
waves. The waves are 4cos x, 3cos x  15 sin x,  3sin x, and 3cos x  4sin x ,
respectively. Which doctor’s destructive drawing has the largest amplitude?
a) Evil
b) Doom
c) Frankenstein
d) Morris
e) NOTA
3. Mrs. Doker is musing that the Dokerset, defined to be   {D, O, K, E, R} , needs only an ‘s’
and a ‘u’ to be able to create the name of her alter-ego, Sudoku. If you choose 6 letters
randomly from   S, U, U , what is the probability that they allow you to spell the word
SUDOKU?
a)
1
14
b)
1
21
c)
1
28
d)
1
35
e) NOTA
4. I never understood what continuous meant until I had a conversation with Mrs.
Woolfenden. She once said, “Not only must the limit be two-sided, but the function value
itself must be preserved on the function’s entire domain.” Hopefully, Mrs. Woolf has now
helped you identify the continuous function below.
 1
,

a) a(x)   x  1

 1,
x  1
,

c) c(x)   x  1

 4,
x1
x1
x  1
x  1
 Tan1x, x  0

b) b( x)  
x0

 x,
 x,
d) d(x)  
 x  1,
x0
x 0
FAMAT January Pre-Calculus Regional
5
5. If sin x  , give the sum of the two possible values of cosx .
7
a) 
2 6
7
b) 0
c)
2 6
7
d) 2
e) NOTA
6. Mr. Nichols told me to put a function composition question on this test that looked easy,
but had a well-known trick to it. If f ( x)  x , and g ( x)  x2 , what is f g ( x) ?
a) x
b)
x
c) x2
d) x2.5
e) NOTA
7. What is the maximum value attained by the graph of y  6x2  12x  1 ?
a) 5
b) -1
c) -12
d) -19
e) NOTA
8. What is the sum of the first whole number, the first natural number, and the first prime
number? (By first, I mean smallest)
a) 3
b) 4
c) 5
d) 6
e) NOTA
9. The legendary Frank Caballero brilliantly summed up an entire section of a book of sums
with the analogy “  :sum ::  :product .” Which of the following expressions is
b
equivalent to
ln j ?
j a
n
a)
 ln ab
j 1
b
b)
 ln j
j a
b
c)
ln(1  j)
j a
 b 
d) ln   j 
 j a 
e) NOTA
1 
e
10. What is the determinant of the matrix 
?
e
 1 ln e 
a) -1
b) 0
c) 1
d) e
e) NOTA
11. Even though he wasn’t my coach, Mr. Propert was disappointed in me when I missed the
question about the rational function’s asymptotes because one was a removable
discontinuity. He said, “The factor that causes you to get zero over zero is a removable
x2  x  2
discontinuity; a hole, which isn’t an asymptote.” Identify the asymptotes of y  2
.
x  x 6
a) x  1, x  3 b) y  1, x  3 c) x  2, x  3, y  1 d) x  2, y  1 e) NOTA
FAMAT January Pre-Calculus Regional
12. Mr. Pease told me that inverse functions were defined from one-to-one functions. He
described it as a function passing a horizontal line test, where regular functions only pass
the vertical line test. Lost without a clue, I asked for an example, and he said: “Think of
y  x3 as opposed to y  x  x  1 x  1 . One of these will not have an inverse. A function is
one-to-one if and only if it has an inverse.” Which of the following trigonometric functions is
one-to-one?
a) y  sin x, 0  x  2


c) y  sec x,   x 
2
2
b) y  cos x, 0  x  2


d) y  csc x,   x 
2
2
e) NOTA
13. Which of the following is not equivalent to cos2x ?
a) cos2 x  sin2 x
c) 1  2cos2 x
b) cos4 x  sin4 x
d) 2cos2 x  1
e) NOTA
14. On the Cartesian Plane, what is the graph of 2x2  y2  4x  2y  2  0 ?
a) Ellipse
b) Point
c) Parabola
d) Hyperbola
e) NOTA
15. On the Cartesian Plane, what is the graph of 2x2  y2  4x  2y  3  0 ?
a) Ellipse
b) Point
c) Parabola
d) Hyperbola
e) NOTA
16. Give the sum of the solutions to 4cos2 x  3  0, 0  x  2 .
a) 2
b)

6
c) 4
d) 3
e) NOTA
17. Solve for x in 2sin x cos x  cos2x .

a)    n, n 
4
 
c)   n, n 
4 2






 

b)   n, n  
8 2



d)    n, n  
8

e) NOTA
FAMAT January Pre-Calculus Regional
18. Identify the focus of the parabola y   x  2   3 .
2
11 

a)  2,  
4

b) (-2, -3)
 7

c)   , 3 
 4

d) (2, -3)
e) NOTA
19. Identify the focus of the parabola y2  8x  2y  7  0 .
a) (1, -1)
b) (1, 1)
c) (1, 0)
d) (-1, 0)
e) NOTA
c) 0, 
d) 0, 
e) NOTA
20. Give the domain of f ( x)  ln x .
a)
1,
b)  1, 
x1
2
and g ( x)  .
x
x
21. Ignoring domain issues, find and simplify g ( f ( x)) .
For questions 21 and 22, f ( x) 
a)
2
x1
b)
2x
x1
c)
x1
2
d)
x1
2x
e) NOTA
22. Give the value of  f g  1 .
a) 1.5
b) 2
c) 2.5
d) 3
e) NOTA
For questions 23 and 24, triangle CMT is such that c  2 7, m  4, t  6 .
23. In degrees, what is the measure of angle C ?
a) 30
b) 45
c) 60
d) 75
e) NOTA
24. Ignoring units, what is the area of triangle CMT ?
a) 3 3
b) 6 3
c) 9 3
d) 6
e) NOTA
25. The function h( x)  e  x is bounded below by all of the following except
a) -2
b) -1
c) 0
d) 1
e) NOTA
FAMAT January Pre-Calculus Regional
26. In the system of equations below, which of the choices evaluates to the reciprocal of x ?
 2x  y  z  5

 x  3y  2z  10
x  2y  3z  15

2
1
1
5
1
1
2
5
1
1
3
2
10
3
2
1
10
2
1 2 3
5 1 1
a)
10
3
b)
15 2 3
2 1 1
2
1
15 2 3
3
c)
2
1
1 2 3
d)
2
1
1
1
3
2
1 2 3
2 5
1
1
10
1 15 3
2 1 1
3
2
1 2 3
e) NOTA
2
1 15 3
1 1 1
27. If a  3  7  11  15  ...  83 and b  1     ... , give  a  1 b .
3 9 27
a) 1200
b) 1256
c) 1356
d) 1476
e) NOTA
28. Find the sum of the integral solutions to 2x  3  5 .
a) -6
b) -9
c) 1
d) 2
e) NOTA
29. Give the equation of the line with positive slope that passes through (1, 0) and when it

strikes the x-axis makes an acute angle of .
6
a)
3x  y  1
b) x  3y  1
c)
x  3y  1
d)
3x  y  1
e) NOTA
30. Give the degree of the expression 3x2 y3z  3x3 y 3z  3x  3y .
a) 3
b) 5
c) 6
d) 7
e) NOTA