1. Which of the Following Is Not a Function
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1. Which of the following is not a function :
(A) y = 9x 2 (B) 1 - y 2 = x 2 (C) y = 2 x 3 (D) x = 2 - y (E) 8y = x 3
If you graph each choice, you can than apply the “vertical line test” to determine if the given relation is a function. If a vertical line intersects the graph more than once, the relation is not a function. Note : 1 - y 2 = x 2 can be rewritten as x 2 + y 2 = 1 which is an equation of a circle whose radius is 1. If you use the vertical line test, you will determine that it is not a function.
2 1 2. If a = 5 , a = 8, and a = 10 , then a = ______. 3 3 4 5 3 1 2
1 2 1 (A) 24 (B) 25 (C) 26 (D) 27 (E) 29 3 3 3
2 1 1 7 5 , 8, 10 , … is an arithmetic series with d = 2 = 3 3 3 3
7 2 Note : To find the 12th term, you must add nine times to 5 3 3
2 7 2 2 5 + 9 = 5 + 21 = 26 3 3 3 3
1 2 3. Find the determinant of the 2 x 2 matrix . 3 4
(A) - 4 (B) - 3 (C) - 2 (D) 0 (E) 4
(- 1)(- 4) - 3(2) = 4 - 6 = - 2 4. The probability of scoring below 70 on a test worth a total of 100 points is 34%. What are the odds of a student scoring more than or equal to 70 on this test?
(A) 33:50 (B) 17:66 (C) 33:17 (D) 35:17 (E) 35:33
F a v o r a b l e 6 6 Probability = = 66% = T o t a l O u t c o m e s 1 0 0
F a v o r a b l e 6 6 3 3 Odds = = = = 33:17 U n f a v o r a b l e 3 4 1 7
5. At Chance Middle School, 18% of all students play football and basketball and 32% of all students play football. What is the probability that a student plays basketball given that the student plays football? (nearest whole percent).
(A) 14% (B) 44% (C) 50% (D) 56% (E) 178%
1 8 = 56.25% = 56% 3 2
6. The graph of y = |x| is shifted left 2 units, vertically stretched by 3, and shifted up 5. Which of the following is the equation for the transformed graph?
(A) y = 3 + 5|x - 2| (B) y = 5 + |3x + 2| (C) y = 2|3x + 5|
(D) y = 3|2x + 5| (E) y = 3|x + 2| + 5
Shifting the graph of y = |x| left 2 units will result in a graph whose equation is y = |x + 2|.
Vertically stretching y = |x + 2| by 3 will result in a graph whose equation is y = 3|x + 2|.
Shifting the graph of y = 3|x + 2| up 5 units will result in a graph whose equation is y = 3|x + 2| + 5. 7. Which of the following numbers is not considered to be a “perfect” number?
(A) 6 (B) 28 (C) 268 (D) 496 (E) 8128
Note : The first three perfect numbers are 6, 28, and 496, therefore 268 can’t be perfect.
If the sum of the positive integral divisors of a number is equal to twice the number, the number is considered to be “perfect”.
8. L, M, and N are the roots of (x 2 + x - 20)(x + 3) = 0. Find L + M + N.
(A) - 17 (B) - 4 (C) 6 (D) 9 (E) 12
(x 2 + x - 20)(x + 3) = 0
(x + 5)(x - 4)(x + 3) = 0
x = - 5, 4, or – 3
L + M + N = (- 5) + 4(- 3) = - 4
9. The graph of 9x 2 + 4y 2 = 36 is a(n) :
(A) ellipse (B) circle (C) parabola (D) line (E) hyperbola
9 x 2 4 y 2 3 6 x 2 y 2 9x 2 + 4y 2 = 36 ; = ; + = 1 (Ellipse) 3 6 3 6 4 9
10. If 7 ( x y ) = 8 and 7 ( x y ) = 6, then 49 y = ?
1 3 (A) 2 (B) 1 (C) (D) 14 (E) 48 3 4
y 49 y = 7 2 = 7 2 y
7 x y = 7 ( x y ) ( x y ) = 7 2 y 7 x y
7 x y 8 4 1 7 2 y = = = = 1 7 x y 6 3 3 11. …, P, Q, 1.2, 3.0, R, … is an arithmetic sequence. Find P + Q + R.
(A) 1.8 (B) 0.6 (C) 4.2 (D) 6.0 (E) 7.8
Find d. 3.0 - 1.2 = 1.8
3.0 + 1.8 = R = 4.8
1.2 - 1.8 = - .6 = Q
- .6 - 1.8 = - 2.4 = P
P + Q + R = (- 2.4) + (- .6) + 4.8 = 1.8
12. A bag contains green golf balls, white gold balls, and yellow golf balls. How many different color combinations can occur if three golf balls are drawn from the bag? Order is not important.
(A) 9 (B) 10 (C) 12 (D) 15 (E) 27
Combinations : G, W, Y G, G, G W, W, W Y, Y, Y G, G, W
G, G, Y W, W, G W, W, Y Y, Y, G Y, Y, W
Answer : 10
13. A short stack of playing cards contains 4 hearts, 3 spades, 2 clubs, and 1 diamond. The deck is shuffled and two cards are randomly selected, one at a time, without replacement. What is the probability that both are spades?
1 1 1 3 3 (A) (B) (C) (D) (E) 1 5 8 4 1 0 7
3 2 6 1 = = 1 0 9 9 0 1 5 14. If log 1 0 2 = m and log 1 0 = n, then log 1 0 12 = ?
(A) m 2 + n (B) 2m + n (C) 4n (D) 2mn (E) m 2 n
log 1 0 12 = log 1 0 (2)(2)(3) = log 1 0 2 + log 1 0 2 + log 1 0 3 = m + m + n = 2m + n
2 a 3 a 2 15. Simplify : a 0 a
3 9 (A) a (B) 2 a (C) a 6 (D) 2a 3 (E) a
1 2 6 2 a a a 1 1 9 9 2 6 4 = a 2 = a 2 = a 2 = a 1
16. When k is divided by 6, the remainder is 5. What is the remainder when 3k is divided by 6?
(A) 1 (B) 2 (C) 3 (D) 4 (E) 5
To find the remainder when 3k is divided by 6, you multiply the remainder when 3 is divided by 6 (which is 3) and k divided by 6 (which is 5). The result is 3(5) which is equal to 15. Since the remainder must be less than 6, find the remainder when 15 is divided by 6. Answer : 3
17. What is the probability that a factor of 234 is also divisible by 3?
Factors of 234 : 1, 2, 3, 6, 9, 13, 18, 234, 117, 78, 39, 26
Factors of 234 that are divisible by 3 : 3, 6, 9, 18, 39, 78, 117, 234
8 2 Probability = = 1 2 3 18. Let x = .333… in base four and y = .111… in base four, find the product of x and y in base ten.
1 1 3 2 3 (A) (B) (C) (D) (E) 4 3 1 6 3 4
a Shortcut : x = .aaa… in base b = , where a is a one-digit positive integer. b 1
3 1 ,333… base four = and .111… base four = 3 3
3 1 1 = 3 3 3
19. Which of the following is not a rational number?
1 (A) - 4 (B) 0.8333… (C) 2 (D) 2.43 (E) 5
Any number that can be expressed as a terminating or repeating decimal is considered to be rational. 1 - 4 and 2.43 are terminating decimals. 0.8333… is a repeating decimal. = . 5 2 is a terminating decimal. Since 2 can’t be expressed as a terminating or repeating decimal, then it is irrational.
2 ( n 1 ) 20. Evaluate : n 2 (n) , n 0
(A) 16 (B) 8 (C) - 2 (D) - 4 (E) - 8
When π appears instead of you must find the product of the terms obtained when substituting from n = - 2 to 2.
1 (- 2) 3 x (- 1) 2 x (1) 0 x 2 1 = (- 8)(1)(1) = - 4 2 21. A, B, and C are the roots of (x + 1)(x 2 - 5x + 6) = 0. Find AB + BC + AC.
(A) - 5 (B) - 4 (C) 1 (D) 5 (E) 6
(x + 1)(x - 3)(x - 2) = 0 ; x = - 1, 3, or 2
(- 1)(3) + (- 1)(2) + (3)(2) = - 3 - 2 + 6 = 1
2 x 5 22. If f(x) = - 3, then f 1 (0) = ? 4 x
(A) 2 (B) 0.5 (C) - 0.5 (D) - 1.5 (E) - 2
Step #1 : Interchange the x an y and solve for y.
2 y 5 2 y 5 x = - 3 ; x + 3 = 4 y 4 y
4xy + 12y = 2y + 5 ; 4xy + 10y = 5
y(4x + 10) = 5
5 y = f 1 (x) = 4 x 1 0
Step #2 : Find f 1 (0).
5 5 = + 0.5 4 ( 0 ) 1 0 1 0
23. How many distinguishable ordered arrangements can be made from the letters of the word LETTERS?
(A) 1,260 (B) 2,520 (C) 5,040 (D) 120 (E) 30
7 ! 5 0 4 0 = = 1,260 2 ! 2 ! 4 24. A box of golf balls contains 3 orange ones, 2 pink ones, and 5 white ones. If a golf ball is chosen at random, what is the probability of not getting a pink one?
(A) 20% (B) 30% (C) 50% (D) 70% (E) 80%
8 = 80% 1 0
25. P and Q are the roots of 6x 2 + x - 2 = 0. Find P 2 - Q 2 when P > Q.
1 1 7 1 1 (A) (B) (C) - (D) - (E) - 1 2 3 3 6 6 3
2 1 (3x + 2)(2x - 1) = 0 ; x = - , 3 2
2 2 1 2 1 4 7 P 2 - Q 2 = - = - = - 2 3 4 9 3 6
x 2 x 6 26. How many vertical asymptotes of y = exist? x 2 4
(A) 0 (B) 1 (C) 2 (D) 3 (E) 4
( x 3 ) ( x 2 ) x 3 y = = ( x 2 ) ( x 2 ) x 2
The vertical asymptote is x - 2 = 0 ; x = 2 ; There is 1 vertical asymptote.
27. A student is chosen at random from a class of 16 girls and 14 boys. What are the odds that the student chosen is not a girl?
8 7 7 1 7 (A) (B) (C) (D) 1 (E) 1 1 5 1 5 8 7 8
F a v o r a b l e 1 4 7 Odds = = = U n f a v o r a b l e 1 6 8 28. The graph of x 2 - 8x - 4y 2 + 16y = 36 is a(n) :
(A) parabola (B) circle (C) ellipse (D) hyperbola (E) cardioid
B 2 - 4AC = 0 2 - 4(1)(- 4) = 16
Since B 2 - 4AC > 0, then the given equation is that of a hyperbola.
29. {2, 1, 3, 4, 7, 11, …} is the set of Lucas numbers. What is the 4th Even Lucas number?
(A) 18 (B) 34 (C) 76 (D) 132 (E) 144
The sum of the two previous numbers is equal to the next number. Extend the list of Lucas numbers until you find the 4th Even Lucas number.
2, 1, 3, 4, 7, 11, 18, 29, 47, 76, …
Answer : 76
30. The number 235 in base 7 is equivalent to the number xyz in base 8, where x, y, and z are digits. Find x + y + z.
(A) 12 (B) 11 (C) 10 (D) 9 (E) 8
7[2(7) + 3] + 5 = 7[17] + 5 = 119 + 5 = 124 (base 10)
124 base 10 = 1(64) + 7(8) + 4(1) = 174 base 8.
x = 1, y = 7 and z = 4
x + y + z = 1 + 7 + 4 = 12
31. Find f(g(h(x))) when f(x) = x - 5, g(x) = 3 - 2x, and h(x) = - x.
(A) 2(x - 1) (B) 0 (C) 2x + 13 (D) 3x - 8 (E) x
g(h(x)) = 3 - 3(- x) = 3 + 2x
f(g(h))) = 3 + 2x - 5 = - 2 + 2x = 2x - 2 = 2(x - 1) 32. The vertex of the function y = 3|x - 6| + 14 is ?
(A) (3, 14) (B) (7, 3) (C) (6, 14) (D) (6, 7) (E) (14, 6)
Use transformations to find the vertex.
First there is a vertical shift of 14 units, thus the y value of the vertex is 14. Second there is a horizontal shift of 6 units, thus the x value of the vertex is 6.
Answer : (6, 14)
1 1 33. Find the multiplicative inverse of . 1 1
1 1 1 1 1 1 1 0 (A) (B) (C) (D) (E) 1 1 1 1 0 0 0 1 does not exist
34. Hi Roller is playing a game with a single fair die. He gets 900 points if he rolls a 6. He loses 200 points if he rolls a 1, 2, or 3 and loses 120 points if he rolls a 4 or 5. How many points should Hi expect to gain or lose on average?
(A) gain 150 (B) gain 10 (C) gain 5 (D) lose 140 (E) lose 290
1 3 2 ( 9 0 0 ) - (200) - (120) = 150 - 100 - 40 = 10 6 6 6
Answer : gain 10
35. The next term in the recursive sequence 2, 5, 11, 23, 47, … is ______.
(A) 95 (B) 88 (C) 81 (D) 70 (E) 59
Notice how they find each successive number : First they add 3, then 6, then 12, then 24. To find the next number you must add 48 to 47. The answer is 95. 36. Which of the following is considered to be an “unhappy” number?
(A) 7 (B) 10 (C) 13 (D) 17 (E) 19
A happy number is defined by the following process. Starting with an positive integer, replace the number by the sum of the squares of its digits, and repeat the process until the number equals 1 (where it will stay), or it loops endlessly in a cycle which does not include 1. Those number for which this process ends in 1 are happy numbers., while those that do not end in 1 are unhappy numbers.
The first few happy numbers are 1, 7, 10, 13, 19, 23, 28, 31, 32, 44, 49, 68, 70, 79, 82, 86, 91, 94, 97, 100, …
Answer : 17
37. Morfolk, Tx has a population on Jan 1, 2005 of 180. On Jan 1, 2006 the population was 210. In what year will the population of Morfolk reach 384, provided the population grows the same amount each year as id did from 2005 to 2006?
(A) 2010 (B) 2011 (C) 2012 (D) 2013 (E) 2014
Use the information given and translate to the following collinear points;
(9, 180), (1, 210), and (x, 384)
Since slopes between any two consecutive collinear points are equal you can set up the following equation.
3 8 4 2 1 0 2 1 0 1 8 0 = x 1 1 0
174 = 30x - 30 ; 30x = 205 ; x = 6.8
2005 + 6.8 = 2011.8
The population of Morfolk will reach 384 in 2011. 38. Use the infinite geometric sequence a, - 6, b, c, .75, … to find a + b + c.
(A) 16.5 (B) 13.5 (C) 7.75 (D) - 4.75 (E) - 10.5
Let x = ratio
. 7 5 . 7 5 1 - 6x 3 = .75 ; x 3 = ; x = 3 = - 6 6 2
1 - 6 = b ; b = 3 2
1 3 = c ; c = - 1.5 2
1 a = - 6 ; a = - 6(- 2) ; a = 12 2
a + b + c = 12 + 3 - 1.5 = 13.5
2 5 39. Find the harmonic mean of and . 3 9
3 7 1 1 1 0 2 0 (A) (B) (C) (D) (E) 1 1 1 2 1 8 2 7 3 3
2 2 5 2 G e o m e t r i c M e a n 3 9 Harmonic Mean = = 2 5 = A r i t h m e t i c M e a n 3 9 2
1 0 1 0 2 7 2 7 1 0 5 4 2 0 = 3 3 = = = 3 3 2 7 3 3 3 3 2 7 5 4 2 40. 24% of the girls at Poly Anna Tech play in the school band and have a laptop computer. 45% of the girls have a laptop. What is the probability that a girl plays in the band given that she has a laptop? (nearest whole percent).
(A) 53% (B) 69% (C) 65% (D) 11% (E) 21%
2 4 x 100 = 53.333…% (Answer is 53%) 4 5
41. If log 1 0 2 = x and log 1 0 3 = y and the log 1 0 5 = z, then log 1 0 60 = ?
(A) 2xyz (B) 2x + yz (C) x 2 yz (D) 2x + y + z (E) x 2 + y + z
log 1 0 60 = log 1 0 (2)(2)(3)(5) = log 1 0 2 + log 1 0 2 + log 1 0 3 + log 1 0 5
= x + x + y + z = 2x + y + z
42. Tick Tock the circular clock showed the time to be 7:25. What was the measure of the acute angle between the big hand and the little hand at that time?
(A) 80° (B) 72.5° (C) 65.25° (D) 60° (E) 36.25°
Make sketch depicting time. Keep in mind that there are 30° between hour hands. Notice that between the hour hand and the minute hand there is 2 hours of time which equates To 60°. In the 25 minutes of time after 7:00 the minute hand traveled 2 5 3 0 = 12.5°. 6 0
Measure of angle = 60° + 12.5° = 72.5° 43. Twenty-five students play at least one sport. Eighteen students play basketball and 11 play tennis. How many students play both basketball and tennis?
(A) 4 (B) 7 (C) 9 (D) 12 (E) 14
Set up a Venn diagram. Let x equal number of students that play both basketball and tennis.
x + (18 - x) + (11 - x) = 25
29 - x = 25 ‘ - x = - 4 ; x = 4
44. 444 5 + 333 4 - 222 3 = ______2 .
(A) 10100001 (B) 111 (C) 10101 (D) 11001101 (E) 1001001
[5(5)(4) + 4) + 4] + [4(3(4) + 3) + 3] - [3(2(3) + 2) + 2] = 124 - 63 - 26 = 161
161 1 0 = 10100001 2
x 4 x 3 B 45. If + is written as the mixed number A , then B is? x 3 x 4 C
(A) 49 (B) 31 (C) 25 (D) 24 (E) 7
Find the remainder when x - 4 is divided by x + 3 and x + 3 is divided by x - 4.
Use long division. (x - 4) ÷ (x + 3) = 1 remainder – 7 and (x + 3) ÷ (x - 4) = 1 Remainder of 7.
B 7 7 7 x 2 8 7 x 2 1 4 9 = + = = ; B = 49 C x 3 x 4 ( x 5 ) ( x 4 ) ( x 3 ) ( x 4 ) 46. If A + B = 12 and A x B = 22, then B A = ______.
(A) 2 6 6 (B) 2 1 4 (C) 5 6 6 (D) 2 1 0 (E) 5 1 4
If A + B = 12, then A = 12 - B.
(12 - B)B = 22
12B - B 2 = 22
B 2 - 12B + 22 = 0
Use the quadratic formula to find B.
( 1 2 ) ( 1 2 ) 2 4 ( 1 ) ( 2 2 ) B = 2 ( 1 )
1 2 5 6 1 2 2 1 4 B = = = 6 + 1 4 or 6 - 1 4 2 2
A = 12 - B = 12 - 6 - 1 4 = 6 - 1 4 Note : A could be either 6 - 1 4 or
6 + 1 4 . B A = 6 1 4 ( 6 1 4 ) = 2 1 4
47. Use the Fibonacci characteristic sequence … p, - 3, q, 1, r, 6, … to find p + q + r.
(A) 17 (B) 16 (C) 15 (D) 14 (E) 13
The third term is equal to the sum of the two previous terms (1st and 2nd). The fourth term is equal to the sum of the two previous terms (2nd and 3rd), and so on.
- 3 + q = 1 ; q = 4
p + (- 3) = q l p + (- 3) = 4 ; p = 7
1 + r = 6 ; r = 5
p + q + r = 7 + 4 + 5 = 16 2 1 x 4 48. Find y if = . 1 3 y 5
(A) - 2.0 (B) - 1.6 (C) 1.2 (D) 2.8 (E) 3.4
2x - y = 4 and - x + 3y = 5
1 7 [3(2x - y) = 3(4)] + [- x + 3y = 5] = 5x = 17 ; x = = 3.4 5
2s - y = 4
2(3.4) - y = 4
- y = 4 - 6.8
-y = - 2.8 ; y = 2.8
49. Ann Teeks purchased a 1935 Plymouth coupe with a rumble seat an old car auction three years ago. The vale of Ann’s coupe fell 11% the first year, rose 20% the second year, and rose 9% the third year. What percent per year did the coupe’s value rise, on average? (nearest tenth)
(A) 13.3% (B) 9.0% (C) 5.2% (D) 12.3% (E) 6.0%
Assume cost is $100.
100(.89)(1.2)(1.09) = $116.4120
100(x) 3 = 116,4120
x = 1.052 ; Percent increase per year is 5.2%
50. Determine the sum of the infinite series : .27 + .0027 + .000027 + .00000027 + …
7 1 2 2 3 (A) 2 (B) 3 (C) 3 (D) (E) 1 0 2 1 1 7 1 1
a . 2 7 . 2 7 2 7 3 Sum of an infinite geometric series = = = = = 1 r 1 . 0 1 . 9 9 9 9 1 1 1 2 5 51. P, Q, and R are positive integers. If P + 1 = , then find P + Q + R. Q 1 9 R
1 9 (A) 25 (B) 18 (C) 10 (D) 6 (E) 2 5
1 1 6 6 P + 1 = 1 ; P = 1 and 1 = ; Q 1 9 Q 1 9 R R
1 1 9 1 1 1 1 Q + = ; Q + = 3 ; Q = 3 and = ; R = 6 R 6 R 6 R 6
P + Q + R = 1 + 3 + 6 = 10
52. Which of the following numbers is considered to be an “happy even” number?
(A) 384 (B) 388 (C) 396 (D) 400 (E) 404
A happy number is defined by the following process. Starting with an positive integer, replace the number by the sum of the squares of its digits, and repeat the process until the number equals 1 (where it will stay), or it loops endlessly in a cycle which does not include 1. Those number for which this process ends in 1 are happy numbers., while those that do not end in 1 are unhappy numbers.
Solution : 404--- 4 2 + 0 2 + 4 2 = 32
32 ---- 3 2 + 2 2 = 13
13--- 1 2 + 3 2 = 10
10--- 1 2 + 0 2 = 1, Thus 404 is a happy number (and it is also even). 53. The zoo has two bears, a polar bear and a grizzly bear. What is the probability that both bears are females if you know that one of them is a female?
1 2 (A) 25% (B) 33 % (C) 50% (D) 66 % (E) 100% 3 3
Since there are only 2 bears, the possibilities are (M, M), (M, F), (F, M) and (F, F) where M represents a male bear and F represents a female bear.
IF you know that one of the bears is female then there are only 3 possibilities :
(M, F), (F, M), (F, F)
1 1 Probability = = 33 % 3 3
54. If f(x) = 1 - 3x, then f 1 f ( f ( 2 ) ) equals :
(A) 16 (B) 15 (C) - 3 (D) - 5 (E) - 11
f(2) = 1 - 3(2) = - 5
f(f(2)) = f(- 5) = 1 - 3(- 5) = 16
f 1 (16)--- 16 = 1 - 3x ; - 3x = 15 ; x = - 5
55. Which of the following is not a one-to-one function :
3 (A) y = |2x + 3| (B) y = 3 + 2x (C) y = 2x 3 (D) y = - 2 x (E) y = 2x - 3
To determine if a function is one-to-one you can use the “horizontal line test”. If a horizontal line intersects the graph of a function more than once, the function is not one-to-one. If a horizontal line intersects the graph less than twice it is a one-to-one,
Note : The graph of y = |2x + 3| is V-shaped, thus a horizontal line will intersect it twice. This function is not one-to-one. 56. Let g(x) = 2x + 1 and h(x) = 2 - 3x. Find [h{g{x - 1))].
(A) 5 - 3x (B) 6x + 1 (C) x + 5 (D) 3 - x (E) 5 - 6x
g(x) = 2x + 1
f(x - 1) = 2(x - 1) + 1 = 2x - 2 + 1 = 2x - 1
h(g(x - 1)) = h(2x - 1) = 2 - 3(3x - 1) = 2 - 6x + 3 = 5 - 6x
57. In the expansion of (2x - y) 5 , the coefficient of the 4th term is :
(A) - 40 (B) - 20 (C) 20 (D) 40 (E) 60
Assume that you are expanding (a + b) 5 .
a 5 + 5a 4 b + 10a 3 b 2 + 10a 2 b 3 + 5ab 4 + b 5
The fourth term is 10a 2 b 3 .
Since a = 2x and b = - y, substitute to find the 4th term in the original expansion.
10(2x) 2 (- y) 3 = - 40x 2 y 3 ; Answer : - 40
58. The geometric mean of 2008 and 8002 is ______% of the arithmetic mean of 2008 and 8002. (nearest whole %)
(A) 125% (B) 120% (C) 80% (D) 40% (E) 25%
2 0 0 8 ( 8 0 0 2 ) 2 0 0 8 8 0 0 2 x 100 = 80% 2 3 59. Evaluate : log 4 (24) - log 4 + log(32) 1 6
(A) 3.5 (B) 4 (C) 6 (D) 8 (E) 13.25
1 6 Solution : log 4 24 (32) = log 4 4096 3
4 x = 4096 ; x = 6 60. How many ways can Frank N. Stine fill bags of candy with 5 pieces of candy if his only 3 choices are sour balls, jawbreakers, and black licorice?
(A) 21 (B) 30 (C) 60 (D) 125 (E) 243
Let the three choices of candy be A, B, and C. The following are the possible combinations.
(1) AAAAA (2) AAAAB (3) AAABB (4) AABBB
(5) ABBBB (6) BBBBB (7) BBBBC (8) BBBCC
(9) BBCCC (10) BCCCC (11) CCCCC (12) CCCCA
(13) CCCAA (14) CCAAA (15) CAAAA (16) AAABC
(17) AABBC (18) ABBBC (19) AABCC (20) ABCCC
(21) BBACC
Answer : 21
61. Which of the following values of x is a vertical asymptotes for f(x) = ( x 3 ) ( x 7 ) ? ( x 4 ) ( x 3 )
(A) 10 (B) 7 (C) 3 (D) - 3 (E) - 4
Find values of x that will make the denominator equal to 0/
(x - 4)(x + 3) = 0 ; x = 4 or - 3
Answer : - 3 62. The roots of the equation 3x 3 - 2x 2 + cx + d = 0 are - 2, 3, and R. Find R.
2 1 2 (A) 1 (B) 1 (C) 0 (D) - (E) - 1 3 3 3
b If ax 3 + bx 2 + cx + d = 0, then the sum of the roots is - . a
b 2 - = - = - 2 + 3 + R a 3
2 = 3{R + 1{
1 3R = - 1 ; R = - 3
63. Which of the following is considered to be a “lucky” number?
(A) 13 (B) 11 (C) 8 (D) 5 (E) 2
In number theory, a lucky number is a natural number in a set which is generated by a “sieve: similar to the Sieve of Eratosthenes that generates the primes.
1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49, 51, 63, 67, 69, 73, 75, 79, 87, 93, 99, …
Answer : 13 64. If the matrix equation - 3x y = x 4 - 8 2 y , then x + y eqls :
(A) 3.25 (B) 2.8 (C) 2.4 (D) 1.2 (E) - .8
-3x y = x 4 - 8 2 y
3 x 3 y = x 8 4 2 y
- 3x = x - 8 ; - 4x = - 8 ; x = 2
4 - 3y = 4 + 2y ; - 5y = 4 ; y = - 5
4 1 x + y = 2 + = 1 = 1.2 5 5
x 2 6 x 8 65. How many vertical asymptotes does y = have? x 2 6 x 8
(A) 4 (B) 3 (C) 2 (D) 1 (E) 0
( x 4 ) ( x 2 ) y = ( x 4 ) ( x 2 )
Find vertical asymptotes.
(x - 4)(x - 2) = 0 ; x = 4 and x = 2
There are 2 vertical asymptotes. 66. The graph of y = x 2 - 2x + 3 is transformed by being shifted down 4 units. Then it is shifted left 2 units. Which of the following is the equation for the transformed graph?
(A) y = x 2 - 6x + 1 (B) y = x 2 + 3x - 1 (C) y = x 2 + 1
(D) y = x 2 - 4x - 1 (E) y = x 2 + 2x - 1
y = x 2 - 2x + 3
y - 2 = x 2 - 2x + 3 - 2
y - 2 = (x - 1) 2
y = (x - 1) 2 + 2
y = (x - 1) 2 + 2 - 4 Vertical shift down 4 units
y = (x - 1 + 2) 2 - 2 Horizontal shift left 2 units
y = (x + 1) 2 - 2
y = x 2 + 2x + 1 - 2
y = x 2 + 2x - 1 67. Students at the University of Prob must take 2 final exams on final exam day. There are 5 class periods available for the finals. What is the probability that a student with a randomly selected schedule has both finals during consecutive class periods?
1 4 (A) 25% (B) 33 % (C) 40% (D) 44 % (E) 50% 3 9
A, B, _____, _____, _____ A, _____, B, _____, _____ A, _____, _____, B, _____
A, _____, _____, _____, ______, A, B, _____, ______, A, _____, B, _____
_____, A, _____, _____, B _____, _____, A, B, ______, _____, A, _____, B
_____, _____, _____, A, B
There are 10 possible arrangements. Four of the arrangements have the classes next to each other.
4 Probability = = 40% 1 0
68. Find x + y from the given system of equations : 2x + 3y = 6
4x - y = 5
1 2 1 1 (A) 3 (B) 2 (C) 1 (D) - (E) - 2 3 3 2
(2x + 3y = 6) + 3(4x - y = 5)
2 1 14x = 21 ; x = = 1,5 1 4
4(1,5) - y = 5 ; 6 - y = 5 ; - y = - 1 ; y = 1
1 x + y = 1.5 + 1 = 2.5 = 2 2 1 69. Simplify : a 2 b 3 a 2 x b 3
(A) b 6 (B) a 4 b 6 (C) a 4 b (D) a 4 b 1 (E) a 4
2 3 1 1 a b 3 4 4 2 x b = a = a a
70. If Hi Score makes 188 on his next UIL number sense test, his average will be 196. If he makes 206, his average will be 202. How many tests has he already takent?
(A) 1 (B) 2 (C) 3 (D) 4 (E) 5
Let x = number of tests taken already
Let y = total points already accumulated
y 1 8 8 y 1 8 8 = 196 l y + 188 = 196(x + 1) ; x + 1 = x 1 1 9 6
y 2 0 6 y 2 0 6 = 202 ; y + 206 = 202(x + 1) ; x + 1 = x 1 2 0 2
y 1 8 8 y 2 0 6 = 1 9 6 2 0 2
202y + 188(202) = 196y + 186(206)
6y = 196(206) - 188(202)
1 9 6 ( 2 0 6 ) 1 8 8 ( 2 0 2 ) y = = 400 6
4 0 0 2 0 6 x + 1 = 2 0 2
x + 1 = 3 ; x = 2 71. The set of odd numbers is closed under which of these standard operations :
(A) addition (B) subtraction (C) multiplication (D) division (E) none of these
The set of odd numbers is closed under multiplication because the product of any two odd numbers is odd.
x 2 i f x 0 72. Find f(2) + f(0) + f(- 2) if f(x) = { 2 i f x 0 2 x i f x 0
(A) - 8 (B) - 6 (C) - 2 (D) 4 (E) 6
f(2) : x > 0 ; f(2) = 2 - 2 = 0
f(0) : x = 0 ; f(0) = - 2
f(- 2) : x < 0 ; f(- 2) = - 2 - 2 = - 4
f(2) + f(0) + f(- 2) = 0 + (- 2) + (- 4) = - 6
73. What is the digit in the tens place of 7 2 1 ?
(A) 0 (B) 1 (C) 3 (D) 4 (E) 9
Find the powers of 7 until you find a pattern.
7 1 = 7 ; 7 2 = 49 ; 7 3 = 343 ; 7 4 = 2401
7 5 = 16807 ; 7 6 = 117,649 ; 7 7 = 823,543 ; 7 8 = 5764801
Notice that the tens and units digits follow the pattern : 07, 49, 43, 01
Note : 7 1 , 7 5 , 7 9 , 7 1 3 , 7 1 7 , 7 2 1 each have 07 as the tens and units digits. Thus the answer is 0. 74. The whole numbers 20, P, 12, Q, are arranged in decreasing order. The numbers 20, P, 12, Q form an arithmetic sequence with a common difference, d. The numbers 32, P, Q, form a geometric sequence with a common ratio, r. Find the product of d and r.
(A) - 8 (B) - 2 (C) 4 (D) 10 (E) 20
20 + 2d = 12 ; 2d = - 8 ; d = - 4
Note : 20 + (- 4) = P ; P = 16 ; 12 + (- 4) = Q ; Q = 8
Numbers in the pattern are 20, 16, 12, 8
1 The geometric sequence 32, P, Q is 32, 15, 8 ; The ratio, r = / 2
1 dr = (- 4) = - 2 2
75. Simplify : log p (2w) + log p (3x) - log p (4y)
(A) log p )2w + 3x - 4y) (B) log p (2wxy) (C) log p (6wx - 4y)
2 ( w x ) 1 (D) log p (E) log p (1.5wxy ) 3 y
2 w ( 3 x ) 3 w x log p (2w) + log p (3x) - log p (4y) = log p = log p = log p 4 y 2 y (1.5wxy 1 )
76. The eccentricity of the ellipse 16x 2 + 7y 2 = 112 is :
1 2 3 7 3 7 (A) (B) (C) (D) (E) 7 1 1 2 1 6 4 4
a 2 = 16 ; a = 4 ; b 2 = 7
a 2 = b 2 + c 2 ; 16 = 7 + c 2 ; c 2 = 9 ; c = 3
c 3 eccentricity = = a 4 77. How many negative real roots will 2x 4 - x 3 + 4x 2 - 5x + 3 = 0 have?
(A) 0 (B) 1 (C) 2 (D) 3 (E) 4
Apply the Descartes’s Rule of Signs.
f(- x) = 2(- x) 4 - (- x) 3 + 4(- x) 2 - 5(- x) + 3
f(- x) = 2x 4 + x 3 + 4x 2 + 5x + 3
Since there is no change of signs, then there are 0 negative real roots.
78. Doug Up charges $3.50 to dig the first fence post hole. He charges $3.70 for the second hole, $3.90 for the third hole, $4.10 for the fourth hole and so on. At this rate what would he charge to dig the twentieth hole?
(A) $8.10 (B) $6.90 (C) $7.50 (D) $8.50 (E) $7.30
Note pattern : 3.50, 3.70, 3.90, 4.10, 4.30, 4.50, 4.70, 4.90, …
There is a difference of $0.20 between successive terms. This is an arithmetic sequence.
To find the 20th term, you must add $0.20 nineteen times to $3.50.
3.50 + 19(.20) = 7.30 79. Find the remainder when 4x 4 - 2x 2 + 1 is divided by x 2 - 1.
(A) 4 (B) 3 (C) 0 (D) - 1 (E) - 2
Find the remainder by using long division.
4x 2 + 2 x 2 - 1 4 x 4 2 x 2 1
- (4x 4 - 4x 2 )
2x 2 + 1
- (2x 2 - 2)
3
80. If a 1 = 1, a 2 = 2, and a n = 2(a n 2 ) - (a n 1 ) where n > 2, then a 6 = ?
(A) 6 (B) - 8 (C) 12 (D) 0 (E) - 4
a 3 = 2(a 3 2 ) - (a 3 1 ) = 2a 1 - a 2 = 2(1) - 2 = 0
Knowing the first 3 terms and using the information given, you can find the first 6 terms.
1, 2, 0, - 4, - 4, 12
Answer : 12 1 81. If 3 + x + x 2 + x 3 + x 4 + … = 3 , then x = ? 2
1 1 1 2 2 (A) (B) (C) (D) (E) 3 4 7 7 3
1 3 + x + x 2 + x 3 + x 4 + … = 3 2
1 x + x 2 + x 3 + x 4 + … = 2
x + x 2 + x 3 + x 4 + … is an infinite geometric series with first term x and common ratio x.
a Sum of an infinite geometric series = where a is the first term and r is the 1 r common ratio.
x 1 = 1 x 2
1 2x = 1 - x ; 3x = 1 ; x = 3 82. Solve the system of equations for x.
x + 2y - z = 5 3x + 2y + z = 11 (x + 2y) 2 - z 2 = 15
(A) - 5 (B) - 1 (C) 0 (D) 4 (E) 5
Rewrite the leftmost equation : x + 2y = z + 5.
Substitute for x + 2y in the right-most equation.
(z + 5) 2 - z 2 = 15
z 2 + 10z + 25 - z 2 = 15
10z = - 10 ; z = - 1
Substitute z = - 1 in the two left-most equations and rewrite :
x + 2y = 4
3x + 2y = 12
Solve for x. Subtract the bottom equation from the top equation.
- 2x = - 8 ; x = 4
83. How many four digit even numbers greater than 2006 and divisible by 3 can be created using the digits 1, 2, 3, 4, and 5? Digits cannot be repeated.
(A) 16 (B) 15 (C) 12 (D) 8 (E) 6
Possible combinations :
2154 2514 5214 5124
5412 5142 4152 4512
Answer : 8 84. If a 1 = - 1, a 2 = 1, a n = a n 1 a n 2 , where n ≥ 3, then a 1 3 + a 1 4 + a 1 5 equals :
(A) 2 (B) 1 (C) 0 (D) - 1 (E) - 2
Find a 3 . a 3 = a 3 1 a 3 2 = a 2 a 1 = (1)(- 1) = - 1
Note : The first three terms are – 1, 1, - 1. Using information given list the first fifteen terms.
First 15 terms : - 1, 1, - 1, - 1, 1, - 1, - 1, 1, - 1, - 1, 1, - 1, - 1, 1. -1
a 1 3 + a 1 4 + a 1 5 = (- 1) + 1 + (- 1) = - 1 Answer is D
85. The directrices of an ellipse are x = 4 and x = - 4. The foci are (1, 0) and (- 1, 0). Find the equation of the ellipse.
(A) x 2 + 4y 2 = 4 (B) 3 x 2 + 2y 2 = 2 3 (C) 4x 2 + 3y 2 = 24
(D) 3x 2 + 4y 2 = 12 (E) 4x 2 + y 2
x 2 y 2 In the ellipse 1 , a 2 = b 2 + c 2 , where a = length of semi-major a 2 b 2 axis, b = length of semi-minor axis, and c = the distance between the center of the ellipse and a focus. In the given ellipse, c = 1
a Note : The distance between the center and the directrix of an ellipse = , e c where e = (e = eccentricity of the ellipse). a
a a 2 a 2 c = ; 4 = ; a = 2 c 1 a
If a 2 = b 2 + c 2 , then 4 = b 2 + 1 ; b 2 = 3
Substitute in the standard equation formula.
x 2 y 2 + = 1 4 3
Multiply both sides of the equation by 12. 3x 2 + 4y 2 = 12 86. Robin Stern enters the archery contest. The target is a circle with a diameter of 3 feet. The bull’s-eye is a circle with a radius of 6 inches in the center of the target. What are the odds of hitting the bull’s-eye if the arrow is equally likely to hit any spot on the target?
1 1 1 1 1 (A) (B) (C) (D) (E) 1 0 9 8 3 2
F a v o r a b l e Odds = U n f a v o r a b l e
Favorable is equal to the area of the bull’s-eye : π (6) 2 = 36π
Unfavorable is equal to the area of the region between the outer circle and the bull;s-eye : π(18) 2 - π (6) 2 = 288π
3 6 1 Odds = = 2 8 8 8
87. Let P, Q, and R be Real numbers. If P + Q = 2 and PQ = R 2 + 1, then PQR = ?
(A) 0 (B) 1 (C) 2 (D) 3 (E) 4
Think of possible combinations.
P = 1, Q = 1 and PQ = 1 ; 1 = R 2 + 1 ; then R = 0 ; PQR = (1)(0) = 0
Thus, the answer is A.
P = 2, Q = 0 and PQ = 0 ; 0 = R 2 + 1 : No real values of R
P = 0, Q = 2 and PQ = 0 ; 0 = R 2 + 1 ; No real values of R 88. One of the factors of 15x 2 + 8ac - 20cx - 6ax is :
(A) 5c - 3x (B) 2a + 5x (C) 3x - 4c (D) 15x + 3a (E) 3x - 2a
15x 2 - 20cx + 8ac - 6ac = 5x(3x - 4c) - 2a(3x - 4c) = (5x - 2a)(3x - 4c)
The answer is (C) 3x - 4c.
89. Rob R. Ball drops a ball from a height of 8 feet. It bounces back to a height of 40% of the distance it fell. How far has it traveled when it hits the ground the fifth time? (nearest inch)
(A) 10’ 5” (B) 13’ 2” (C) 14’ 2” (D) 17’ 11” (E) 18’ 5”
8 + 2[8(.4) + 8(.4) 2 + 8(.4) 3 + 8(.4) 4 ) = 18.3936’ = 18’ 5”
90. Betty Wont has a bag that contains blank CDs are red, 4 are white, and 5 are blue. Willy Wheel randomly selects a CD from Betty’s bag and places it on the table. He selects a Second CD and places it on the table next to the other one. What is the probability that one of the two CDs is red and the other is blue?
1 2 8 1 0 4 (A) 9 % (B) 18 % (C) 22 % (D) 30 % (E) 53 % 1 1 1 1 1 1 3 3 7
3 5 5 3 3 0 7 2 8 + = = 22 % = 22 % 1 2 1 1 1 2 1 1 1 3 2 9 9 1 1
91. P, Q, and R are real numbers. If two of the roots of x 3 + Px 2 + Qx + R = 0 are equal in absolute value but opposite in sign, then PQ equals :
(A) - PQR (B) - P (C) R (D) PR + QR (E) - R
Let the roots of the cubic equation be a, - a, and b.
Sum of the roots = P = (- a) + a + b = b; P = b
Product of the roots = - R = (- a)(a)(b) ; R = a 2 b
Sum of product of the roots taken two at a time = (a)(- a) + (a)(b) + (- a)(b) = Q ; Q = - a 2
PQ = (b)(- a 2 ) = - a 2 b and PQ = - R 92. If (3 + 2i) ÷ (4 - i) = (a + bi), then a + b equals ______.
1 2 8 2 4 (A) - 1 (B) - 1 (C) 1 (D) 1 (E) 1 4 3 1 7 5 1 7
3 2 i 4 i 1 2 1 1 i 2 1 0 1 1 = = + i 4 i 4 i 1 6 1 1 7 1 7
1 0 1 1 2 1 4 a + b = + = = 1 1 7 1 7 1 7 1 7
3 2 93. R 1 , R 2 , and R 3 are the roots of the equation 2x + 9x - 6x - 5 = 0. R 1 2 and R 2 are the roots of the equation x + 4x - 5 = 0 as well. Find R 3 .
(A) 2.5 (B) 1 (C) - .5 (D) - 4.5 (E) - 5
b If ax 2 + bx + c = 0, then the sum of the roots = - . The sum of the roots a of x 2 + 4x - 5 = 0 is - 4. If ax 3 + bx 2 + cx + d = 0, then the sum of the roots = b - . The sum of the roots of 2x 3 + 9x 2 - 6x - 5 = 0 is - 4.5. a
The sum of the roots of the quadratic equation = R 1 + R 2 = - 4, The sum of the roots of the cubic equation = R 1 + R 2 + R 3 = - 4.5
Thus, R 3 = - .5 ; Answer is C
94. The expression x 4 + 4x 3 y + 6x 2 y 2 + 4xy 3 is equal to r 4 if we substitute ______for x.
(A) r - y (B) r + y (C) - r (D) - r + y (E) None of these
x 4 + 4x 3 y + 6x 2 y 2 + 4xy 3 = (x + y) 4
(x + y) 4 = r 4
x + y = r ; x = r - y 95. If x = .24, base 10, then x equals ______, base 5.
(A) .11 (B) .12 (C) .21 (D) .42 (E) None of these
2 4 x y 2 0 x 4 y = + = + ; Note : x = y = 1 1 0 0 5 2 5 1 0 0 1 0 0
Answer : .11
96. In the sequence 2, 11, 28, 53, …, if the nth term is a n and a n = kn 2 + jn + c, then k equals :
(A) 1 (B) 2 (C) 3 (D) 4 (E) None of these
Use given formula for a n .
2 a 1 = k(1) + j(1) + c = 2 ; k + j + c = 2
2 a 2 = k(2) + j(2) + c = 11 ; 4k + 2j + c = 11
2 a 3 = k(3) + j(3) + c = 28 ‘ 9k + 3j + c = 28
You now have three equations with three unknowns.
k + j + c = 2
4k + 2j + c = 11
9k + 3j + c = 28
Subtract the first equation from the last two equations.
3k + j = 9
8k + 2j = 26
Subtract the product of 2 times the top equation, from the bottom equation.
8k + 2j = 26 - 2(3k + j = 9) ; 2k = 8 l k = 4
Note : This problem could be solved by using Linear Regression on a TI- Calculator. 97. Find the length of a tangent to the circle x 2 + y 2 + 4x - 2y = 31 from the point (6, 7).
(A) 6.5 (B) 7 (C) 7.4 (D) 8 (E) None of these
Complete the square.
x 2 + 4x + 4 + y 2 - 2y + 1 = 31 + 4 + 1
(x + 2) 2 + (y - 1) 2 = 36
Center of (x - h) 2 + (y - h) 2 = r 2 is (h, k) and ra dius = r.
Center : (- 2, 1) and radius = 6
Make a sketch using given information.
Find the distance between the center (- 2, 1) and the point (6, 7).
( 2 6 ) 2 ( 1 7 ) 2 = 10
The triangle with vertices at the center (2, - 1), the point (6,7) and the point of tangency is a right triangle. Use the Pythagorean formula to find the length of the tangent.
10 2 = 6 2 + (tangent) 2
(tangent) 2 = 64. tangent = 8 98. Find the sum of the values of k such that 4x 2 + 2(1 - k)x + k + 2 = 0 has only one real root.
(A) 5 (B) 6 (C) 7 (D) 8 (E) None of these
Use the discriminant to determine one a quadratic equation has only one root.
If B 2 - 4AC = 0, then Ax 2 + Bx + C = 0 has only one real root.
2 ( 1 k ) 2 - 44 k 2 = 0
2 2 k 2 - 1 6 k 3 2 = 0
4 - 8k + 4k 2 - 16k - 32 = 0
4k 2 - 24k - 28 = 0
k 2 - 6k - 7 = 0
(k - 7)(k + 1) = 0
k = 7 or k = - 1
Sum of k-values : 7 + (- 1) = 6
99. Find f(g(1 - x)) when g(x) = 4x - 2 and f(x) = 3 + 6x.
(A) 15 - 24x (B) - 33 - 24x (C) 28 - 24x (D) 15 - 6x (E) 15 - 4x
g(1 - x) = 4(1 - x) - 2 = 4 - 4x - 2 = 2 - 4x
f(g(1 - x)) = 3 + 6(2 - 4x) = 3 + 12 - 24x = - 24x + 15 = 15 - 24x 100. Let x vary directly as the product of m and n and inversely as the product of r and 1 the square of s. If x = 3 when m = 4, n = 5, r = 2, and s = 3, find x when 3 m = - 2, n = 2, r = 3, and s = 5.
(A) .32 (B) .048 (C) - .05333… (D) - .16 (E) - .4
Let M = constantl
M ( m n ) x = r s 2
1 M ( 4 ) ( 5 ) 3 = 3 2 ( 3 2 )
1 0 2 0 M = 3 1 8
1 0 2 0 1 8 0 M = = = 3 3 1 8 6 0
3 ( m n ) 3 ( 2 ) ( 2 ) 1 2 Thus, x = 2 and x = 2 = - = - .16 r s 3 5 7 5 1 2 4 7 101. Find the coefficient of the x term of (x 2 - x 3 ) .
(A) - 21 (B) - 35 (C) 7 (D) 21 (E) 35
1 2 Let a = x 2 and b = - x 3
1 2 7 7 (x 2 - x 3 ) = (a + b)
(a + b) 7 = a 7 + 7a 6 b + 21a 5 b 2 + 35a 4 b 3 + 35a 3 b 4 + 21a 2 b 5 + 7ab 6 + b 7
1 4 Since they are looking for the coefficient of the x term, let a = x 2 and b = - 2 4 x 3 in each term until you get the x term.
4 3 1 2 Note : 35a 4 b 3 = 35 x 2 x 3 = - 35x 4
The coefficient of the x 4 term is - 35.
102. Find the sum of the coefficients of the quotient : (5x 4 - 3x 2 - 68) ÷ (x + 2)
(A) - 35 (B) - 22 (C) - 2 (D) 12 (E) 66
Use Synthetic Division to find the quotient.
| 5 0 - 3 0 - 68 - 2 | 5 - 10 17 - 34 0
The quotient is 5x 3 - 10x 2 + 17x - 34
Thus the sum of the coefficients is equal to 5 + (- 10) + 17 + (- 34) = - 22 103. The integers – 27, P, - 15, Q, and R are arranged in decreasing order and form an arithmetic sequence. The integers - 27, Q, and R form a geometric sequence. Find the sum of the next term of the arithmetic sequence and the next term of the geometric sequence.
(A) - 6 (B) - 3 (C) 0 (D) 2 (E) 4
If - 27, P, - 15, Q, R are terms of an arithmetic series, then
- 27 + 2d = - 15 ; 2d = 12 ; d = 6
Since the common difference is 6, find the terms of the arithmetic sequence.
- 27, - 21, - 15, - 9, - 3 ; P = - 21, Q = - 9, R = - 3
The next term of the arithmetic series is 3.
Use the values of Q and R to find the terms of the geometric sequence - 27, Q, R.
- 27, - 9, - 3
The next term of the geometric series is – 1.
Sum of the next term of the arithmetic series and the geometric series is 3 + (- 1) = 2.
104. The number 2007 in base 9 is equivalent to the number wxyz in base 8, where w, x, y, and z are digits. Find w + x + y + z.
(A) 16 (B) 9 (C) 17 (D) 8 (E) 18
Convert 2007 9 to base 10.
9(9(9(2) + 0) + 0) + 7 = 1465.
Convert 1465 1 0 to base 8.
2(512) + 6(64) + 7(8) + 1 ; 2671 8
w = 2, x = 6, y = 7, z = 1
w + x + y + z = 2 + 6 + 7 + 1 = 16 | x 2 | i f x 0 105. Find f(4) - f(0) - f(- 2) if f(x) = { 3 i f x 0 4 x i f x 0
(A) - 1 (B) 1 (C) 0 (D) 3 (E) - 7
Find f(4). Note : 4 > 0.
f(4) = 4 - 4 = 0
Find f(0). Note : if x = 0, then f(x) = - 3.
f(0) = - 3
Find f(- 2). Note - 2 < 0.
F(- 2) = |- 2 - 2| = |- 4| = 4
F(4) - f(0) - f(- 2) = 0 - (- 3) - 4 = - 1
106. An operation “ ” is defined by : a b = (ab) b . What is the value of 4 (2 1) ?
3 2 2 (A) 64 (B) (C) 16 (D) 1 (E) 4 2
1 2 1 = 2 ( 1 ) 1 = 2
1 1 2 1 1 1 2 4 (2 1) = 4 = = 2 = = 4 2 2 2 2 2
107. Given the function f(x) = x 4 + 3x 3 - 8x 2 + 5x - 2 there is a possibility of how many positive real zeroes?
(A) 4, 2, or 0 (B) 3 or 1 (C) 2 or 0 (D) 1 (E) 0
Apply the Descartes Rule of Signs. Determine how many sign changes f(x) has.
f(x) has 3 sign changes. Therefore there are 3 or 1 possible positive real roots. 108. The roots of the equation ax 3 + 3x 2 + cx - 6 = 0 are - 3, 2, and P. Find a.
(A) 3 (B) 2 (C) 1 (D) - 1 (E) - 2
b If ax 3 + bx 2 + cx + d = 0, then the sum of the roots is - and the product a d of the roots is - . a
3 3 Sum of the roots : - = - 3 + 2 + P ; - = P - 1 ; - 3 = Pa - a a a 6 6 Product of the roots : - = (- 3)(2)(P) ; = - 6P ; 6 = - 6aP ; aP = - 1 a a
- 3 = - 1 - a ; - 2 = - a ; a = 2
109. The value of Fran Teck’s super computer rose 15% the first year, 22% the second year, and fell 18% the third year. What percent per year did the computer value rise, on average? (nearest hundredth)
(A) 19.00% (B) 18.33% (C) 6.33% (D) 5.96% (E) 4.78%
Let x = initial cost of the computer ; r = average rise in percent per year
x(r) 3 = (.82)(1.22)(1.15)x
r 3 = (.82)(1.22)(1.15)
r = 3 ( . 8 2 ) ( 1 . 2 2 ) ( 1 . 1 5 ) = 1.0478
Answer : 4.78%
110. Boxes A, B, and C each contain four congruent poker chips colored red, white, blue, and green. One chip is selected at random from each of the boxes. Find the probability of getting three different colors.
(A) 62.5% (B) 37.5% (C) 12.5% (D) 9.375% (E) 1.5625%
1 3 2 6 3 1 4 = = = 37 % 4 4 4 1 6 8 2 111. The function f(x) = 5x 5 - 28x 4 + 50x 3 - 46x 2 + 45x - 18 has how many negative real roots?
(A) 0 (B) 1 (C) 0, 2, or 4 (D) 1 or 3 (E) 1, 3, or 5
Apply the Descartes Rule of Signs.
Find f(- x), f(- x) = - 5x 5 - 28x 4 - 50x 3 - 46x 2 - 45x - 18
Determine the number of sign changes of f(- x). Since there are none, then f(x) has 0 negative real roots.
112. Eighty-two percent of the licensed drivers in Texas wear seat belts. Barney Fife randomly selects two drivers to stop to see if they are wearing seat belts. What is the probability that neither of them are wearing seat belts? (nearest whole percent)
(A) 3% (B) 5% (C) 18% (D) 33% (E) 67%
1 8 1 8 ( 1 0 0 ) = 3.24% = 3% 1 0 0 1 0 0
1 113. Simplify : a b 3 x a b 2 a 3 b x ( a b ) 0
(A) a 4 b 2 (B) a 2 b 4 (C) a 1 b 2 (D) a 5 (E) ab 2
1 a b 3 ( a b 2 ) ( a b ) 0 3 a b
a 3 b 3 a b 2 ( 1 ) a 3 b
1 a b 2
a 1 b 2 1 x x 114. Solve for x, where x is a Real number : 6 + log 1 0 9 = log3 2
(A) 6 (B) 3 (C) 2 (D) infinite solutions (E) no solutions
1 x x : 6 + log 1 0 9 = log3 2
1 x x 6 + log 1 0 9 2 = log 1 0 3
x x 6 + log 1 0 3 = log 1 0 3
6 = 0
No solution
115. If 4 + 2x - 4x 2 + 8x 3 - 16x 4 + … = 4.375, then x = ?
(A) .75 (B) .625 (C) .3 (D) .25 (E) .125
4 + 2x - 4x 2 + 8x 3 - 16x 4 + … = 4.375
2x - 4x 2 + 8x 3 - 16x 4 + … = .375
The ratio of the infinite geometric series is - 2x
a Use the formula S = formula to find the sum of the terms on the left side 1 r of the equation.
2 x = .375 1 ( 2 x )
2 x 3 = 1 2 x 8
16x = 3 + 6x
10x = 3 ; x = .3 116. x + 1, 2x - 1, and 3x + 2 are factors of which of the following cubic polynomials?
(A) 6x 3 + 7x 2 - x - 2 (B) 6x 3 - 7x 2 + x - 2 (C) 6x 3 - 7x 2 - x + 2
(D) 6x 3 + 7x 2 - x + 2 (E) 6x 3 + 7x 2 - 5x - 2
(x + 1)(2x - 1)(3x + 2)
(x + 1)(6x 2 + 4x - 3x - 2)
(x + 1)(6x 2 + x - 2)
6x 3 + x 2 - 2x + 6x 2 + x - 2
6x 3 + 7x 2 - x - 2 ; Answer : A
x 117. The domain of the f(x) = is : 2 x 4
(A) {all reals} (B) [0, 2) (2, ) (C) (- , 2) (2, ) (D) (2, ) (E) (0, 2] (2, )
Since f(x) contains the term x , then the domain has to be greater than or equal to 0. Since the denominator of f(x) is 2x - 4, then x can/t be equal to 0. Thus the domain of x is B.
118. The length of an elliptical table measures 3 feet and the width measures 2 feet. What is the eccentricity of the ellipse? (nearest hundredth)
(A) 1.50 (B) .44 (C) .67 (D) 2.25 (E) .75
Since the major axis = 2a = 3, then a = 1.5. The minor axis = 2b = 2 and b = 1.
On an ellipse, a 2 = b 2 + c 2 .
1.5 2 = 1 2 + c 2
2.25 = 1 + c 2
c = 1 . 2 5
c 1 . 2 5 Eccentricity = = = .7454 = .75 a 1 . 5 119. Let f(x) = 3x - 1 and g(x) = 2 - x. Find g(f(2x)) + f(g(- 2x)).
(A) 12x (B) 3 - 6x (C) 2 (D) 8 (E) x6 + 5
f(2x) = 3(2x - 1) = 6x - 1
g(f(2x)) = 2 - (6x - 1) = 3 - 6x
g(- 2x) = 2 - (- 2x) = 2 + 2x
f(g(- 2x)) = 3(2 + 2x) - 1 = 5 + 6x
g(f(2x)) + f(g(- 2x))
3 - 6x + 5 + 6x = 8
120. Determine the type of conic section this equation y 2 - 4x 2 - 6y = 11 will produce.
(A) cardioid (B) circle (C) ellipse (D) hyperbola (E) parabola
B 2 - 4AC = 0 2 - 4(1)(-4) - 16
B 2 - 4AC > 0 ; Conic is a hyperbola
3 1 121. Let = 2x - 1. Find x. 2 5
(A) 9 (B) 6.5 (C) 5 (D) 1 (E) 13
3(5) - (2)(- 1) = 2x - 1
15 + 2 = 2x - 1
2x - 1 = 17 ; 2x = 18 ; x = 9 122. The graph of x 2 + 6x + y 2 - 6y + 15 = 0 lies in quadrant(s) :
(A) IV (B) III (C) II (D) I (E) I, II, III, & IV
x 2 + 6x + 9 + y 2 - 6y + 9 = - 15 + 9 + 9
(x + 3) 2 + (y - 3) 2 = 3
Center of the circle is (- 3, 3) and the radius is 3 = 1.732
Plot center (- 3, 3). It is located in the 2nd quadrant. The distance from the center to the x-axis is 3 units and the distance from the center to the y-axis is 3 units. Since the radius of the circle is approximately 1.732, then the circle is located entirely in the 2nd quadrant.
123. Determine the range the function f(x) = 2 - 3 x 5 /
2 2 (A) [- 1 , ) (B) (- , - 2] (C) (- , ) (D) (- , 2] (E) [2, - 1 ] 3 3
Use transformation techniques to sketch graph (or use TI graphing calculator). The graph can be thought of as starting at the point (- 5, 2). The graph has a y- intercept of 2 - 5 .
The range is (- , 2] 3 2 124. R 1 , R 2 , and R 3 are the roots of the equation 6x - 11x - x + 6 = 0. R 1 2 and R 2 are the roots of the equation 6x - 5x - 6 = 0 as well. Find R 3 .
(A) 1.5 (B) 1 (C) - .5 (D) - 1.5 (E) - 6
D If Ax 3 + Bx 2 + CX + D = 0, the product of the roots is - . A
6 The product of the roots of 6x 3 - 11x 2 - x + 6 = 0 is - = - 1. 6
R 1 R 2 R 3 = - 1
C If Ax 2 + Bx + C = 0, the product of the roots is . A
6 The product of the roots of 6x 2 - 5x - 6 = 0 is - = - 1 6
R 1 R 2 = - 1
(- 1)R 3 = - 1 ; R 3 = 1
125. The focus of the figure given by the equation y 2 + 4y + 12x + 16 = 0 is (x, y). Find x.
(A) - 1 (B) - 2 (C) 2.5 (D) 3 (E) 4
y 2 + 4y + 4 = - 12x - 16 + 4
(y + 2) 2 = - 12x - 12
(y + 2) 2 = - 12(x + 1)
If (y - k) 2 = 4p(x - h), then (h, k) is the vertex. This parabola is opening to the left. The distance from the vertex to the focus is p.
|4p| = |- 12) ; p = 3
vertex : (- 1, - 2)
The focus is to the left of the vertex. 3 units to the left of the vertex (- 1, - 2) is (- 4, - 2).
x = - 4 126. Willy Luze is playing a game with a fair dice. If he rolls a 1, 2, or 3 he loses 200 points. If he rolls a 4 or 5 he loses 100 points. If he rolls a 6 he wins 1000 points. How many points should Willy expect on average to gain or lose?
1 2 2 1 2 (A) gain 33 (B) gain 66 (C) lose 16 (D) lose 33 (E) lose 66 3 3 3 3 3
3 2 1 ( 2 0 0 ) + ( 1 0 0 ) + (1000) 6 6 6
1 0 0 1 0 0 0 6 0 0 2 0 0 1 0 0 0 2 0 0 1 0 0 1 - 100 - + = = = = 33 gain 3 6 6 6 3 3
127. Two girls and two boys are lined up randomly in a row. What is the probability there is exactly one girl between the two boys?
1 2 (A) 25% (B) 33 % (C) 50% (D) 66 % (E) 75% 3 3 Let two girls be G1 and G2 and both boys be B1 and B2.
Total number of outcomes = 4! = 24
Find number of arrangements where there is exactly one girl is between the two boys.
G1, B1, G2, B2 G2, B1, G1, B2 G1, B2, G2, B1 G2, B2, G1, B1
B1. G1, B2, G2 B2, G1, B1, G2 B1, G2, B2, G1 B2, G2, B1, G1
8 1 Probability = = 33 % 2 4 3 128. Which of the following numbers is considered to be an “unlucky abundant” number?
(A) 128 (B) 112 (C) 87 (D) 63 (E) 45
The first few lucky numbers are : 1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49, 51, 63, 67, 69, 73, 75, 79, 87, 93, 99, … Notice that all of the lucky numbers are odd. Thus, the only choices that are unlucky are (A) 128 and (B) 112. Determine which of these two numbers is abundant.
The sum of the positive integral divisors of 128 is
2 7 1 1 = 255. Since the sum of the positive integral divisors is less 2 1 than twice the number, then 128 is deficient.
The sum of the positive integral divisors of 112 is
2 4 1 1 7 1 1 1 = (31)(8) = 248. Since the sum of the positive 2 1 7 1 integral divisors is more than twice the number, then 112 is abundant.
Answer : 112
128. Vector v = (8, 6) and vector u = (- 3, 4). Find the dot product of vectors u and v.
(A) 0 (B) 14 (C) 15 (D) 24 (E) 36
(8)(- 3) + (6)(4) = - 24 + 24 = 0
130. Which of the following sets of numbers does not satisfy the closure property of addition?
(A) Evens (B) Integers (C) Irrationals (D) Rationals (E) Reals
A set of numbers satisfy the closure property of addition if the sum of any element in the set is equal to one of the members of the set?
Irrational numbers : 2 + 2 = 0 This example shows the sum of the two Irrational numbers is equal to a real number, not an irrational number.
Answer : (C) Irrationals. 131. One of the factors of x 3 + 7x 2 + 2x + 14 is :
(A) x - 7 (B) x 2 + 2 (C) x + 14 (D) x 2 - 2 (E) x 2 - 7
Method #1 : Use synthetic division to determine root(s) of polynomial.
0 | 1 7 2 14 - 1 | 1 6 - 4 18 - 2 | 1 5 - 8 30 - 3 | 1 4 - 10 44 - 4 | 1 3 - 10 54 - 5 | 1 2 - 8 54 - 6 | 1 1 - 4 38 - 7 | 1 0 2 0
Since - 7 is a root, x + 7 is a factor. Looking at the last row in the synthetic divisions shown above, x 3 + 7x 2 + 2x + 14 = (x + 7)(x 2 + 2) = 0.
Answer : (B) x 2 + 2
Method #2 : Use the graphing calculator to determine if there are any integral roots of the polynomial. You will find out that there is a root at x = - 7. You will than be able to factor the polynomial.
132. The sum of the real solutions of 3 2 x = 4 is :
(A) 4 (B) 16 (C) - 2 (D) - 5 (E) 9
3 - 2 x = 4 ; - 2 x = 1 ; 2 x = - 1, Since the absolute value of 2 - x is equal to – 1, there is not solution.
3 - 2 x = - 4 ; - 2 x = - 7 ; 2 x = 7
2 - x = 7 or 2 - x = - 7 - x = 5 or - x = - 9 x = - 5 or x = 9
The sum of the solutions is (- 5) + 9 = 4 133. The number 2008 in base 9 is equivalent to the number wxy2 in base 8, where w, x, and y are digits. Find w + x + y.
(A) 27 (B) 24 (C) 18 (D) 16 (E) 15
Convert 2008 base 9 to base 10.
9(0(9(2) + 0) + 0) + 8 = 1466 in base 10
Convert 1466 base 10 to base 8.
2(512) + 6(64) + 7(8) + 2(1)
1466 base 10 = 2672 base 8.
w = 2, x = 6, y = 7 ; w + x + y = 2 + 6 + 7 = 15
134. Which of the following is NOT a solution to |1 - 4x| - 7 < - 2?
(A) 1.742 (B) 1.427 (C) 0.247 (D) - 0.472 (E) - 0.742
|1 - 4x| < 5
1 - 4x < 5 and 1 - 4x > - 5
- 4x < 4 amd - 4x > - 6
x > - 1 and x < 1.5 l - 1 < x < 1.5
Determine which answer choice is not between – 1 and 1.5. The answer is (A) 1.742. 135. The ellipse 4x 2 + y 2 - 8x + 4y - 8 = 0 has a major axis with a length of :
(A) 20 (B) 16 (C) 12 (D) 8 (E) 4
4x 2 - 8x + y 2 + 4y - 8 = 0
4(x 2 - 2x + 1) + y 2 + 4y + 4 = 8 + 4 + 4
4(x - 1) 2 + (y + 2) 2 = 16
( x 1 ) 2 ( y 2 ) 2 + = 1 4 1 6
( x h ) 2 ( y k ) 2 In the ellipse + = 1, a = the length of the semi-major axis. b 2 a 2
Since a 2 = 16 and a = 4, then the length of the major axis is 2a = 2(4) = 8.
136. If a 1 = 2, a 2 = 1 and a n 2 + a n 1 , where n ≥ 3, then a 6 equals :
(A) 5 (B) 6 (C) 7 (D) 11 (E) 18
a 3 = a 3 2 + a 3 1 = a 1 + a 2 + 2 + 1 = 3
Note : The first 3 terms of the sequence are 2, 1, 3,… The third term is equal to the sum of the two previous terms, Extend the sequence to find the 6th term :
2, 1, 3, 4, 7, 11
Answer is 11.
137. Simplify to the form a + bi : 3 5 0 + 7 2
(A) 11 + 3 2 I (B) 0 + 21 2 I (C) - 21 2 + 0i
(D) 15 2 + 6 2 I (E) 0 - 2 2 i
3 2 5 ( 2 ) ( 1 ) + 3 6 ( 2 ) ( 1 )
15i 2 + 6i 2 = 21i 2 = 0 + 21 2 i 138. The Cheep Shoppe sells baby chicks for ten cents each and sells adult hens for seventy-five cents each. One week they sold a total of 346 baby chicks and adult hens for a total of $123.00. How many baby chicks did they sell that week?
(A) 74 (B) 85 (C) 136 (D) 210 (E) 223
Let x = number of baby chicks and 346 - x = number of adult chicks
.10x _ .75(346 - x) = 123
10x + 75(346 - x) = 12300
10x + 75(346) - 75x = 12300
- 65x = 12300 - 75(346)
1 2 3 0 0 7 5 ( 3 4 6 ) x = = 210 6 5