Pensacola Invitational 2014: April 12, 2014 Algebra 1 Individual Test

You have 50 minutes to complete this test. Select the best answer to each question. The choice e) NOTA" denotes "None of the Above."

Remember, these questions are meant to be challenging even for top students. Good luck!

1) Evaluate (2+ 8÷ 4!)×2− 1

a) 4 b) 7 c) !!

d) !!

e) NOTA

2) The operation Δ is defined such that aΔb = a+ b!. What is 16∆4?

a) 16 b) 20 c) 64 d) 258 e) NOTA

3) Shayli is a good note taker. So far this school year, she has taken 326 pages of notes. If she wants to number all the pages in her notebook, how many digits will she need to write in total?

a) 326 b) 868 c) 870 d) 872 e) NOTA

4) What is the product of the solutions of the quadratic equation 2x! − 5x+ 3 = 0

a) 1 b) !!

c) !!

d) 3 e) NOTA

5) The sum of three consecutive integers is 78. What is the value of the smallest integer?

a) 26 b) 27 c) 28 d) 29 e) NOTA

6) If (a, b) is a solution to the following system of equations, what is a+ b?

3y− 2x = 11 −2x− 3y = −7

a) -1 b) 0 c) 1 d) 2 e) NOTA

7) If (2,−5) is a solution to the following system of equations, what is a! − b!?

ax− y = 9 x+ by = −18

a) 12 b) -12 c) 2 d) 6 e) NOTA

Pensacola Invitational 2014: April 12, 2014 Algebra 1 Individual Test

8) What is the sum of the solutions to the equation 3x− 2 = 7 ?

a) − 143 b) − !!

c) !!

d) 3 e) NOTA

9) In two years, Chris will be twice Anish’s age. Four years ago, Chris was two years younger than three times Anish’s age. What is the sum of Chris’s age and Anish’s age four years after today?

a) 30 b) 38 c) 46 d) 48 e) NOTA

10) The polynomial 6x! + x− 2 can be factored into the form ax+ b cx+ d , where a, b, c, and d are integers. What is the arithmetic mean of a, b, c, and d?

a) !!

b) !!

c) 2 d)  3 e) NOTA

11) If xy = 16, yz = 2, and xz = 8, and x, y, and z, are all greater than 0, what is the value of xyz?

a) 4 b) 16 c) 32 d) 256 e) NOTA

12) A function, f x , is a line where f −4 = 5 and f 2 = 2. What is the distance between the x-intercept and the y-intercept of this function?

a) 4 b) 5 c) 9 5 d) 5 3 e) NOTA

13) If x < y, and x ≠ y ≠ 0, how many of the following statements must be true?

I)  x+ 3 < y+ 3 II)  x! < y! III)  !!< !

!

IV)  x! < y! V)   x+ y x− y < 0

a) 2 b) 3 c) 4 d) 5 e) NOTA

14) Which of the following lines is perpendicular to the line y = 3 x− 3 + 6

a) 2x+ 6y = 7 b) 3x− y = 4 c) x− 3y = 9 d) 4x+ 3y = 5 e) NOTA

Pensacola Invitational 2014: April 12, 2014 Algebra 1 Individual Test

15) Krisha’s school is 40 miles away. She drove at a rate of 40 mph (miles per hour) for the first half of the distance, then 60 mph for the remainder of the distance. What was her average speed for the entire trip?

a) 40 b) 48 c) 50 d) 52 e) NOTA

16) If a parabola is in the form f x = ax! + bx+ c, and it passes through the points 0,−2 , 1, 3 , and   −1,−1 , what is the product of a, b, and c?

a) 2 b) 6 c) 12 d) -12 e) NOTA

17) If x − !!

!= !

! , which of the following expresses v in terms of k,w, x, and  y?

(Hint: Isolate v)

a) kxwy! b) !

!"!!! c)

!"!"!!!

d) !"!!!!

e) NOTA

18) If x + !!= 3, what is x! + !

!!?

a) 9 b) 47 c) 49 d) 81 e) NOTA

19). Robert has two stock mixtures: Mixture A is 100% NaCl and Mixture B is 50% NaCl. What is the ratio of the volume of Mixture A to the volume of Mixture B if he wants to make 100 mL of a 90% solution of NaCl?

a) 2 b) 3 c) 4 d) 5 e) NOTA

20) A circle circumscribes a square. What is the ratio of the area of the circle to the area of the square?

a) !!

b) 2π c) !!

d) 2 e) NOTA

21) If a+ b = 8 and a! + b! = 60, what is a!b!?

a) 2 b) 4 c) 16 d) 64 e) NOTA

Pensacola Invitational 2014: April 12, 2014 Algebra 1 Individual Test

22) Kimberly is not very good at Spanish, but thankfully she is good at math. After four Spanish tests, Kimberly has a failing test average of 57. What is the average test score she needs to make on her next 6 tests to boost her overall score average of all 10 tests to a passing grade of 72?

a) 74 b) 78 c) 82 d) 84 e) NOTA

23) The operation ! is defined such that n! = n× n− 1 × n− 2 … 2×1. For example, 5! = 5×4×3×2×1 = 120. How many consecutive zeroes are there at the end of 15!

a) 1 b) 2 c) 3 d) 4 e) NOTA

24) Your friend Luke poses you a difficult question: What is an expression for the sum of the roots of quadratic ax! + bx+ c? (Hint: use the quadratic formula!)

a) 0 b) !!

c)  − !!"

d) − !!

e) NOTA

25) There are 25 questions on this test labeled {1, 2, 3, 4, 5, ...}. How many of these numbers are prime?

a) 5 b) 6 c) 7 d) 8 e) NOTA