Grissom High School Math Tournament 2007 Algebra II / Comprehensive Ciphering Algebra II questions will be on top Comprehensive questions will be on bottom Algebra II please sit on the left side of the auditorium, facing the stage Comprehensive please sit on the right side of the auditorium, facing the stage

Ciphering Rules Scoring: First Time Frame (30 sec) 5 Points Second Time Frame (30 sec) 3 Points Third Time Frame (30 sec) 1 Point ** There will be a sliding scale on scoring. A five second warning will be given before the end of each time frame. Checkers will not take your answer sheets until they are folded and raised in the air. Place your answers in the answer box provided on the ciphering sheet or your answer will not receive any points. Answers must be given in exact, reduced, and simplified form. Simplification includes, but is not limited to, the following: All fractions should be written in lowest terms with denominators rationalized. When appropriate, leave answers as simplified improper fractions. Leave answers in terms of ,e, or i.

Sample: Find the coordinates of the y-intercept: Algebra II Ciphering Sample: Find the coordinates of the y-intercept: Comprehensive Ciphering Sample: Find the focal width of the parabola: y = 2(x -1)2 + 3

Question 1

Comprehensive Ciphering Algebra II Ciphering 1. Evaluate: Comprehensive Ciphering 1. Given that a>0, write the following in simplest radical form:

Question 2

Comprehensive Ciphering Algebra II Ciphering 2. Find the product of the real roots of the equation: x2(x – 5) – 24 = 6(x – 4). Comprehensive Ciphering 2. Evaluate the following determinant:

Question 3

3. Find the maximum value of the function f(x) = -3x2 – 42x + 11. Algebra II Ciphering 3. Find the maximum value of the function f(x) = -3x2 – 42x + 11. Comprehensive Ciphering 3. Find the maximum integral value of x in the domain of the function:

Question 4

4. If [x] = the greatest integer < x, then evaluate: Algebra II Ciphering 4. If [x] = the greatest integer < x, then evaluate: Comprehensive Ciphering 4. Find the sum of the values of x on the interval such that

Question 5

5. How many multiples of 7 lie between 1007 and 2007? Algebra II Ciphering 5. How many multiples of 7 lie between 1007 and 2007? Comprehensive Ciphering 5. Describe the shape of the curve described by the given set of parametric equations:

Round 2 Please send up your next team member. Algebra II Ciphering Round 2 Please send up your next team member. Comprehensive Ciphering Round 2 Please send up your next team member.

6. Solve the following for x: Algebra II Ciphering 6. Solve the following for x: Comprehensive Ciphering 6. Find the sum of the first four terms of the geometric sequence with second term = 2007 and common ratio = .

Question 7

Comprehensive Ciphering Algebra II Ciphering 7. Given , and , find Comprehensive Ciphering 7. Solve the following for x:

Question 8

8. Find the larger value of t that is a solution to the equation: . Algebra II Ciphering 8. Find the larger value of t that is a solution to the equation: . Comprehensive Ciphering 8. Find the length of the minor axis of 7x2 + 16y2 – 14x – 105 = 0.

Question 9

Comprehensive Ciphering Algebra II Ciphering 9. Solve the following over the complex numbers: 2x3 – 3x2 + 18x – 27 = 0 Comprehensive Ciphering 9. Lumpy is wandering along the graph of and reaches a hole. Find the coordinates of the hole he needs to patch to make his path continuous.

Question 10

Comprehensive Ciphering Algebra II Ciphering 10. If A = and B = , find the element in the second row, third column of AB. Comprehensive Ciphering 10. How many integral values of x are solutions to the inequality: ?

Round 3 Please send up your next team member. Algebra II Ciphering Round 3 Please send up your next team member. Comprehensive Ciphering Round 3 Please send up your next team member.

11. Find the slope of the line perpendicular to the line 4x = 5y -17. Algebra II Ciphering 11. Find the slope of the line perpendicular to the line 4x = 5y -17. Comprehensive Ciphering 11. Write in a + bi form: (1 – i)7.

Question 12

12. Solve the following for x: Algebra II Ciphering 12. Solve the following for x: Comprehensive Ciphering 12. If , and cos x > 0, find the value of tan 2x.

Question 13

Comprehensive Ciphering Algebra II Ciphering 13. If , find . Comprehensive Ciphering 13. James and John are playing catch. James is standing at the point with polar coordinates (4, 220°) while John is at the point (8, 280°). How far apart are they?

Question 14

Comprehensive Ciphering Algebra II Ciphering 14. If A, B, and C are single digit numbers and 29A86BC is divisible by 720, find 2A + B – C. Comprehensive Ciphering 14. Brownie the puppy has a new carpet for his home, and its shape is described by . Find the area of his new carpet.

Question 15

Comprehensive Ciphering Algebra II Ciphering 15. If , and , find . Comprehensive Ciphering 15. How many pairs of positive integers (x, y) are solutions to the equation: 9x + 15y = 2007?

Round 4 Please send up your next team member. Algebra II Ciphering Round 4 Please send up your next team member. Comprehensive Ciphering Round 4 Please send up your next team member.

Comprehensive Ciphering Algebra II Ciphering 16. Simplify: Comprehensive Ciphering 16. Find the value of:

Question 17

Comprehensive Ciphering Algebra II Ciphering 17. Tom defined a new function a☼b = . Find the smaller value of a for which a☼2 =a. Comprehensive Ciphering 17. Find the slope of the line tangent to the circle with equation x2 + y2 = 4 at the point .

Question 18

Comprehensive Ciphering Algebra II Ciphering 18. If three cards are drawn at random without replacement, what is the probability they are all the same suit? Comprehensive Ciphering 18. If the graph of f(x) = |x3 – 1| is increasing on the interval (a, b), find the value of a.

Question 19

Comprehensive Ciphering Algebra II Ciphering 19. If the solution to the system of equations given is (a, b), find the value of 3ab. Comprehensive Ciphering 19. Find the sum of the real solutions to the equation:

Question 20

Comprehensive Ciphering Algebra II Ciphering 20. Write the range of the graph of y = |x + 1| + | x – 2| in interval form. Comprehensive Ciphering 20. Given three points in space G (2, 3, 7), H (1, -2, a), and S (-3, -1, 5), find the value(s) of a that make .

This concludes ciphering Algebra II Ciphering Comprehensive Ciphering This concludes ciphering

X1. Solve the following for x: Algebra II Ciphering X1. Solve the following for x: Comprehensive Ciphering X1. Find the smallest positive value of θ such that

Extra Question 2

Comprehensive Ciphering Algebra II Ciphering X2. Find the probability of rolling at least one odd number in two throws of a standard six-sided die. Comprehensive Ciphering X2. If 1+i is one of the solutions to the equation x3 – 6x2 + 14x – 24 = 0, find the value of the real solution to the equation.

Comprehensive Ciphering Algebra II Ciphering X3. Solve for x: Comprehensive Ciphering X3. Write the following in terms of a single trig function: (sin θ – 1)(tan θ + sec θ).