ALGEBRA I January 24, 2007
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- Cameron Golden
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1 ALGEBRA I January 4, When a number is added to twice the sum of the number and 8, the result is 4 more than the number. Find the number.. Tickets to the Super Bowl were selling for $35. After the Chicago Bears won the NFC Championship game, ticket prices skyrocketed to $3055. What was the percent of increase? 5 3. A quadratic equation in the form ax bx c 0 has roots of 7 a, b and c where a, b, and c are integers with no common factors? and 3. What is the sum of 4. Find the ordered pair ab, that makes the following true: 5y x 3y 4 x y 3x ax by 5. Solve for x: 4 x 3 6. If x y, x y z 1 and 7y z 6, what is the value of x? 7. Write the equation of a line that passes through,5 and is perpendicular to the line that contains the points 3,7 and 5,6. Write your answer in slope-intercept form. 8. Solve for x: x 7 x 3 9. y Evaluate: x 3xz 9yx z y for x 5, y 1, and z 3. Write your answer as a simplified single fraction % of 10 more than a number is 1 more than 5% of 4 more than the number. What is the number?
2 WRITTEN COMPETITION ANSWER SHEET COMPETITION
3 GEOMETRY January 4, Two angles of a quadrilateral have angles with measures of 46 and 10. The remaining two angles are congruent. If an angle is selected at random, what is the probability that it is an obtuse angle? Give your answer as a fraction in lowest terms.. The two opposite sides of a square are increased by 10 and the other two sides are decrease by 8, forming a rectangle with an area of 63 square units. What is the area of the original square? 3. The ratio of the complement and angle to the supplement of the angle is 1:3. Find the degree measure of the angle. 4. Rectangle ABCD is inscribed in circle O, which has a radius of 5. If BC has length 8, what is the area of rectangle ABCD? B C O A D 5. An isosceles right triangle has a leg of 4. A circle is inscribed in the triangle. What is the radius of the circle? 6. Triangles ABC and DEF are similar right triangles, with right angles C and F. If m B = 30 and the shorter leg in triangle ABC is the same length as the longer leg in triangle DEF, what is the ratio of the area of ABC to the area of DEF? Write your answer in the form a:b where a and b are integers with no common factors. 7. The legs of a right triangle are x and 3 x 1 and the hypotenuse has length 4x 3. Find the numeric length of the hypotenuse.
4 8. The leg of an isosceles triangle is contained in the line with equation x y. If the base of the triangle is along the x-axis, and the vertex is the point 10, 8, what is the perimeter of the triangle? 9. If the measure of THM 46, how many degrees are in the measure of TAM? T A H M 10. Given O, AE BC, and m EC 80, find the number of degrees in m BFC. A B O F G C E D
5 WRITTEN COMPETITION ANSWER SHEET COMPETITION
6 ALGEBRA II January 4, If 4 x y 8 x 5y, find the ratio of x to y where x and y are positive integers with no common factors.. Solve for x: log6 13x x 3. Find all values of k for which the system k x 9y 5 4x k y 11 has no solution. 4. What is the largest integer a that makes the inequality 3a a 6 true? 5. What is the value of the sum x y for the following: x y 4 3 x y 4 3? 6. Solve for x: 4x 9 13x 4 7. Find the product xy, given that x y 6 and x y What is the maximum area of the given rectangle? 5 x x 3 9. Solve for x: x 7 x A line passes through the first, third and fourth quadrants and makes an angle of 30 with the x-axis. What is the slope of the line?
7 WRITTEN COMPETITION ANSWER SHEET COMPETITION
8 PRE-CALCULUS January 4, Simplify the expression: 3/ 3/9 1/3 3/ Solve for x: 1 x 8x Simplify to an expression of the form log a b with a and b are the smallest possible values: log 5 3 log 5 4. Solve for, : 8cos sin 1 4sin 6sin 5. Find the sum of all integral values 10 x 10 for which x x x 3 x If 3x is a factor of fraction. 9x 11x k, find the value of k. Give your answer as a single simplified 7. Given function f where f 4 6, f 6 and f 4, what is 1 1 f f f 1? 8. Solve for x: 0 x : cot x cot x Find the point that is 3 5 of the way from A to B where A 4, 3 and B 19, 10. What is the sum ?
9 WRITTEN COMPETITION ANSWER SHEET COMPETITION
10 FRESHMAN SOPHOMORE 8 PERSON TEAM January 4, Given f x x 1 and g x 3x 5. If f k g k 1 14, what is the value of k?. Points A, B, C and D are collinear and are located in that order (A, B, C, D) along the line. AB 6, AC = x 3, BC x 1 and CD x 4. What is the length of AD? 3. Define a * b a ab b. What is the value of 5* 3*6? 4. If a 3b x and 4a b y and k x m y 6a 4b, what is the ordered pair km,? 5. What is the order quadruple a, b, c, d that will make the following matrix equation true? 3 c 4 1 a 11 5 d 3 b 3 6. Two concentric circles have radii of 4 and 8. What is the length of a chord of the larger circle that is tangent to the smaller circle? 7. Solve for the base x for which the following is true: x Three numbers have a mode of 3. The largest number is 5 times the smallest number. Find all possible values for the mean of the three numbers. 9. What is the perimeter of the shaded region, where each arc is half of a circle?
11 10. The equation x 5 c has solutions of 1 and 4. What is the value of c? 11. Polygon ALGEBRY is similar to polygon GEOMTRY. The area of ALGEBRY is 7 and the area of GEOMTRY is 16. If side LG in ALGEBRY has a length of 8, what is the length of side EO in GEOMTRY? 1. What is the product of 0.7 and 4.8? Give your answer in fraction form. 13. What is the area of the circle x y 4x y 11 0? 14. Given BF CE and CD ED. If m EDC = 78, find the degree measure of EFG. G F E B C D A 15. In triangle ABC, AB = 10 and BC = 17. What is the length of the altitude from B to AC? B A 3, 0 C 18, 0
12 A B 3x Find the ordered pair A, B so that x 1 x 1 x Line l is perpendicular to the line m where the equation of m is x 5y 1. The x-intercept of l is twice the x-intercept of line m. What is the y-intercept of line l? 18. Four consecutive integers have a sum of 78. What is the largest of these four integers? 19. In Q, HW = 7, TW = x, MW = x +, and AW = y + 5. If MT HA, find the smallest possible value for x. M x+ Q H 7 x W y+5 A T 0. An integer is selected at random from the possible values for A, where 5 A 10. What is the probability that the selected integer is greater than the sum of the x and y coordinates of the point of intersection of the lines x y 7 and 4x y 11?
13 TEAM COMPETITION ANSWER SHEET COMPETITION
14 JUNIOR SENIOR 8 PERSON TEAM January 4, Solve for x, 0 x : 8sin xcos x. What is the average of the first one million odd numbers? 3. If a line is selected at random from the lines y, x 3, x 1, y 0 and x 3, what is the x probability that is an asymptote of the graph of f x? x log 3 log log x 4. Solve for x: Find the value of x if x 1, 3x 3, 4x 4 is a geometric sequence. 6. The quadratic equation What is the value of c? x 11x c 0 has roots 1 r and r where r is twice the sum of r 1 and Solve for x: x 1 3 x 4 x An ellipse has a center at 5,, a vertex at 5, 8 and a focus at 5, 6. What is the length of the minor axis? 9. What is the equation of the line of symmetry for the graph x 3y 1? 10. Find the value of sin tan In an arithmetic sequence, the nd term is 5, the 7 th term is 5 times the first term and the 10 th term is 3 times the 3 rd term. What is the 5 th term?
15 1. Hyperbolic Sine (sinh) and hyperbolic Cosine (cosh) defined as follows: x x x x e e e e y sinh x and y cosh x What is the value of cosh x sinh x? 13. When the denominator of the expression completely simplified to an expression of the form 4 3 is rationalized and the resulting fraction is a b c, find the ordered triple abc.,, 14. Find the distance between the vertices of the parabolas y 4x 16x 3 and y x 8x cot x tan x tan kx. What is the value of k? 16. What is the area of the region enclosed by the graph x 3 y 1? 17. Define x 1 3 x 1 f x x 1 x 4. Let k be the smallest integer in the range of f x and x 8 4 x 9 let m be the largest integer in the range of f x. What is the sum k m? 18. What are the rectangular x, y coordinates of the point described by the polar coordinates 6;10? 19. The smallest root of the equation 3 x x 8x 4 0 is the largest root of a quadratic equation whose roots have a product of 6. Write the quadratic equation in the form ax bx c 0 0. What is the common ratio in a geometric sequence where the 5 th term is log3 43 and the 6 th term is log8 9?
16 TEAM COMPETITION ANSWER SHEET COMPETITION
17 CALCULATOR January 4, If the angle of elevation from the end of a tree s shadow to the top of the tree is 48, and if the length of the shadow is 100 feet, how many feet tall is the tree?. If x.007, evaluate the expression x 5 3 x 1 4 x 3. A student has scores of 80, 8, 81, 85, 84 and 86 on her 6 unit tests. The first 4 tests each count for 10% of her semester grade, and the last tests each count for 15% of her semester grade. The final exam counts for the other 30% of the semester grade. What is the lowest final exam grade (rounded to 4 significant digits) that the student can score and have a semester grade of at least 83%? 4. The point 5,11 is on the graph of f x 3 a x. What is the value of f 16? 5. A triangle has sides of 3.8, 7.9 and 8.3. What is the degree measure of the smallest angle? 6. You are preparing for your Super Bowl party and can t decide what to buy. Three cases of pop and 5 bags of chips cost $3.8. Four cases of pop and 3 bags of chips cost $ How much would cases of pop and 4 bags of chips cost? 7. A small circle has area k. A larger circle has a radius that is 3.1 units larger than the small circle and an area of What is the value of k? people were surveyed to determine their blood type. 13 were type A, 14 were type B, 17 were type AB and 16 were type O. If three people are selected at random from this sample, what is the probability that they are all of type O? 9. Given isosceles triangle ABC with vertex angle A, find the area of the triangle if m A 30 and the altitude to BC has length Of 70 applicants for job openings at a company, 38 were women. If 4 applicants were hired, what is the probability that all 4 are of the same gender? 11. Solve for x if x Write your answer in scientific notation.
18 1. A new element is discovered with a half-life of 007 years. How many grams of a 00 gram sample of this new element will be left 100 years from now? 13. Find the number of square units in the area of quadrilateral ABCD. 41 A 15 D 1 C f x is a quartic function with f 13, f , f 3 53, f and f find the value of f Find the smallest possible real number x so that sin 14 cos3 tan x 16. In triangle ABC, AB = 4, BC = 9 and m BAC 73. What is the area of triangle ABC? 17. Solve for x: 5 8 x 1 3x 18. Given the diagram as shown, find the length, x, of the angle bisector x 19. A triangle with sides of 8 and 9 is inscribed in a circle with diameter 16 as shown. What is the length of the third side of the triangle? 0. If an n 4n 8, what is the value of 30 n 1 a? n
19 TEAM COMPETITION ANSWER SHEET COMPETITION
20 FRESHMAN SOPHOMORE PERSON TEAM January 4, The lines ax 3y 11, 4x by 14 and 3x y c all pass through the point, 3. What is the ordered triple abc?,,. The ratio of the angles of triangle 1 is 1::6. The ratio of the angles of triangle is 5:9:. What is the positive number of degrees in the difference between the measure of the largest angle in triangle 1 and the measure of the largest angle in triangle? 3. k 50, 3 m 75 and 6 p 4. What is the product kmp? 4. Angles A and B are complementary. m A m B 16 Angles C and D are supplementary. m C m D 3 Find the ordered quadruple m A, m B, m C, m D 5. The equation of the perpendicular bisector of AB, where A, 6 and B 8, 6 is y ax b. The equation of the perpendicular bisector of CD, where C 1, 4 and D 7, is y cx d What is the sum a c?
21 Person Team School Names Level: Fr/So Jr/Sr Total Score (see below*) = Team # Answer Score (to be filled in by proctor) TOTAL SCORE: (*enter in box above) * Scoring rules: Correct in 1 st minute 6 points Correct in nd minute 4 points Correct in 3 rd minute 3 points PLUS: point bonus for being first In round with correct answer
22 JUNIOR - SENIOR PERSON TEAM January 4, A is the amplitude of the graph of y sin 3x B is the period of the graph of y 4 cos What is the sum A B? x. Let k = the sum of the roots of Let l = the product of the roots of Let m = the sum of the roots of Let n = the product of the roots of x 5x x x 0 x 7x 6 0 3x 10x 8 0 What is the sum k l m n? Give your answer as a reduced fraction 3. How many integral solutions are there to the inequality 5 3x 1 0? 4. 3i 4 i a bi 5 i 3 4i c di What is the sum of the complex numbers a bi c di? 5. The point, 3 is reflected over the line y to the point A. The point 3, 5 is reflected over the line x 1 to the point B. What is the length of AB? Give your answer rounded to 4 significant digits.
23 Person Team School Names Level: Fr/So Jr/Sr Total Score (see below*) = Team # Answer Score (to be filled in by proctor) TOTAL SCORE: (*enter in box above) * Scoring rules: Correct in 1 st minute 6 points Correct in nd minute 4 points Correct in 3 rd minute 3 points PLUS: point bonus for being first In round with correct answer
24 FRESHMAN SOPHOMORE RELAY January 4, 007 ROUND 1 1. The ratio x: y is equal to 14.86: If x 4, what is the value of y?. For how many integer values is the inequality x 4 ANS? 3. Solve for x given l m (8 ANS) l m (9 ANS + x) 4. Solve for x given that m AC (5 ANS 9) C (x + 4) A x B
25 RELAY TEAM ANSWER SHEET SCHOOL: LEVEL: FR/SO JR/SR TEAM # ROUND # ANSWER: SCORE: TOTAL ANSWER POINTS: ( points per problem) BONUS POINTS: (+5 all correct at 3 minutes +3 all correct at 6 minutes) TOTAL SCORE FOR ROUND:
26 FRESHMAN SOPHOMORE RELAY January 4, 007 ROUND 1. Solve for x: x 3 x 4 x x 3. If the points 4,, ANS, 5 and x,3 are collinear, find the value of x. 3. If EF is a midline of the trapezoid shown, find its length A ANS + 9 B E F D 30 - ANS C 4. Find the length of AD, given that AB 5 and AC ANS C A D B
27 RELAY TEAM ANSWER SHEET SCHOOL: LEVEL: FR/SO JR/SR TEAM # ROUND # ANSWER: SCORE: TOTAL ANSWER POINTS: ( points per problem) BONUS POINTS: (+5 all correct at 3 minutes +3 all correct at 6 minutes) TOTAL SCORE FOR ROUND:
28 FRESHMAN SOPHOMORE RELAY January 4, 007 ROUND 3 1. What is the smallest integer a that makes the following true: 16a 7 10a 6 4a 8. Find the y-intercept of the line that is parallel to the line 10x 5y 4 and passes through the point ANS,1. Write your answer as an ordered pair x, y. 3. Line l is perpendicular to the line through the points 5, and 4, 8. Line l also contains the midpoint of AB where point A is ANS and point B is, 1. What is the x-intercept of line l? 4. In circle D with diameter AC, find the length of AB given that mbc ANS +13 circle D is 36. and the area of A B D C
29 RELAY TEAM ANSWER SHEET SCHOOL: LEVEL: FR/SO JR/SR TEAM # ROUND # ANSWER: SCORE: TOTAL ANSWER POINTS: ( points per problem) BONUS POINTS: (+5 all correct at 3 minutes +3 all correct at 6 minutes) TOTAL SCORE FOR ROUND:
30 JUNIOR - SENIOR RELAY January 4, 007 ROUND 1 1. What is the largest solution to the equation 15x x 6? Give your answer in simplest fractional form.. Simplify: ANS ANS 3. Solve for x: log3ans log3 x log8. Write your answer as a fraction reduced to lowest terms. 4. ANS is a fraction of the form a b. Let k a b 1. What value of x gives you the minimum value for the graph of y 3x k x 15?
31 RELAY TEAM ANSWER SHEET SCHOOL: LEVEL: FR/SO JR/SR TEAM # ROUND # ANSWER: SCORE: TOTAL ANSWER POINTS: ( points per problem) BONUS POINTS: (+5 all correct at 3 minutes +3 all correct at 6 minutes) TOTAL SCORE FOR ROUND:
32 JUNIOR - SENIOR RELAY January 4, 007 ROUND 1. What is the point of intersection of the lines y x 3 and 4x 3y 1?. The points 4,, 7, k and ANS are collinear. What is the value of k? 3. In an arithmetic sequence t 4 13 and t 8 9. What is t ANS? 5 4. Given sin ANS and 5 cos 13 What is the value of sin? where and are both acute angles.
33 RELAY TEAM ANSWER SHEET SCHOOL: LEVEL: FR/SO JR/SR TEAM # ROUND # ANSWER: SCORE: TOTAL ANSWER POINTS: ( points per problem) BONUS POINTS: (+5 all correct at 3 minutes +3 all correct at 6 minutes) TOTAL SCORE FOR ROUND:
34 JUNIOR - SENIOR RELAY January 4, 007 ROUND 3 1. Solve for x: 13 3x x 3. Simplify the expression log9 3 log ANS log 5 0. log 8 3. If 5x 3 f x, find the value of f 1 ANS 4. x varies directly as the square of y. If x 10 when y 4, and if y represents a positive value, find the value of y when x ANS
35 RELAY TEAM ANSWER SHEET SCHOOL: LEVEL: FR/SO JR/SR TEAM # ROUND # ANSWER: SCORE: TOTAL ANSWER POINTS: ( points per problem) BONUS POINTS: (+5 all correct at 3 minutes +3 all correct at 6 minutes) TOTAL SCORE FOR ROUND:
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