Algebra Questions

Find all of the real solutions to the following equation Write your answers in order from least to greatest 5x x 4x X X X

Use the given display to find the median Round to the nearest tenth if necessary Frequency Listen 20 10 5 4 N 21 7 201 10 20 30 40 50 Score

Solve the equation for all exact solutions where appropriate Round approximate answers in degrees to the nearest tenth Write answers using the least possible nonnegati csc 0 1 0 Choose the correct answer below O A 270 360 n 315 360 n where n is any integer O B 270 360 n where n is any integer OC 270 180 n where n is any integer OD 270 180 n 315 180 n where n is any integer

Use the given display to find the mean Round to the nearest tenth if necessary A 10 11 12 13 14 15 Score Frequency 13 5 4 12 5 12 6 12 4

Solve the equation for all exact solutions where appropriate Round approximate answers in degrees to the nearest tenth Write answers using the least poss 3 secx 2 3 0 Choose the correct answer below O A OB O C O D 2nx 2nt where n is any integer 6 76 11x 6 11x 6 6 11x n nt where n is any integer 6 nx where n is any integer 2nt where n is any integer www

K Solve the equation in radians for all exact solutions where appropriate Round approximate answers to four decimal places Write answers using the least possible nonnegative ang 2 sin 2x 3 sin x 1 0 Choose the correct answer below OA The solution set is OB The solution set is OC The solution set is OD The solution set is 3A 2 7 6 3x 2 2nx 2nx 2n 7T 75 6 2nx 11x 6 12x 5 12x 5 11x 6 2n 2nx 2nx 2n 2nx where n is any integer integer 2 3n 3 2 3x 2 F 3 2 2nx where n is any integer eger 2nx where n is any integer 2nx where n is any integer COLLE

The ages of 25 patients who suffered strokes are as follows 69 72 46 81 45 50 57 61 48 56 77 58 74 73 80 47 53 60 58 46 61 66 55 79 84 Fill in the blanks in the grouped frequency distribution for the data belov Classes Frequency 45 49 50 54 55 59 60 64 65 69 70 74 75 79 80 84

The city council of a small town needs to determine if the to support the building of a new library The council decides to conduct a survey of a sample of the town s residents Which one of the following procedures would be most appropriate for obtaining a sample of the town s residents Survey a random sample of persons within each neighborhood of the town Survey every 14th person who enters the old library on a given day Survey a random sample of librarians who live in the town Survey 400 individuals who are randomly selected from a list of all people liv

A record club has found that the marginal profit P x in cents is given by P x 0 0008x 0 35x 36 7x for x 400 where x is the number of members currently enrolled in the club Approximate the total profit when 240 members are enrolled by computing the sum 6 P x Ax with Ax 40 i 1 The total profit when 240 members are enrolled is approximately Round to the nearest cent as needed Marginal profit in cents 22 000 17 600 13 200 8 800 4 400 0 0 x 0 0008x 0 35x 36 7x IV V VI 100 200 Number of members 300 400 Q G

A researcher records the number of employees of each of the IT companies in th town of Westmoore The results are summarized in the table Number of Employees 0 749 750 1499 1500 2249 2250 2999 3000 3749 Find the class width 750 749 5 5 Number of IT Companies 30 24 6 5 5

Describe the population whose properties are analyzed by the given data During 2001 there were 574 crimes in a certain city per 100 000 residents criminals in the country residents of the country O residents of the city criminals in the city

Find the orders of all the nonzero elements of GF 3 x 2x 2

Listen Use the following frequency distribution to determine the class limits of the third class Class 9 11 12 14 15 17 18 20 21 23 Frequency 7 11 8 5 9 lower limit 14 upper limit 18 lower limit 15 upper limit 17 lower limit 14 upper limit 17 Olower limit 15 upper limit 18

Numbered disks are placed in a box and one disk is selected at random If there are 6 red disks numbered 1 through 6 and 2 yellow disks numbered 7 through 8 find the probability of selecting a red disk given that an odd numbered disk is selected O 3 4 01 8 300

Exercise 14 3 1 Compute a u v where u 1 b x y where x 10 c 2u v v using u and v above 1 the following inner products 2 2 T v 3 5 14 y 0 49 0 0

oblem 10 Which of the following sets are closed under addition i The set of all vectors in R of the form a b where b a ii The set of all 3 3 matrices that have the vector 1 3 2 as an eigenvector iii The set of all polynomials in P of the form ao a x a2x where ao A iii only B ii only C i and iii only D i only E i and ii only F ii and iii onl G all of them H none of them

Solve the triangle with the given parts a 23 9 m b 13 3 m c 21 6 m What is the degree measure of angle A 82 8 Round to the nearest tenth as needed What is the degree measure of angle B Round to the nearest tenth as needed

oblem 9 Which of the following sets are closed under scalar multiplication i The set of all vectors in R of the form a b where a 5b 0 ii The set of all 2 2 matrices whose trace is equal to 0 iii The set of all polynomials in P2 of the form ao a x a2x where the product ao a a2 0 Problem 9 A none of them B all of them C i and ii only D ii and iii only E iii only F 1 only G ii only H i and iii only Select v

oblem 11 Let v 1 2 1 and v 3 2 0 Which of the following vectors are in span V V i 2 4 1 ii 5 6 0 iii 7 10 2 Problem 11 A all of them B none of them C i and iii only D i only E ii and iii only F ii only G iii only H i and ii only Select v

oblem 7 Which of the following statements are always true for vectors in R i If u vxw 4 then w vxu 4 ii 5u v x u 6v 31 u V iii If u is orthogonal to v and w then u is also orthogonal to w v v u A all of them B iii only C i and iii only D i only E none of them F G ii and iii only H i and ii only only

1 Suppose you invest 5 000 in an account that carns 1 interest on the amount in the account month If you make no withdrawals or deposits to the account how much will you have in the

Exercise 13 5 5 Verify the following diagonalizations starting from the characteristic polynomial solving for the eigenspaces and then inverting the matrix of eigenvectors A 1 B 22 1 0 1 0 2 1 2 X X 0 1 0 0 1 3 1 1 1 2 1 C 2 4 2 3 6 3 1 YA Y ZACZ 1 1 3 2 1 2 5 3 673 6 96 0 0 00 1 1 0 1 1 1 3 2 3 1 3 2 1 2 1 3 1 1 2 0 1 1 0 2 00 0 10 0 02 200 000 000 12 5 3 9 8 0 3 3 6 3 6 2 2 3 5 2 0 3

oblem 8 Let V be the set of all ordered pairs of real numbers u with u 0 Consider the following addition and scalar multiplication operations on u u U and v v V u v u v 2 6u2v2 ku ku ku Use the above operations for the following parts Problem 8 a Problem 8 b Problem 8 c a Compute u v for u 6 2 and v 4 7 b If the set V satisfies Axiom 4 of a vector space the existence of a zero vector what would be the zero vector c If u 3 8 what would be the negative of the vector u referred to in Axiom 5 of a vector space Don t forget to use your answer to part b here enter your answer in the form a b Enter your answer symbolically as in these examples Enter your answer symbolically as in these examples enter your answer in the form a b enter your answer in the form a b

lem 5 Find the parametric equations of the line in R that passes through the points 5 5 2 and 1 4 9 B x 6 6t y 5 2t 2 11t C x 5 6t y 5 t 2 11t E x 5 6t y 5 A 6 6t y 5 t 2 12t F x 5 6t y 4 t 2 11t G x 6 6t y 5 t 2 12t H x 5 5t y 5 t 2 11t D x 5 6t y 5 t 3 11t 2 12t

se sin x 0 2938 cos x 0 9559 tan x 0 3074 sin x 0 2938 cos x 0 9559 tan x 0 3074 and

roblem 2 Let u v and w be vectors in R Determine which of the following statements are always true i If u 4 v 5 and u v 8 then u v 4 ii If u 2 and v 3 then u v 5 iii The expression vw u is both meaningful and defined A none of them B all of them C i and ii only D i only E i and iii only F ii only G iii only H ii and iii only

et u 1 2 1 and a 3 2 2 Find the vector component of u orthogonal to a Enter your answer symbolically wor in the formah

inner products T a u v where u 12 T v 5 b x y where x 10 14 3 y 0 49 00 c 2u v v using u and u above Compute the norms of u v x and y above simplify only where reasonable Compute the distance from u to v above Find the angle between u and u above calculator needed Draw two vectors in R2 which are not perpendicular Verify this by showing that a their dot product nonzero and b the Pythagorean for dot products does not hold for these vectors Prove that the standard dot product we re using actually satisfies all of the properties of an inner produc

Problem 1 Let P be the point 1 3 3 Suppose that the point Po 3 3 3 is of the way from P to Q Find the point Q Problem 1 Enter the point Q a b c as a b c

List the roots of the parabola y x 4x 29 In other words list the solutions of the equation 0 2 4x 29 Step 1 Identify the parameters A B and C A 1 B 4 C 29 Step 2 Compute the discriminant B 4AC the value of the discriminant is so we know that there will be no real solutions Step 3 Compute the other pieces of the quadratic formula What is B What is 24 Step 4 Finish computing the quadratic formula to get the root s B B 4AC 2A Note Your answers must be in exact form do not give their decimal values x B B2 4 2A

Problem 6 Find the area of the triangle with vertices 2 3 1 1 4 3 and 3 3 0 Enter your answer symbolically as in these examples 5 2

9 1 7 99x 119x 76x 96 11x 12 2 A 9x x 6 11x 12 3 11x 12 B 9x x 8 C 9x x 8 D 9x x 7 2 11x 12 9 21 154p 104 p 172p 2 14p 2 A 11p 9p 12 1 7p 4 5 B 11p 7p 10 C 11p 6p 8 D 11p 9p 11 2 14p 1 2 14p 1 2 14p 11 5x 24x 61x 30 5x 4 6 A x 4x 9 5x 4 8 B x 4x 9 C x 4x 9 D x 4x 7 5x 4 1 5x 4 5 5x 4 8 8 12 153y 48v 45 12 zoom 4 A v 12v 8 3 4v 3 2 B v 12v 4 3 4v 3 1 C v 12v 7 4v 3 D v 12v 5 10 12 10 224x 49x 90 14x 14x 7 A x 4x 14 8 14x 7 1 B x 4x 15 C x 4x 11 D x 4x 13 2x 1 4 14x 7 10 14x 7 12 16 92r 178r 116 8r 10 11 A 2r10r 12 8r 10 3 4r 5 7 B 2r 9r 11 C 2 10r 8 D 22 9r 13 8r 10 11 8r 10

1 9 9 21 154 p 104p 172p 2 14p 2 A 11p 9p 12 1 7p 5 B 11p 7p 10 C 1lp 6p 8 D 11p 9p 11 40 14p 2 14p C x 4x 9 1 D x 4x 7 11 5x 24x 61x 30 5x 4 6 A x 4x 9 5x 4 8 B x 4x 9 5x 4 1 2 14p 5x 4 5 5x 4 10 12 10 224x 49x 90 14x 14x 7 A x 4x 14 B x 4x 15 Cx 4x 11 1 D x 4x 13 8 14x 7 1 2x 1 4 14x 7 10 14x 7 12 16r 92r 178r 116 8r 10 A 2 10r 12 11 B 2 9r 11 C 2 10r 8 D 2r 9r 13 8r 10 3 4r 5 7 8r 10 11 8r 10

7 99x119x 76x 96 11x 12 8 8 12 153 48v 45 12 2 A 9x x 6 A 12v 8 2 11x 12 B 12v 4 B 9x x 8 3 C 9x x 8 D 9x x 7 11x 12 2 11x 12 4 9 21 154p 104 p2 172p 2 14p 2 A 11p 9p 12 1 7p 5 B 11p 7p 10 2 14p C 11p 6p 8 D 11p 9p 11 2 14p 1 2 14p 10 C 12v 7 D 12v 5 3 4v 3 2 3 4v 3 1 4v 3 10 224x 49x 90 14x 14x 1 A x 4x 14 7 8 14x 7 B x 4x 15 7 2x 1 4 14x 7 10 14x 7 C x 4x 11 D x 4x 13 ZOC

9 21 154p 104 p 172p 2 14p 2 A 11p 9p 12 1 7p 5 B 11p 7p 10 2 14p C 11p 6p 8 D 11p 9p 11 2 14p 1 2 14p 11 5x 24x2 61x 30 5x 4 A x 4x 9 B x 4x 9 C x 4x 9 D x 4x 7 6 5x 4 8 5x 4 1 5x 4 5 5x 4 12 10 10 224x 49x 90 14x 14x 7 A x 4x 14 B x 4x 15 C x 4x 11 D x 4x 13 4 8 14x 7 1 2x 1 4 14x 7 10 14x 7 12 16r 92r 178r 116 8r 10 11 A 2 10r 12 8r 10 3 B 2r 9r 11 2 C 2 10r 8 8r 10 D 2r 9r 13 11 4r 5 7 8r 10

7 99x 119x 76x 96 11x 12 2 A 9x x 6 11x 12 B 9x x 8 3 C 9x x 8 D 9x x 7 11x 12 2 11x 12 9 21 154p 104p 172p 2 14p 2 A 11p 9p 12 1 7p 5 B 11p 7p 10 C 11p 6p 8 D 11p 9p 11 2 14p 1 2 14p 1 2 14p 8 10 8 12 153 48v 45 12v 9 4 A 12v 8 3 4v 3 2 B 12v 4 3 4v 3 C v 12v 7 D v 12v 5 4 4v 3 10 224x 49x 90 14x 14x 7 8 A x 4x 14 14x 7 B x 4x 15 2x 1 4 C x 4x 11 D x 4x 13 14x 7 10 14x 7

4 Evaluate x 4 x 2 for x 5

3 Evaluate 3x 3 for x 2

2 Which of the following is equivalent to 3x 2x 4 a x b 5x 4 c 6x 4 d 1

7 Determine the interest rate necessary for 1500 to amount to 2075 in 6 years with interest compounded semi annually

5 Evaluate 3 x 4 2 4x 2 for the value x 10

4 Evaluate the expression 2 3x 20 10 for the value x 4

3 Simplify the expression 2 4x 6y 3 2x 5y by distributing and combining like

2 Simplify the expression 2 x 4 10 y 10 by distributing and combining like terms

Homework 1 Simplify the expression 2 3x 3 3 x 10 by distributing and combining like terms

Independent Practice 1 Simplify the expression 2 3x 2 3x 8

3 b 3b 32b 28 b 7 B b 46 1 C b 4b 4 D b 4b 4 1 b 7 5 n 4n 128n 143 n 13 A n 9n 11 B n 9n 8 C n 9n 10 D n 9n 8 1 n 13 3 n 13 n 13 C 4 4 6 4 K 13k 33k 30 k 10 1 k 10 A K 3k 4 B k 3k 3 C K 3K S D k 3k 6 k 10 6 k k 130k 24 k 12 A K 11k 2 B k 11k 5 k 12 C k 11k 4 5 k 12 D K 11k 4

e 13 5 1 Diagonalize 1 0

1 3 W Divide 1 a 11a 13a 27 a 10 6 A a a 3 a 10 7 B a a 3 a 10 5 a 10 C a a 3 a 10 D a a 3 3 b 36 32b 28 b 7 2 b 7 A b 4b 3 B b 4b 1 C b 4b 4 D b 4b 4 4 b 7 5 n 4n 128n 143 n 13 AL O 11 c 4 6 2 2 x 14x 56x 59 x 8 A x 6x 9 6 9 1 8 B x 6x 9 C x 6x 6 D x 6x 8 A k 3k 4 B k 3k 3 C k 3k 5 4 k 13k 33k 30 k 10 1 k 10 8 x 8 x 8 5 x 8 D k 3k 6 5 k 10 6 k k 130k 24 k 12 A 22 116 2

1 3 Long Division Divide 1 a 11a 13a 27 a 10 6 A a a 3 a 10 7 B a a 3 C a a 5 a 10 a 10 Duket D a a 3 3 a 10 3 6 3b 32b 28 b 7 A b 4b 3 7 B b 4b 1 C b 4b 4 D b 4b 4 1 BROM 2 b 7 4 b 7 4 4 2 2 x 14x 56x 59 x 8 6 A x 6x 9 x 8 8 B x 6x 9 C x 6x 6 D x 6x 8 A K 3k 4 B k 3k 3 C k 3k 5 X 8 1 4 k 13k 33k 30 k 10 1 k 10 x 8 5 X 8 5 k 10 D k 3k 6