Algebra Questions

17 Let W be the set of all vectors in R of the form a 1 0 Which of the following statemen is true i W is not closed under additon and not closed under scalar multiplication ii W is closed under addition but not closed under scalar multiplication iii W is closed under scalar multiplication but not closed under addition iv W is a subspace a ii b iii c iv d i e none of them

10 Let V be the subset of R consisting of all vectors of the form a b c d where d a b c Find a basis for V a 1 0 0 0 0 1 0 0 0 0 1 0 b 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 c 1 1 1 3 d 1 1 1 3 1 0 1 2 e 1 0 0 1 0 1 0 1 0 0 1 1

7 The displacement from equilibrium of an oscillating weight suspended by a spring and subject to the damping effect of friction is given by All changes saved y 2e cos 6t 1 where y is the displacement in centimeters and t is the time in seconds see figure Find the displacement when t 4 Round your answer two decimal places

4 Suppose that V1 V2 V3 is a linearly independent set of vectors Which of the foll sets are also linearly independent i V V1 V2 V V V3 ii V1 2v2 2V3 iii V1 V2 V2 V3 V1 V3 a i and ii only b all of them c ii only d i only e iii only

6 Which of the following sets of polynomials form a basis for P 1 1 2 3x ii iii 1 1 a i and ii only b i only c none of them d all of them e ii and iii only 1 2 1 1 1

Decem 1 Let V be a vector space with dimension n 2 Which of the following statements are true 1 Every set of n vectors in V is a basis for V ii Every set with less than n vectors must be linearly independent iii Every set with more than n vectors must span V a all of them b none of them c ii and iii only d i and iii only e i and ii only

5 Let V be the set of all ordered pairs of real numbers a y where a and y are both nonzero Consider the following addition and scalar multiplication operations on u and v 01 02 u v 1 v ku u k u k Although V is not a vector space each vector in V does have a negative as defined in the axioms of a vector space If u is in V what is the negative of the vector u a u b u 1 u 1 c d u 1 2 1 e u 1 1

2 Find the Hill cipher of the word AT using the following enciphering matrix 3 A B C D E F G H I J K L M N O P Q R S T U V W X Y 2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 0 a OP b BC c BR d OX e ZP

Find a basis for the null space of A 80A D 0 48840 e a d 1 2 4 5 A 0 1 3 2 1 0 0 1 2 b c

6 The displacement from equilibrium of an oscillating weight suspended by a spring is given by 1 y 2cos 6t where y is the displacement in centimeters and t is the time in seconds see figure Find the displacement when t answers to two decimal places 4 A 3 2 wwwwww Equilibrium Displacement Round yo

10 You deposit 8000 in an account earning 7 interest compounded continuously a Write the equation that models the amount of money in the account after t years b How much will you have in the account in 8 years c How long does it take for the money in the account to double 0

2 Find the center and the radius of the circle x y 4x 3 0

10 A 20 meter line is used to tether a helium filled balloon Because of a breeze the line makes an angle of approximately 85 with the ground Use a trigonometric function to write an equation involving the unknown quantity and explain what you did to set this up and solve

Di intercent s and y intercept s of the function f x 3r 2 7

f x is a one to one function How many x intercepts Extra Credit Suppose y can the graph of the function f 1 inverse of the function f x have Explain your answer

8 Determine whether the statement is true or false Justify your answer In each of the four quadrants the signs of the cosec function and cosine function will be the same True False In each of the four quadrants the signs of the cosec function and the sine function will be the same because these functions are reciproc

3 Will Barrow wanted to know how tall the flagpole was in front of his school To find its height he drove a stake into the ground at the tip of flagpole s shadow and recorded the angle of elevation at two different times during the day He then measured the distance between the stake Will s data is below Stake A B Distance between stakes A B Time 2 00 PM 3 00 PM 12 feet Angle of Elevation 80 59

14 Let y 3 sin x a Determine the amplitude and period b Sketch the graph of this function Make sure to indicate key points on your graph 6 5 4 3 2 1 87 77 6 5 4 3 27 1 2 3 4 5 56 6 K 2 3 4 5 6 7 8

6 Conside the following piece wise function x 2 if x 0 e llestradogle onlinresis a if 0 x 2 4 if x 2 a Calculate the following values f 3 f 2 and f 1 f x x 2 PIGI b Graph the piece wise function Use the grid provided below N 5 4 3 2 1 5 4 3 2 1 tyd habbom si absti mill c State the domain and range of f x 1 2 3 4 5 90097 Hib ade brid ch I 2 3 4 5 under of

9 Using properties of logarithms express log c 5 rithms as a s

4 Find the values of 0 in degrees 0 0 90 and radians s 0 0 7 2 sec Not defined and cot 0 0 without the aid of a calculat

2 Using the figure below if 0 21 and y 8 determine the exact value of x y D

5 The revenue R x of producing and selling AirPads is modeled by the function R x 5x 175x a How many AirPads need to be produced and sold in order to maximize the revenue Round to the nearest whole number ant of boilans enojnolans thean ainer d most juzex b What is the maximum revenue Round to the nearest cent

5 Use the given function value s and trigonometric identities including the cofunction identities to find the indicated trigonometric functi sin 0 0 tan 0 0 in 00 and ton 008

1 A biologist wants to know the width w of a river so that instruments for studying the pollutants in the water can be set properly From point A th biologist walks downstream x 150 feet and sights to point C see figure From this sighting it is determined that 0 54 How wide is the river 0 54 AXI

Video The graph shows g x which is a translation of f x x Write the function rule for g x 2 10 8 6 4 2 g x 1 0 8 6 4 2 O 2 4 6 8 10 y 2 4 6 8 Write your answer in the form a x h k where a h and k are integers or simplified fractions

3 Given f x and g x ln x 3 a What is the domain of g h Find the domain of f o g

b log x 4 log x 7 2

13 Solve the following equation 2 sin 0 cos 0 2 for 0 0 2 v atqso i a viem wolnoitonuri sno of onos si s t que botix moy aislyx Torsd 2 noionit sit to setovat oitonut oil to dagodt i

12 Prove the following identity tan x sin x sec x cos x sec x

10 Find the point x y on the unit circle that corresponds to the real number t 57 How did you find that

9 Explain how you can use the unit triangle 45 45 90 and it s side measures to find exact values for sin 45 and cos 45

7 6 8 Find the point x y on the unit circle that corresponds to the real number Use your results to evaluate cos t

Let W be the subspace spanned by u and u and write y as the sum of a vector in W and a vector orthogonal to W 7 U 1 4 3 5 The sum is y z where y is in W and z Simplif SIDD is orthogonal to W

1 Evaluate if possible the sine cosine and tangent of the real number

5 Find the exact values of the three trigonometric functions sine cosine tangent of the real number t B a 1 b 1 3 b 5 13 a b

4 Evaluate the trigonometric function using its period as an aid 11

3 Use the value of the trigonometric function to find the sin t sin t 1

7 Evaluate if possible the sine cosine and tangent of the real number 7

6 Use the value of the trigonometric function to find cost 1

2 Evaluate the trigonometric function using its period as an aid COS 25

1 y x 6x 9x 1 find I

ABEN AB 6 A B A B

40 ABC CBA EN A B C ABC CBA 693 min ABC 1

37 AB BA EN AB BA 63 A B 9 AB

e the system of equations 2x y 5 and 3x 4y 42 by ations try 5 4y 42 2x y 3x 4y 2x Y 3x 4y x y ANABANG 5 42

EXPONENTS a a 5 so Estimating Square Roots A number is called a if its square root is an integer For example 25 and 36 are both perfect squares since 25 52 and 36 6 so 25 5 and 36 6 If we don t know the square root of some number x and don t have a calculator handy we can estimate the square root by bracketing it betweer two perfect squares for which we do know the square roots For example to estimate IO we know and 10 10 8 is between 2 Estimate the following square 23 Therefore 10 lies somewhere between and closer to 9 than to 16 According to a calculator 10 and 642 roots where and are perfect squares probably closer to because 10 is 95 is between and

EX 3 Find the value of each expression A log 32 B log 18

EX 2 Expand each logarithm 8x 5y A log c log 9x log 37 B loga D log 4

L Simplify the following expressions and express your answers with positive exponents a x x 5 d 13 8 JAL M Before we can apply the rules of exponents to expressions in the form of a n where the exponent is a fraction we need to consider what such an expression means Simplify the following expressions a x1 2 x 1 2 c m 2 m 2 9 then m d h 2 h 2 If h The expression a called a b a xxx h1 2 Simplify the following expressions 1 2 is often written as If h 5 5 6 d h 3 h 3 h1 3 1 2 c 5 5 5 5 then x h1 3 1 3 C 1 2 16 broc com what s another way to write h 23 x 2 and is called the ola 1 3 The nth Root If a is a real number and n is a positive integer a a b 41 2 41 2 what s another way to write h 32 b 1 3 27 the nth root of a e v 2 w5 v of a The symbol 8 is