Practice Problems for Mathematics Placement Eamination (MPE) Revised June, 011 When ou come to New Meico State Universit, ou ma be asked to take the Mathematics Placement Eamination (MPE) Your inital placement level is based on our Math ACT (or SAT) score and our high school grade point average The MPE will help to determine in which mathematics course ou will be placed If ou are dissatisfied with our initial placement, ou ma choose to take the MPE and achieve a qualifing score If ou wish to enroll in Math 191 (Calculus I), ou must take the MPE and achieve a qualifing score The problems on this practice eam will help ou review our mathematical skills and give ou an idea of what is on the test The actual test is multiple-choice and has 0 problems broken down as follows: Part I 1-10 Algebra I Part II 11-0 Algebra II Part III 1-0 Precalculus Algebra Part IV 1-0 Trigonometr These sample problems are grouped in a similar manner There are more practice problems than are on the actual eam Placement Based on the Mathematics Placement Eam Placement Eam Score Placement Level Math Course Placement at least 6 on Part I Math 10, 10G a total of at least 1 on Parts 1 and Math 111, 11G, Stat 1, 71 a total of at least 19 on Parts 1, and Math 1G, 190G, Math/Hon 7G at least 6 on each of the parts Math 191G,, 78, 79 Note: An ACT Math score of at least 16 is required to take the Mathematics Placement Eam A calculator is neither required nor necessar However ou will be allowed to use a non-graphing scientific calculator like the TI-0 PRACTICE PROBLEMS-Review of Basic Skills The first 1 problems given here are a review of basic arithmetic skills while not directl tested on the MPE, the knowledge of these skills is essential for the concepts that are tested 1 Simplif 9 Simplif + 10 Find all the positive factors of Add + 6 1
Subtract 8 6 7 6 Multipl 1 81 7 Divide 1 1 8 A carpenter is dividing a board that is piece in ards? d long into 9 equal pieces What is the length of each 9 If 1 d of paper are cut from a roll that is 18 d long, how man ards of paper are left? 10 True or False: 11 Simplif: ( ) 8 1 < 7 1 ( 7 + 1 1 Simplif 7 1 08 1 What is 7% of? 1 What percent of 6 is 7? ) 1 A store sells 1 oz of Brand A for $10 and 16 oz of Brand B for $10 Which is the more economical bu? 16 Solve for : PRACTICE PROBLEMS FOR PART I - Algebra I 8 17 = 6 17 Solve for : 17 = ( ) 18 If z = ( 1) and = 9, then what is z? 19 Simplif + 1 0 Simplif + 1 Rationalize the denominator: Translate the following statement into an equation: S equals the quotient of r and the sum of r and 8 Collect like terms: a bc ab c + ab c + 6a bc ab c Simplif and write with positive eponents: ( 6 ) 18
Combine into a single fraction: 7 + 1 6 Find the equation of the line through the points ( 1, ) and (, 7) Write our answer in slope-intercept form 7 Find the equation of the line that has slope 8 Simplif 10 + 7 ( z) z 9 Simplif 1 6 0 Factor 9 + 0 1 Factor + + 8 For the function f () = +, find f () and f ( ) and that goes through the point (, ) The formula for converting Centigrade (C) into Fahrenheit (F ) is given b the formula F = 9 C + Solve for C in terms of F Thirteen more than eight times a number is the same as two less than eleven times the number Set up an appropriate equation and solve for the number A T-shirt compan has $000 per da in fied costs and $ per T-shirt in production costs Find the cost function C () that gives the cost of producing T-shirts in das 6 Solve for and : + = 10 = + 8 7 Solve for a and b : a b = a + b = PRACTICE PROBLEMS FOR PART II - Algebra II 8 Simplif + 1 6 16 9 Simplif 0 Solve for : + 6 6 + + 6 1 6 1 = 1 1 Solve for a in terms of the other variables: Simplif ( z) (10 1/ / z / ) 1 A = 1 a + 1 b Simplif 1 7 Simplif 6 z 6
Write under a single radical and simplif: a7 d ad 6 Graph the line and label the and -intercepts: = 6 7 Find the equation of the line that contains the point (, ) and that is parallel to the line + = 6 Write our answer in slope-intercept form 8 Find the equation of the line that contains the point (, ) and that is perpendicular to the line + = 6 Write our answer in slope-intercept form 9 Find the verte and the and -intercepts of the function f () = + Graph the function, labelling the verte and intercepts 0 The height above ground (in feet) of a to rocket launched upward from the top of a building is given b S (t) = 16t + 96t + 6 a) What is the height of the building? b) What is the maimum height attained b the rocket? c) Find the time when the rocket strikes the ground 1 Find the domain of f () = + Find the domain of g () = Solve for : = 1 6 A colon of 100 termites invades a house Assuming that the colon doubles in number ever weeks and grows continuousl, the appropriate mathematical model for the population P (t) of termites t weeks in the future is P (t) = P 0 e kt a) Find the constants P 0 and k b) How man termites will there be in 10 weeks? c) When will there be a billion termites? The amount remaining (in grams) of a radioactive substance after t hours is given b A (t) = 100e kt After 1 hours the initial amount has decreased b 7% a) Solve for the deca constant k b) How much of the substance remains after 8 hours? c) What is the half-life of the substance? PRACTICE PROBLEMS FOR PART III - Precalculus Algebra 6 The volume of a sphere of radius r is given b V = πr Solve for r in terms of V 7 Find the distance d between the points (, 1) and (1, 6) in the rectangular coordinate sstem 8 Solve the inequalit + 8 9 Solve the inequalities a) and b) < 60 If f () = + + and g () = 1, find f (g ()) 61 Factor the following, if possible, over the real numbers: a) a b b) a + b c) a b d) a + b 6 Epress the surface area of a clinder with a closed bottom and open top in terms of the radius r and the height h
6 Epress the surface area of a cube in terms of its side length 6 Graph the parabola f () = For which values of do we have a) f () < 0, b) f () = 0, and c) f () > 0? 6 + + 6 = is the equation of a circle Put the equation in standard form and state the center and radius of the circle 66 Graph the functions f () = and g () = on the same coordinate aes 67 Graph the function f () = 68 The length L of a certain rectangle is ft more than three times the width, W The perimeter of the rectangle is 00 ft Set up a sstem of equations that will allow ou to solve for the length and the width, and then find those dimensions 69 The area of a right triangle is 66 in The height h of the triangle is in greater than three times the base b Set up a sstem of equations that will allow ou to solve for the base and height, and then find those dimensions 70 The length L of a rectangle is twice as much as the width W The area of the rectangle is 0 ft Set up a sstem of equations that will allow ou to solve for the length and the width, and then find those dimensions 71 For the function f () = 7, simplif the epression f ( + h) f () h 7 For the function g () = g ( + h) g (), simplif the epression 1 h 7 Solve the equation log ( ) = 7 Solve the equation log + log = 1 PRACTICE PROBLEMS FOR PART IV - Trigonometr 7 Given sin θ = 1 and 0 θ π, find the other five trigonometric functions 76 a) Convert 1 to radians b) Convert π to degrees 77 Find the following ranges for the value of θ in radians: a) 0 θ 90 b) 90 θ 180 c) 180 θ 70 d) 70 θ 60 78 a) What is the amplitude of f () = sin ()? c) What is the period of h () = sin (π)? b) What is the amplitude of g () = cos ()? d) What is the period of k () = 6 tan ()? 79 Simplif the epression sin θ cot θ sec θ 80 Graph two periods of = cos () where is in radians 81 Find the amplitude, period, horizontal and vertical shifts of f () = cos graph the function for two periods ( ( π )) +1 Then
8 Complete the following identities: a) sin ( θ + cos θ = b) tan θ + 1 = c) 1 + cot θ = π ) ( π ) ( π ) d) sin θ = e) cos θ = f) sec θ = g) sin θ = h) cos θ = ( ) ( a) Find the eact value of cos 8 1 b) Find the eact value of sin 1 1 ) c) Find the eact value of tan 1 (1) ( ( )) 8 a) Simplif tan arccos b) Epress cos ( sin 1 ) in terms of without trig functions 8 Find all the solutions of sin () = 1 for 0 π 86 Find all the solutions of cos () = 1 for 0 π 87 Solve the following for θ, with 0 θ < π: cos (θ) sin θ = 0 88 Suppose that in a triangle, the side a is opposite the angle α and the side b is opposite the angle β Use the Law of Sines to solve for b, given that α =, β = 60, and a = 6 89 Recall that the Law of Cosines states b = a + c ac cos β, where b is the side opposite the angle β Given that a =, b =, and c = 6, solve for β Leave our answer in terms of an inverse trig function 90 An observer is standing mi from a ver steep mountain She measures the angle between the ground and the top of the mountain and finds it to be How high above her ground level is the mountain? 91 The diameter of the base of a cone is 10 ft Its height is 1 ft Find the angle of inclination of the side of the cone Leave our answer in terms of an inverse trig function Practice Problem Answers 1) 16 ) 17 ) 1,,,, 6, 8, 1, ) 11 1 6) 19 1 7) 8) 1 9 1 d 9) 1 1 1 ) 17 1 d 10) True 11) 7 16) 76 17) = 7 6 1) ) S = r r + 8 6) = + 1 7) = + 1 1) 6 1) 9 1) 11% 1) Brand B 18) z = 18 19) 0 0) 11 1 ) 11a bc ab c + ab c ) 6 9 ) 7 + + + 8) 17 8z 9) 7 0) ( ) ( ) 1) ( 7) ( + ) ) f () = 7, f ( ) = 9 ) C = 9 (F ) = 9 F 160 9 )8 + 1 = 11, = ) C () = 1000 + 6) = 1, = 9 ( ) 7) a =, b = 8) + 0) = 1) a = Ab = 6 + b A ) ) z ) a d d 6 9) ) 1 ( 6) ( + 6) = 1 6
6) Intercepts are (, 0) and (0, ), 1 1 7) = 6 8) = -intercepts: (1, 0) and (, 0) -intercept: (0, ) 9) verte: (, 1) 0) a) 6 ft, b) 00 ft c) t = 8 s 1) 1, Equivalentl, (, 1) ( 1, ) (, ) ) Equivalentl, (, ] ) = ) a) P 0 = 100, k = ln = 010 b) P (10) = 100e ln (10) = 100 10 = 1008 (rounding up) 1 ln 10 c) t = = 6976 wks ln ln (09) ) a) k = = 00060 1 b) A (8) = 100e ln(09) = 100 (09) = 781 g ln (0) 1 ln (0) c) t = = k ln (09) = 116 h V 6) r = 7) d = 1 8) 11 ( π 8 Equivalentl,, 11 ] 8 9) a) 9 or 1 Equivalentl, (, 1] [9, ) b) 1 < < 9 Equivalentl, (1, 9) 60) 8 1 + 1 = 8 + 1 1 61) a) (a b) (a + b) b) Not factorable over the reals c) (a b) (a + ab + b ) d) (a + b) (a ab + b ) 6) S = πr + πrh 6) S = 6 6) f () = b) = 1 1, = c) < or > 6) ( ) + ( + ) = 16, center is (, ), radius is a) < < Equivalentl, ( ) Equivalentl,, ( ), ( ), 7
0 10 66) f () =, g () = 1 0 1 67) f () = 1 0 1 68) L = W +, W + L = 00, L = 7 69) 1 bh = 66 h = b + b = 1 in, h = 1 in 70) L = W, LW = 0, L = 1 ft, W = 1 ft 6 71) + h 7) ( 1) ( + h 1) 1 7)cos θ =, tan θ = 10 ft, W = 7 ft 7) = 11 1, csc θ =, sec θ = 1 7) = 1 7 = 9 1, cot θ = 1 76) a) 7π b) 0 77) a) 0 θ π b) π θ π c) π θ π d) π θ π 78) a) b) c) 1 d) π 79) sin θ cos θ = sin θ tan θ 1 80) = cos () 1 81) Amp =, period= π, horizshift= π right, vert shift= 1 up a) 1 e) sin θ b) sec 8) θ f) csc θ c) csc θ g) sin θ cos θ d) cos θ h) cos θ sin θ = cos θ 1 = 1 sin θ 8) a) π 6 b) π 6 c) π 8) a) b) 1 8) = π 6, π 6, 1π 6, 17π 6 86) = π 9, π 9, 8π 9, 10π 9, 1π 9, 16π 87) θ = π 9 6, π 6, π 88) b = ( ) ( ) 0 89) β = cos 1 = cos 1 90) h = mi 18 ( ) 9 ( ) 1 91) θ = tan 1 () = sin 1 = cos 1 10 10 8