Mathematics Placement Examination (MPE)


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1 Practice Problems for Mathematics Placement Eamination (MPE) Revised August, 04 When you come to New Meico State University, you may be asked to take the Mathematics Placement Eamination (MPE) Your inital placement level is based on your Math ACT (or SAT) score and your high school grade point average The MPE will help to determine in which mathematics course you will be placed If you are dissatisfied with your initial placement, you may choose to take the MPE and achieve a qualifying score If you wish to enroll in Math 9 (Calculus I), you must take the MPE and achieve a qualifying score The problems on this practice eam will help you review your mathematical skills and give you an idea of what is on the test The actual test is multiplechoice and has 40 problems broken down as follows: Part I 0 Algebra I Part II 0 Algebra II Part III 0 Precalculus Algebra Part IV 40 Trigonometry These sample problems are grouped in a similar manner There are more problems here than there are on the actual eam Placement Based on the Mathematics Placement Eam Placement Eam Score Placement Level Math Course Placement 0,, or on Part I 0 CCDM 0N 4 or on Part I CCDM 4N at least 6 on Part I Math 0, 0G a total of at least on Parts and Math, G, Stat, 7 a total of at least 9 on Parts, and 4 Math 4G, 90G, Math/Hon 7G at least 6 on each of the 4 parts Math 9G,, 78, 79 Note: A calculator is neither required nor necessary However you will be allowed to use a nongraphing scientic calculator like the TI0 PRACTICE PROBLEMSReview of Basic Skills The first problems given here are a review of basic arithmetic skills while not directly tested on the MPE, the knowledge of these skills is essential for the concepts that are tested Simplify 9 4 Simplify Find all the positive factors of 4
2 4 Add Subtract Multiply Divide 4 8 A carpenter is dividing a board that is 4 piece in yards? yd long into 9 equal pieces What is the length of each 9 If 4 yd of paper are cut from a roll that is 8 yd long, how many yards of paper are left? 0 True or False: Simplify: ( ) 8 < 7 ( Simplify 7 08 What is 7% of 4? 4 What percent of 64 is 7? ) A store sells oz of Brand A for $0 and 6 oz of Brand B for $0 Which is the more economical buy? 6 Solve for : PRACTICE PROBLEMS FOR PART I  Algebra I 8 7 = 6 7 Solve for y : 4y 7 = ( y) 8 If z = ( ) and = 9, then what is z? 9 Simplify Simplify + 4 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 4ab c + ab c + 6a bc ab c
3 4 Simplify and write with positive eponents: ( 6 y ) Combine into a single fraction: 7 y + 8y 4 6 Find the equation of the line through the points (, ) and (4, 7) Write your answer in slopeintercept form 7 Find the equation of the line that has slope 4 8 Simplify ( z) z 9 Simplify Factor Factor For the function f () = +, find f () and f ( 4) and that goes through the point (, ) The formula for converting Centigrade (C) into Fahrenheit (F ) is given by the formula F = 9 C + Solve for C in terms of F 4 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 Tshirt company has $000 per day in fied costs and $4 per Tshirt in production costs Find the cost function C () that gives the cost of producing Tshirts in days 6 Solve for and y : + y = 0 y = Solve for a and b : a b = a + b = 8 Simplify y + y 6 y 6 9 Simplify 40 Solve for : PRACTICE PROBLEMS FOR PART II  Algebra II y 4 y = 4 Solve for a in terms of the other variables: ( y z) 4 Simplify (0 / / yz / ) A = a + b
4 4 Simplify y y 7 44 Simplify 64 y 4 z 6 4 Write under a single radical and simplify: 4a7 d ad 46 Graph the line and label the and yintercepts: y = 6 47 Find the equation of the line that contains the point (, ) and that is parallel to the line + y = 6 Write your answer in slopeintercept form 48 Find the equation of the line that contains the point (, ) and that is perpendicular to the line + y = 6 Write your answer in slopeintercept form 49 Find the verte and the y and intercepts of the function f () = + 4 Graph the function, labelling the verte and intercepts 0 The height above ground (in feet) of a toy rocket launched upward from the top of a building is given by S (t) = 6t + 96t + 6 a) What is the height of the building? b) What is the maimum height attained by the rocket? c) Find the time when the rocket strikes the ground Find the domain of f () = + 4 Find the domain of g () = 4 Solve for : 4 = 64 4 A colony of 00 termites invades a house Assuming that the colony doubles in number every weeks and grows continuously, 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 many termites will there be in 0 weeks? c) When will there be a billion termites? The amount remaining (in grams) of a radioactive substance after t hours is given by A (t) = 00e kt After hours the initial amount has decreased by 7% a) Solve for the decay constant k b) How much of the substance remains after 48 hours? c) What is the halflife of the substance? PRACTICE PROBLEMS FOR PART III  Precalculus Algebra 6 The volume of a sphere of radius r is given by V = 4 πr Solve for r in terms of V 7 Find the distance d between the points (, ) and (, 6) in the rectangular coordinate system 8 Solve the inequality Solve the inequalities a) 4 and b) < 4 60 If f () = and g () =, find f (g ()) 6 Factor the following, if possible, over the real numbers: a) a b b) a + b c) a b d) a + b 4
5 6 Epress the surface area of a cylinder 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 64 Graph the parabola f () = 4 For which values of do we have a) f () < 0, b) f () = 0, and c) f () > 0? y + 6y = is the equation of a circle Put the equation in standard form and state the center and radius of the circle 66 Graph the function f () = 67 Graph the function f () =/(+4) 68 The length L of a certain rectangle is 4 ft more than three times the width, W The perimeter of the rectangle is 00 ft Set up a system of equations that will allow you 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 system of equations that will allow you 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 system of equations that will allow you to solve for the length and the width, and then find those dimensions 7 For the function f () = 7, simplify the epression f ( + h) f () h 7 For the function g () = g ( + h) g (), simplify the epression h 7 Solve the equation log ( ) = 74 Solve the equation log y + log y = 4 PRACTICE PROBLEMS FOR PART IV  Trigonometry 7 Given sin θ = and 0 θ π, find the other five trigonometric functions 76 a) Convert to radians b) Convert 4π to degrees 77 Find the following ranges for the value of θ in radians: a) 0 θ 90 b) 90 θ 80 c) 80 θ 70 d) 70 θ a) What is the amplitude of f () = 4 sin ()? c) What is the period of h () = sin (4π)? b) What is the amplitude of g () = cos (4)? d) What is the period of k () = 6 tan ()? 79 Simplify the epression sin θ cot θ sec θ 80 Graph two periods of y = cos () where is in radians
6 8 Find the amplitude and period of f () = cos() and graph the function for two periods 8 Complete the following identities: a) sin ( θ + cos θ = b) tan θ + = c) + 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 b) Find the eact value of sin ) c) Find the eact value of tan () ( ( )) 84 a) Simplify tan arccos b) Epress cos ( sin ) in terms of without trig functions 8 Find all the solutions of sin () = for 0 4π 86 Find all the solutions of cos () = for 0 π 87 Solve the following for θ, with 0 θ < π: cos (θ) sin θ = 0 88 An observer is standing mi from a very steep mountain She measures the angle between the ground and the top of the mountain and finds it to be 4 How high above her ground level is the mountain? 89 The diameter of the base of a cone is 0 ft Its height is ft Find the angle of inclination of the side of the cone Leave your answer in terms of an inverse trig function Practice Problem Answers ) 6 ) 7 ),,, 4, 6, 8,, 4 4) 6) 9 7) 4 8) 4 9 yd 9) 4 ) 7 yd 0) True ) 7 ) 6 ) 94 4) % ) Brand B 8) z = 8 9) 0 0) 6) 76 7) y = 7 6 ) ) S = r r + 8 6) y = + 7) y = ) a bc 4ab c + ab c 4) 6 y 9 ) 7 + y + y + 8) 7 8z 9) 7 0) ( ) ( 4) 6
7 ) ( 7) ( + 4) ) f () = 7, f ( 4) = 49 ) C = 9 (F ) = 9 F )8 + =, = ) C () = ) =, y = 9 (y ) 7) a =, b = 8) = y 6 4 9) y + 4 y + 4 ( 6) ( + 6) = ) = 4 4) a = Ab 4) b A y 4) y 44) 4yz y 4) a d d y 4 46) Intercepts are (, 0) and (0, ), 4 47) y = 6 48) y = y intercepts: (, 0) and (, 0) 4 yintercept: (0, ) 4 49) verte: (, ) 0) a) 6 ft, b) 400 ft c) t = 8 s ), Equivalently, (, ) (, ) (, ) ) 4 Equivalently, (, 4 ] ) = 4) a) P 0 = 00, k = ln = 00 b) P (0) = 00e ln (0) = 00 0 = 008 (rounding up) ln 0 c) t = = 6976 wks ln ln (09) ) a) k = = b) A (48) = 00e 4 ln(09) = 00 (09) 4 = 748 g ln (0) ln (0) c) t = = k ln (09) = 46 h V 6) r = 7) d = 4 8) ( 4π 8 Equivalently,, ] 8 9) a) 9 or Equivalently, (, ] [9, ) b) < < 9 Equivalently, (, 9) 60) 8 + = 8 + 6) 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 7
8 y 64) f () = 4 b) =, = c) < or > 6) ( ) + (y + ) = 6, center is (, ), radius is 4 y 0 0 a) < < Equivalently, ( ) Equivalently,, ( ), ( ), 66) ) L = W + 4, W + L = 00, L = 7 69) bh = 66 h = b + b = in, h = 4 in 70) L = W, LW = 0, L = ft, W = ft 6 7) 4 + h 7) ( ) ( + h ) 7)cos θ =, tan θ = ft, W = 7 ft 7) =, csc θ =, sec θ = 74) y = 4 7 = 9, cot θ = 76) a) 7π 4 b) 40 77) a) 0 θ π b) π θ π c) π θ π d) π θ π 78) a) 4 b) c) d) π 79) sin θ cos θ = sin θ tan θ y 80) y = cos () 4 4 8) c) a csc θ g) sin θ cos θ d) cos θ h) cos θ sin θ = cos θ = sin θ 8
9 8) a) π 6 b) π 6 c) π 84) a) 4 b) 8) = π 6, π 6, π 6, 7π 6 86) = π 9, 4π 9, 8π 9, 0π 9, 4π 9, 6π 87) θ = π 9 6, π 6, π 88) b = ( ) ( ) 88 h = mi ( ) ( ) 89) θ = tan () = sin = cos 0 0 9
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