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Transcript
Math 1302- Test I Review The following questions give you an idea of how I ask questions as well as a few examples. This test review may be similar to the actual exam – it may also be quite different. The topics should be similar in both. You are responsible for everything we studied in class ; notes, lectures, HW problems, and quizzes. In some cases ( one or two problems), the ideas that we covered have been extended somewhat and you may not have seen that type of question but are asked to use what you have learned to answer the question. No make-up on Exam I. Do not be alarmed contains as many types of problems as possible. about the length of this exam – it is a review and it As stated in syllabus – you can only attend the review session ( 7:00 – 8:00 PM – Wed ) if you have missed at most one day and if you have attempted this review – a blank review is not acceptable. If you have a good reason for having two absences but you have attempted the test review – go ahead and come. This is only for the first exam – on the other exams – only one absence is acceptable since the last exam. When you answer the questions – keep in mind that you are studying the topics , the ideas – you are not memorizing the question. I have a tendency to change the way the question is asked or modify the topic. ( instead of integers I may ask about whole numbers – instead of commutative, I may ask about distributive, instead of additive inverse, I may ask about multiplicative identity, ... ) Answers Should be non calculator based – use it to help you but not to provide an answer. ( 3/7, 2 , π are all good answers – their decimal representations are not 1. Identify each of the following sets; { 1, 2, 3, ... } → ________________________ set of real numbers that can not be written as fractions → __________________________ Whole numbers greater than 1 that are divisible only by 1 and the number itself → ___________ 2. Write down the set of whole numbers: ___________ of integers: ________________ Find the smallest nonnegative integer _________ smallest counting number ___________ 3. Give me an example of the commutative law of addition. __________________ Is putting on your left shoe and then your right shoe commutative ? _____________________ Why or why not ? Give me an example of something in real life ( like above ) that is not commutative . ___________________ 1 4. Which property is being used; 3 ( 4 • 5 ) = ( 3 • 4 ) • 5 ______________ Is ( 3 + 4 ) + 5 = ( 4 + 3 ) + 5 the associative law? ___________ 5. Show by example that the set of whole numbers is not associative under division -( division is not associative with whole numbers ) 6. What does it mean when we say the set of real numbers is closed under addition, subtraction, multiplication, and division (with one exception) Show by example that the set of integers is not closed under the division operation. 7. Find the sum of the first three prime numbers. _________ Find the product of the first three prime numbers ___________ Find xn . if n is the smallest whole number and x is not zero. _________ xn if n is the smallest counting number and x is any real number. _________ 8. Find each of the following absolute values. a) | x 2 | = __________ c) | 4 - 15 | = __________ b) | 4 + x | = _____________ if x is a whole number d) x + | x | = __________ if x is a whole number x - | x | = ____________ if x is an integer e) x = ______ if x is a real number |x| f) | 5 • 5 - 6 | = ________ 2 9. Give me an example of an irrational number different than numbers of the form π and e. __________ of a rational number that is not positive. ____________ Since 4 is a rational number and 2 is not, then what kind of number is 2 + make a conclusion of the statement above → “ the sum of .... is always ....” 2 Is the product of two rational numbers a rational number ? _____________ provide examples or a counterexample Is the product of two irrational numbers an irrational number ? _________ provide examples to convince me. 10. Find the solution of each of the following equations. 3 + x2 = 3 → _______________ 2 – 3 ( x + 1 ) = 1 → __________ x x2 → __________ 3 4 1 - x( 1 – x ) = x2 → ____________ 11. Is the set of whole numbers closed under multiplication ? ___________ Explain what this means. Why is the set { 0, 2, -2 } not closed under addition ? 12. Which is bigger 3/11 or 4/23 ? Show without the use of a calculator - 3 13. Find the additive identity of the set of real numbers . _________________ multiplicative identity ? ________ Find the additive inverse of ( - 4 ) . ______________ The multiplicative inverse of ( - 4 ) ? ______ 14. Use the rules of exponents to simplify. Simplest type – exponents should be nonnegative. a) 2x3 • 5x2y4 = ___________ ( - 3 x2y4 ) 4 = ___________ c) xy(2xy4) = ___________ 12 xy 5 4x 7 y 2 = __________________ 15. More exponent problems. More difficult – exponents in final solution should be nonnegative. a) ( 2x -1 ) -3 = _________ 3x 2 y 2 c) 2 7 12 x y e) 3 = _________ 3x1 / 2 y 1 / 2 = _________ 6 x 2 y 1 / 3 3x 7 g) 2y b) ( -2x-1y4)2 (4x-2y3 )-1 = _____________ d) ( x1/3 ) ( -2x-1/2)(4x1/4y) = ____________ f) ( 2x1/2y-1/3 )-6 0 = _______ h) ( -2/7 )-2 = ________ 4 16. What is the x intercept of 2x – y = 4 ? ______________ the y – intercept ? ____________ Give me three points that lie on the line 2x – y = 4. _______________________ Use them to graph the line. You should include the intercepts above. 17. What is the slope of the lines 2x – y = 4 ? __________ 18. 0•x + y = 2 ? __________ 3x – 0 •y = 12? _____ simplify – reduce to simplest form 12 • 5/9 = ___________ 1– 3/5 - 4/7 = ____________ 1/ 2 = ________ 3/ 6 4 - 3/5 = ______________ 3 x( x 2) = _________ x2 (1 + 1/x) ÷ ( 1 + 1/x) = _______ 19. Write in simpler form x • 1 = _________ 0 + x = _______ x • x = _________ x ÷ x = ______ 20. Find all whole number solutions of the equation 2x + 4 = 0. ________________ Find all integer solutions of the equation x ( 2x + 3 ) = 0 . _________________ Find all real solutions of x2 = 1 – x ( 1-x) , _____________________ 5 21. Find all positive solutions of x4 + 2x2 + 4 = 0 . ______________ 22. Find the GCF of a) 12 and 30 → ___________ b) 240 and 420 → _______________ c) 8x2y6 and 12x3y → _________ d) 5x and 23y → _____________ 23. Find the LCM of (problems above) a) LCM( 12, 30) = _________ b) LCM(240, 420 ) = _______________ c) LCM ( 8x2y6, 12x3y ) = __________ d) LCM ( 5x, 23y ) = ________________ 24. Use a number line to sketch all real numbers that satisfy the following inequalities (three different lines) x ≤ -2 2x + 3 > 2 -2 < x ≤ 4 25. If x is a number in ( 3, 10] ( this is interval notation), a) then what is the smallest value of x ? Is there a smallest value ? b) what is the largest value of x ? Is there a largest value ? 6 26. What if anything is wrong with the statement below ? - 4 < x < - 12 27. How many terms does 3x – y have ? _______________ How many factors does ? ________ 28. Which one does a monomial have more of (usually ) terms or factors ? _____________ 29. Find a polynomial with three factors and only 1 term. _____________ A polynomial with three terms is called a _________________ 30. Which one of these is not a polynomial ? (circle your best solution(s) ) -3 2x + 3 all are 21/2 x2y + 1 x2 y 2 none are polynomials 31. What is the degree of the following polynomials 2 + x → _________ -2 → ___________ 4 + x + x7 → ____________ ½ x + x4y3 - y5 → ___________ 7 32. Simplify by performing the given operation and combining similar terms. 3 2 4 8 = ___________ a) 2i ( 1 – 2i ) = _____________ b) c) ( 1+2i)(3 – 2i) = ______________ d) ( x + 3)2 = ________________ e) ( 1-x)(1-x)(1-x) = _____________ f) ( 3 – 2i ) - ( 4 – 2i ) = ________ g) a + b ( a + b ) = _______________ h) ( 2x –y)(2x + y) = _________ 33. Write in scientific notation 321 = _________________ 0.0021 = _______________ 34. Find the answer to the following two problems and write your final answer in scientific notation. Try to control yourself and not use a calculator. a) ( 3 x 102 ) ( 5 x 103 ) = _______________ b) ( 1.2 x 10-3 ) ÷ ( 1.6 x 102 ) = _______________ 8 36. Simplify the following by using the properties of radicals. ( simplest ) a) c) 64 x12 = ________ y3 3 6 = ________ 3 f) 4 3 3 18 = _________ 1 h) = ______________ (16x-8y4 )-1/2 = ________ b) d) 8x 5 = ____________ 2x g) x2 8 i) 36 x 16x 6 e) ( 9/25)3/2 = ________ = __________ = _________ 37. More radicals. a) c) e) 5 43 4 x 2 = ___________ b) 2x = ___________ 8 x5 y 4 d) 6 • 3 6 = ____________ f) 3 4 x 3 2 x 2 = _________ 3 16 x 3 4 2x 2 3 32 = ____________ = ___________ 9 39. Other problems a) which is bigger x 2 or | x | ? explain your answer. b) Find (assume that x is any real number) – without the use of a calculator. 2 + 3π = ___________ c) 3 x 2 = ______ x 3 = | x | True or False. _________ 40. Word Problems a) 10 % of all students in summer classes miss more than a week’s worth of school. A class consists of 60 students – how many missed one week or less of school ? Actual number ! b) A child can walk around a track in 10 minutes – stops along the way several times. The parent walks at 4 minutes per lap and will not stop until they both meet again at the starting line. Use the concepts we studied to find how long it will be before they meet at the starting line. A single answer will be counted wrong – a list of numbers (guessing) will also be counted wrong. c) Use the given examples to find the missing numbers on the third example What’s next in the pattern - 5, -10, 10, 100 4, 8, -8, 64 -3, _____, _______, _______ d) When we divide the sum of a number and one by 3 we get 23, what was the original number ? I want an equation – I do not want the actual number. Set up an equation that correctly models the statement. 10 41. Which of these are perfect squares ? x8, 9, y25, x2 + 9, (2x – 3)2, 1 + 2x + x2, x2 + 10x + 25 x2 + 12x - 9, none of them all are We will probably be selective on this last two problems – depends on how far we get. We probably will get enough of #42 below but not #43 or #44. 42. Factor the following polynomials. - whatever factoring we get to. x2 - 2x = _____________ 12x2y3 - 8xy4 = _____________ 3x + xy = _____________ 12x + 36x2 = __________ x2 – 49 = _____________ 3y (x + 2) + 2x( x + 2 ) = ____________ 4x2 – 9y2 = ____________ 1 + x2 = __________ x3 + 8 = _____________________ x4 - 16 = ______________ 1 – 27y3 = _____________________ 5x4 + 40 = ______________ x2 - 4x - 5 = ____________ x3 + 8 = ________________ 5 + x3 = _____________ x6 - 64 = ____________________ x2 + 2x – 35 = _____________ x2 + x + 100 = __________ x2 + 12x + 36 = ________________ 8x3 – 18x = ______________ x2 +12x + 32 = _______ x3 – 2x2 – 3x = ___________ x2 – 6x + 9 = ______________ x3 - 27y3 = ______________ 11 43. More Factoring more complicated problems plus trinomials of the form ax2 + bx + c 44. Algebraic fractions – if we get to this section - sum, difference, product, quotient 12