Grade 10 / Set of numbers and calculations 90 exercises (100% corrected)

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ChingQuizz : 7 exercises available for Quizz assessment : 1. Computation in N , Z and Q E.11326 Definitions: The following numbers are classified according to their na-ture : the set of rational numbers , denoted by Q , is the set of numbers that can be written as a fraction. The set of decimal numbers , noted D , is the set of numbers that can be written as a decimal fraction (whose denominator is a power of 10 ) . 1 Show that each of the following numbers belongs to Q and to D : a 5 4 b 0 ; 54 c 12 2 Show that each of the following numbers belongs to Q , but not to D : a 1 3 b 1 + 4 7 E.8270 Consider the two expressions : A = 5 − 4 x ; B = 2 · x + 2 x − 1 1 Evaluate each of the two expressions A and B for each of the numbers x =2 and x = − 5 . 2 Complete the table below with the sets N , Z , or Q to indicate the smallest set to which each of these numbers belongs : x =2 x = − 5 A =5 − 4 x B = 2 · x +2 x − 1 E.8271 Consider the two expressions : A = x 2 − 3 x + 1 ; B = − 6 x 2 + x + 3 1 Evaluate each of the two expressions A and B for each of the numbers x = − 1 and x = 1 2 . 2 Complete the table below with the sets N , Z , D , or Q to indicate the smallest set to which each of these numbers belongs : x = − 1 x = 1 2 A = x 2 − 3 x +1 B = − 6 x 2 + x +3 E.8505 Let n be a natural integer ( n ∈ N ) . 1 Establish identity for any n ∈ N : 4 − 6 n + 2 = 4 n + 2 n + 2 2 Deduce the three values of n for which the integer n +2 is a factor of the integer 4 n +2 2. Set D and frame E.8272 Definition: un A decimal number is a number written as a decimal fraction (where the denominator is a power of 10 ) . The set of all decimal numbers is denoted by D . 1 Consider the number ı , whose approximate value is : ı ≈ 3 ; 14159265 a Give the range of the number ı to the nearest tenth. b Give the range of the number ı to the nearest thou-sandth 2 Consider the number 2 , whose approximate value is : 2 ≈ 1 ; 4142136 a Give the range of the number 2 to the nearest tenth. b Give the range of the number 2 to the nearest thou- sandth E.8506 1 Give a number d belonging to the set D and verifying the square : 4 49 <d< 5 49 2 Give a number d such that d ∈ D and 2 <d < 3 . https://chingmath.fr chapExoCorrec/11326 sacados/11326 chapExoCorrec/8270 sacados/8270 chapExoCorrec/8271 sacados/8271 chapExoCorrec/8505 sacados/8505 chapExoCorrec/8272 sacados/8272 chapExoCorrec/8506 sacados/8506
Aı×r2rAc2c E.8273 Consider the square and disk below : 1 Knowing that the area of the square is 5 m 2 , give a frame-work, to the nearest hundredth, of the measure of the side of this square. 2 Knowing that the area of the disk is 4 m 2 , give a frame, to the nearest hundredth, for the measure of the radius of the disk. E.1735 for a 12-digit calculator (I don’t think it’s suitable for today’s calculators). Proposition: (allowed) A fraction a b irreducible belongs to the set D if, and only if, its denominator admits a decomposition into products of prime factors of the form 2 m × 5 n with m and n natural integers. Consider the three numbers : A = 36 359 363 519 ; B = 0.1000195313 ; C = 5 121 51 200 1 Using the calculator, compare the approximate values of these three numbers. What conjecture can be made? 2 a Verify the equality: 363 519 = 3 2 × 13 2 × 239 Then justify that the number A does not belong to D . b Justify that the numbers A and B are distinct. 3 a Give the units digit of the product : 1 000 195 313 × 512 b Justify that the numbers B and C are distinct. 3. Set Q and membership in D E.8511 Definition: l’ The set of rational numbers , denoted Q , is the set of numbers that can be expressed as a quotient a b where a and b are two relative integers. This exercise is a multiple-choice questionnaire. (QCM) . There is only one correct answer per question. 1 For the number 1 3 : 1 3 =0 ; 33 1 3 =0 ; 34 1 3 =0 ; 3333 1 3 =0 ; 3334 the previous answers are incorrect. 2 The decimal representation of the number 1 3 has a deci-mal part that is composed of : 10 digits 100 digits 10 0000 digits the previous answers are incorrect. 3 For the number 1 3 , we have : 1 3 ∈ N 1 3 ∈ Z 1 3 ∈ D 1 3 ∈ Q E.8274 1 Consider the number x =0.333333 . Is the assertion ˇ 3 × x = 1 ı true or false? 2 Consider the number y =0.333334 . Is the assertion ˇ 3 × y = 1 ı true or false? 3 Consider the equation : ( E ): 3 × z =1 . a Solve equation ( E ) in Q b What can be said about solving the equation ( E ) in D . E.8510 Consider a quotient a b where a ∈ N , b ∈ N ∗ . 1 Give a quotient such that : a b ∈ D et b a ∈ D 2 Give a quotient such that : a b ∈ D et b a ∈ D 4. Comparison of quotients E.346 Without using a calculator, compare the following quotients : a 6 5 and 9 5 b 5 8 and 5 9 c − 7 3 and − 10 3 d 10 27 and 10 31 E.9443 Let n be an integer greater than or equal to 2, compare the two quotients : n n + 1 and n n − 1 E.291 For n a natural number, n ∈ N , com-pare the following rationals : n + 1 n + 2 ; n + 6 n + 3 ; n + 7 n + 3 https://chingmath.fr chapExoCorrec/8273 sacados/8273 Aı×r2rAc2c chapExoCorrec/1735 sacados/1735 chapExoCorrec/8511 sacados/8511 chapExoCorrec/8274 sacados/8274 chapExoCorrec/8510 sacados/8510 chapExoCorrec/346 sacados/346 chapExoCorrec/9443 sacados/9443 chapExoCorrec/291 sacados/291
×2?5?−2?÷25 ?×43?−5?;4−2;2 E.243 1 Perform the following calculations : a 1 2 − 1 3 b 1 3 − 1 4 c 1 4 − 1 5 d 1 5 − 1 6 2 a Let m be a strictly positive integer, make a conjec-ture about writing the following difference : 1 m − 1 m + 1 b Demonstrate this conjecture. E.313 Let a> 1 : 1 Compare : a a − 1 and a 1+ a 2 Arrange these numbers in ascending order : 0 ; a 1 + a ; a a − 1 ; 1 ; a 1 − a ; − a 1 + a E.314 For x a positive number. Com-pare the numbers : x x + 1 ; x + 1 x + 2 E.10220 Compare the following quotients : a 3 ı and 5 − ı b 2 7 and 2 5 5. Calculations in Q E.8275 Calculate and give the result in sim-plified fractions. a 3 4 + 2 6 b 2 15 + 3 20 c 5 12 − 9 8 d 5 6 − 13 9 e 5 12 − 2 15 f 15 66 − 10 44 E.11289 Calculate and give the simplified ex-pression of the calculations : a 1 3 + 35 3 × 5 14 b 5 9 − 2 3 5 4 − 7 6 E.8276 Calculate the following fractions and write them in irreducible form : a 3 4 + 5 6 × 3 2 b 1 2 − 1 3 × 5 6 c 5 − 3 × 7 5 + 9 × 3 E.8313 Leaving the calculation steps in your essay, perform the calculations below, giving the result as a reduced fraction : a 2 7 + 5 14 b 3 4 − 5 6 c 1 3 + 5 3 × 2 4 d 3 7 − 2 7 × 21 8 E.8277 Perform the following calculations : a 5 7 + 1 7 × 5 + 1 2 b 1 3 + 4 3 10 9 c 2 3 + 1 2 17 9 − 1 3 d 2 13 − 5 13 ÷ 10 16 E.4418 Perform the following calculations : a 1 + 3 7 2 − 8 3 b 5 2 + 7 2 11 3 − 5 2 c 1 1 + 1 1 + 1 2 E.1717 Perform the calculations below ; note that a fraction can only be simplified when its numerator and denominator are fully determined : a 1 + 1 2 2 − 23 7 b 5 − 2 − 3 5 − 9 3 + 1 4 + 9 − 4 3 c 1 1 + 1 1 + 1 1 + 1 1 + 1 2 E.241 Perform the calculations : a 2 2 1 4 + 2 − 1 b 1 − 3 2 3 3 5 3 − 101 c 52 1 1 + 1 2 + 5 6 − 3 6. Irrational numbers E.776 Below are ˇ diagrams switching ı. Find missing values and inverse operations. https://chingmath.fr chapExoCorrec/243 sacados/243 chapExoCorrec/313 sacados/313 chapExoCorrec/314 sacados/314 Trop dur chapExoCorrec/10220 sacados/10220 chapExoCorrec/8275 sacados/8275 chapExoCorrec/11289 sacados/11289 chapExoCorrec/8276 sacados/8276 chapExoCorrec/8313 sacados/8313 chapExoCorrec/8277 sacados/8277 chapExoCorrec/4418 sacados/4418 chapExoCorrec/1717 sacados/1717 chapExoCorrec/241 sacados/241 chapExoCorrec/776 sacados/776 ×2?5?−2?÷25 ?×43?−5?;4−2;2
2√32√252?1 ?√42√1;442?2 0;65−3√243NZDQR E.265 In this exercise, we’ll show that, for any natural number a , a is either an integer or a non-decimal number. A decimal number X (in base 10) is a number admitting the writing: X = x 0 ;x 1 x 2 x 3 · · · x n where x 0 is its integer part, n is the number of decimal places of X x 1 , x 2 ,. . . , x n integers between 0 and 9 The first observation is that since X is a number with n dec-imal places, then x n is non-zero. Example: for X =25.153 . We have : x 0 =25 n =3 x 1 =1 , x 2 =5 , x 3 =3 Let a be a natural number, assume that a is a non-integer decimal number: a ∈ D ; a ∈ N Thus, a admits the following decimal decomposition : a = x 0 ;x 1 x 2 x 3 : : : x n where n is non-zero. 1 a Carry out, by putting them down, the following op-erations : 1.1 × 1.1 ; 1.2 × 1.2 ; 1.3 × 1.3 ; . . . ; 1.9 × 1.9 b Justify that the square of a decimal number with a single digit in the decimal part cannot be an integer. 2 Deduce that a cannot be a decimal number. E.779 We wish to show that 2 is an irrational number. That is, the number 2 cannot be written as an irreducible quotient a b where a and b would be two integers. We will then note 2 = ∈ Q . We will reason by the absurd . Assume the existence of two integers a and b verifying the equality 2= a b and such that this fraction is irreducible. And we’re going to develop our reasoning until we reach a contradiction, thus defeating our initial hypothesis. 1 Show that a and b verify the equality: a 2 =2 b 2 2 Deduce that a is an even integer. Thus, there exists an integer c such that : a =2 × c . 3 Using question 1 , deduce the parity of b . 4 Is the fraction a b irreducible? We’ve just reached a contradiction. Since our reasoning is cor-rect, we have to question our starting hypothesis : the number 2 cannot be written as an irreducible fraction. 7. Set of real numbers R E.8023 Definitions: The following numbers are classified according to their na-ture : All positive integers or zero form the set of natural numbers denoted by N . All positive integers, zero, and negative integers form the set of relative numbers denoted by Z . All numbers that can be written as a decimal fraction (whose denominator is a power of 10 ) form the set of decimal numbers denoted by D . All numbers that can be written as a quotient of two integers form the set of rational numbers noted Q . All existing numbers form the set of real numbers denoted by R . Connect each number to the smallest set to which it belongs : https://chingmath.fr 2√32√252?1 ?√42√1;442?2 chapExoCorrec/265 sacados/265 chapExoCorrec/779 sacados/779 Utilise l'expression des nombres pairs chapExoCorrec/8023 sacados/8023 0;65−3√243NZDQR
N1055Z-1-4-101D0,25-5,72,4Q1357211R√2ı2√ 28−7ı3−6−2−43−32NZDQR a∞a∞−∞a−∞aababababxax>axax<aaxbax<ba<xba<x<bl’ensembledesnombressupérieurouégalàal’ensembledesnombresstrictementsupérieuràal’ensembledesnombresinférieurouégalàal’ensembledesnombresstrictementinférieureàal’ensembledesnombressupérieursouégalàaetinférieurouégalàbl’ensembledesnombressupérieursouégalàaetstrictementinférieuràbl’ensembledesnombresstrictementsupérieursàaetinférieurouégalàbl’ensembledesnombresstrictementsupérieursàaetstrictementinférieuràb∞a∞a−∞a−∞aabababab E.8028 Vocabulary : Below are the five most well-known sets of numbers : the set of natural numbers ( N ) , the set of rela-tive numbers ( Z ) , the set of decimal numbers ( D ) , the set of rational numbers ( Q ) , the set of real numbers ( R ) , Match each of the numbers below to the smallest set to which it belongs : E.8315 For each of the numbers below, give the smallest set of numbers to which it belongs : a − 4 + 2 × 5 2 b − 9 + 8 4 c 1 ı d 8 × 2 − 2 3 E.11407 Among N , Z , D , Q , and R , indicate for each number the smallest set to which it belongs : a − 56 8 b √ 64 c 12 9 d √ 2 e 6 ; 25 E.11429 Among N , Z , D , Q , and R , indicate for each number the smallest set to which it belongs : a 24 6 b − 72 8 c √ 3 d √ 64 e 0 ; 49 E.269 For each of the numbers below, de-termine the smallest set of numbers to which it belongs : a 3 4 b 5 3 c 0.3 2.4 d 5.1 1.7 e 18 f 121 g 24 6 h 1.44 E.1726 Nombre Nature On écrit 1 Entier naturel 1 ∈ N − 5 − 5 ∈ − 3.12 − 3.12 ∈ 1 3 1 3 ∈ 4 5 4 5 ∈ 2 2 ∈ 2+1 2 − 1 2 2+1 2 − 1 2 ∈ 3 6 × 4 4 × 15 2 3 7 × 2 3 3 6 × 4 4 × 15 2 3 7 × 2 3 ∈ E.9528 Give the nature of each of the fol-lowing numbers : a 2 b 4 × 10 10 c 6 2 − 3 2 d − 5 2 e 3 × 10 5 × 14 × 10 12 21 × 10 4 f 7 − 3 7+ 3 E.278 For each of the numbers below, indi-cate the smallest set to which it belongs (indicate your calcu-lations if necessary) : a 1 + 1 3 b 5 3 2 9 c 2 d 7 500 e 2 12 f 1+ ı g 1+ 2 2 h cos 60 o 2 8. Interval E.8314 Definition: An interval is any subset of numbers that can be defined by the delimitation of one or two numbers called its boundaries . Here are the different types of intervals and their notations : https://chingmath.fr chapExoCorrec/8028 sacados/8028 N1055Z-1-4-101D0,25-5,72,4Q1357211R√2ı2√ 28−7ı3−6−2−43−32NZDQR chapExoCorrec/8315 sacados/8315 chapExoCorrec/11407 sacados/11407 chapExoCorrec/11429 sacados/11429 chapExoCorrec/269 sacados/269 chapExoCorrec/1726 sacados/1726 chapExoCorrec/9528 sacados/9528 chapExoCorrec/278 sacados/278 chapExoCorrec/8314 sacados/8314 a∞a∞−∞a−∞aababababxax>axax<aaxbax<ba<xba<x<bl’ensembledesnombressupérieurouégalàal’ensembledesnombresstrictementsupérieuràal’ensembledesnombresinférieurouégalàal’ensembledesnombresstrictementinférieureàal’ensembledesnombressupérieursouégalàaetinférieurouégalàbl’ensembledesnombressupérieursouégalàaetstrictementinférieuràbl’ensembledesnombresstrictementsupérieursàaetinférieurouégalàbl’ensembledesnombresstrictementsupérieursàaetstrictementinférieuràb∞a∞a−∞a−∞aabababab
−14 −14 −14 −14 −3 −14 1 -∞∞−41 -∞∞−4 -∞∞02 ABCD 1 Which of the intervals below represents the set of num-bers x satisfying the inequality: − 1 x< 4 : a − 1 ; 4 b − 1 ; 4 c − 1 ; 4 d − 1 ; 4 2 Which of the intervals given below represents the set of numbers x satisfying the inequality x> 4 : a −∞ ; 4 b −∞ ; 4 c 4 ; + ∞ d 4 ; + ∞ E.4202 Four sets of numbers are represented below on a graduated line: a b c d For each of these sets of numbers, associate the frame that is verified by all the numbers in the set : 1 − 1 x 4 2 − 1 <x < 4 3 − 1 x < 4 4 − 1 <x 4 E.4201 On each line below, a set of numbers is represented : a b c Use an interval to describe each of these sets. E.6512 Copy the missing information on your copy: − 4 x < 1 a b c x < 2 d − 3 <x 1 9. Interval and membership E.316 Fill in the blanks using the symbols ∈ and = ∈ : a 3 : : : 0 ; 5 2 b 0 ; 33 : : : 1 3 ; 1 c − 3 : : : 2 ; 4 d 1 : : : − 0 ; 2 ; 3 E.11430 Fill in the blanks with the symbols ∈ and = ∈ : a − 2 ::: − 5 ; − 3 b 1 ; 41 ::: 1 ; 5 ; 4 c 8 9 ::: 1 ; 1 ; 5 d −√ 3 ::: − 6 ; − 1 ; 7 E.11408 Fill in the blanks with the symbols ∈ and = ∈ : a 3 ::: 2 ; 4 b − 3 ; 2 ::: − 7 ; − 4 c 5 3 ::: 2 ; 6 ; 5 d −√ 2 ::: − 3 ; − 1 ; 2 E.8571 Using the symbols for membership ( ∈ ) and non-membership ( ∈ ) , indicate the numbers belonging to the interval [ − 2; 1] : 0 ; − 2 ; 3 ; 4 3 ; ı 4 E.9328 Complete the blanks with the sym-bols ∈ or ∈ : a ı : : : ]3.14; 5] c 2 : : : [2; 3] b ı : : : 0.5 ; 3.1 d ı : : : 3.1 ; 4 e 1 3 : : : 0 ; 0.33 E.311 Copy and complete with the symbol of membership ( ∈ ) and non-membership the following lines : a 2 ::: ]1 ; 3[ b 2 2 ::: [ 2 ; 5] c 1 −√ 11 √ 11 ::: ] −∞ ; 0[ d 16 4 : : : − 4 ; 4 10. Inclusion of intervals E.9475 Below is the universe of outcomes Ω of a random experiment and four such events A , B , C and D https://chingmath.fr chapExoCorrec/4202 sacados/4202 −14 −14 −14 −14 chapExoCorrec/4201 sacados/4201 −3 −14 1 chapExoCorrec/6512 sacados/6512 -∞∞−41 -∞∞−4 -∞∞02 chapExoCorrec/316 sacados/316 chapExoCorrec/11430 sacados/11430 chapExoCorrec/11408 sacados/11408 chapExoCorrec/8571 sacados/8571 chapExoCorrec/9328 sacados/9328 chapExoCorrec/311 sacados/311 chapExoCorrec/9475 sacados/9475 ABCD
-4-3-2-1012345 -4-3-2-1012345 -4-3-2-1012345 -4-3-2-1012345 R−413 R−3;5−101 R−2;5ı√21 R−4103 Using the symbol ⊂ , write the inclusion relationship induced by the above digram. E.601 Say whether the following inclusions are true or false : a 3 ; 17 ⊂ −∞ ; 4 b − 2 3 ; 2 2 ⊂ − 1 ; 1 2 E.337 Give two real numbers a and b veri-fying the following two conditions : b − a = 1 and [ a ; b ] ⊂ 3 4 ; 5 2 . 11. Meeting and intersection of intervals E.5243 1 a Show, on the graduated line below, the two intervals − 1 ; 3 and 0 ; 4 : b Give the interval obtained by joining the intervals − 1 ; 3 and 0 ; 4 . 2 a Show, on the graduated line below, the two intervals − 3 ; 1 and − 1 ; 2 : b Give the interval obtained by the intersection of the intervals − 3 ; 1 and − 1 ; 2 . E.8394 1 a Represent, on the graduated line below, the two in-tervals 0 ; 4 and − 2 ; 5 : b Give the interval obtained by joining the intervals 0 ; 4 and − 2 ; 5 . 2 a Show, on the graduated line below, the two intervals − 2 ; 0 and − 1 ; 2 : b Give the interval obtained by the intersection of the intervals − 2 ; 0 and − 1 ; 2 . E.8425 Give the simplified expression for each of the sets below : a 2 ; 5 ∪ 0 ; 4 b − 1 ; 2 ∩ 3 ; 5 c 2 ; 4 ∩ − 1 ; 3 E.298 In each case, plot the two intervals on a graduated line. Then determine their intersections and meetings : a 0 ; 2 ; 1 ; 3 b 0 ; 2 ; 2 ; 3 c 0.33 ; 2] ; 0.5 ; 1 E.8168 1 Give, if possible, a simplified expression for the following interval unions : a 3 ; 5 ∪ 0 ; 4 b − 3 ; 3 ∪ − 2 ; 2 c − 1 ; 2 ∪ 4 ; 7 2 Give the expression for interval intersections : a 3 ; 5 ∩ 0 ; 4 b − 3 ; 3 ∩ − 2 ; 2 c − 1 ; 2 ∩ 4 ; 7 E.1935 Before performing the operation on the requested intervals, plot each of the two intervals on a graduated line, then give the resulting set. a 2 ; 5 ∪ − 1 ; 7 b 3 ; + ∞ ∪ 0 ; 3 ∪ 3 c 2 ; 5 ∩ − 1 ; 7 d −∞ ; 3 ∩ 3 ; + ∞ E.9347 For each question, plot the resulting set on a graduated line, then, if possible, give a simplified form of this set. a [ − 1 ; 1] ∪ [1 ; 4] b [1 ; 4] ∪ [ − 4 ; − 1] c [4 ; 5] ∩ [ − 1 ; 4] d [ − 1 ; 1] ∩ [2 ; 3] E.1794 Shown below are subsets of R : by hatching the intervals making up this subset ; by marking with a cross the isolated points belonging to it. Using set notation, describe each of these subsets : a b c d E.1905 Represent each of the sets below on a graduated line and give their algebraic form : a − 1 ; ı ∪ 2 ; 5 b −∞ ; 2 ∪ − 1.5 ; + ∞ c − 2 ; 8 ∩ −∞ ; 3 d −∞ ; − 3 ∩ − 3 ; + ∞ https://chingmath.fr chapExoCorrec/601 sacados/601 chapExoCorrec/337 sacados/337 chapExoCorrec/5243 sacados/5243 -4-3-2-1012345 -4-3-2-1012345 chapExoCorrec/8394 sacados/8394 -4-3-2-1012345 -4-3-2-1012345 chapExoCorrec/8425 sacados/8425 chapExoCorrec/298 sacados/298 chapExoCorrec/8168 sacados/8168 chapExoCorrec/1935 sacados/1935 chapExoCorrec/9347 sacados/9347 chapExoCorrec/1794 sacados/1794 R−413 R−3;5−101 R−2;5ı√21 R−4103 chapExoCorrec/1905 sacados/1905
-5-4-3-2-1012345678 E.8632 For each question, plot the resulting set on a graduated line, then, if possible, give a simplified form of this set. a [1 ; 2] ∪ 3 2 ; 14 8 b − 2 ; 5 4 ∩ [1 ; 100] E.2711 Simplify the writing of the following sets : a −∞ ; 3 ∩ − 2 ; 5 b 5 2 ; 10 ∩ 3 ; ı c − 12 5 ; 3 ∪ − 3 ; 9 4 E.8546 For each pair of intervals, give the result set of their intersection and union : a 1 ; 6 et 3 ; 8 b − 2 ; 1 3 et 1 3 ; 5 c −∞ ; ı et 1 ; + ∞ E.859 Determine the set of numbers simul-taneously realizing the two inequalities below and represent this set on a graduated line: 2 x + 4 < 3 x − 2 3 x − 5 < 2 x + 2 12. Absolute values and distance E.354 Definition: Let d ( x ; y ) denote the distance, on a grad-uated line, between the respective abscissa points x and y . Note: As a direct consequence, we have the following prop-erty: for any numbers x and y : d ( x ; y ) = d ( y ; x ) 1 Calculate the distances indicated below : a d (5 ; 2) = : : : : : : b d (1 ; 7) = : : : : : : c d (0 ; 5) = : : : : : : e d ( − 2 ; 5 ; 0) = : : : : : : f d ( − 1 ; 5) = : : : : : : g d ( − 3 ; − 4) = : : : : : : h d ( − 1 ; − 5) = : : : : : : i d (4 ; − 2) = : : : : : : j d ( − 2 ; 5 ; − 1 ; 5) = : : : : : : You can use the graduated ruler below : 2 a Complete the following table : x y x − y d ( x ; y ) 5 2 3 7 − 2 5 1 − 3 − 1 − 6 b Compare x − y et d ( x ; y ) ? 13. Absolute values E.321 Algebraically, calculate the following expressions : a | 2 − 3 | b | 5 + 3 | c | 2 × (4 − 5) | d | 4 × 2 − 5 × 7 | e | 7 + 2 |×| 4 − 6 | f | 2 − 3 |× 2 g | 5.5 | + |− 5.5 | h |− 5.5 | − | 4.5 | i | 2 × 3 − 7 | E.11409 Perform the following calculations : a ⏐ ⏐ − 5 ⏐ ⏐ − ⏐ ⏐ 7 ⏐ ⏐ − 6 b ⏐ ⏐ 2 × 3 − 4 × 5 ⏐ ⏐ − ⏐ ⏐ 3 − 3 × 2 ⏐ ⏐ E.11431 Perform the following calculations : a ⏐ ⏐ − 3 ⏐ ⏐ − 10 − ⏐ ⏐ 3 ⏐ ⏐ b ⏐ ⏐ 5 − 5 × 4 ⏐ ⏐ − ⏐ ⏐ 3 × 2 − 6 × 4 ⏐ ⏐ E.1595 Perform the following calculations : A 2 ×| 3 × 2 − 7 | − | 5 − 3 | b | 3 × 2 − 4 |×| 3 − 5 | c | 8 − 11 × 2 | | +5 | + |− 5 | d | 2 × 4 − 7 | | 3 × 3 − 12 | E.1936 Perform the following calculations : a 2 × ⏐ ⏐ ⏐ 3 × 1 4 − 2 ⏐ ⏐ ⏐ + 1 b | 3 | + |− 3 | ⏐ ⏐ ⏐ 2 − 1 3 ⏐ ⏐ ⏐ c ⏐ ⏐ ⏐ 2 ×| 2 × 5 − 12 | − 7 ⏐ ⏐ ⏐ E.11290 Perform the following calculations : a ⏐ ⏐ 2 − 3 × 2 ⏐ ⏐ − ⏐ ⏐ 2 − 5 2 ⏐ ⏐ b ⏐ ⏐ 5 × 2 − 2 × 9 ⏐ ⏐ ⏐ ⏐ 5 × 2 − 2 ⏐ ⏐ E.4376 Perform the following calculations : a ⏐ ⏐ ⏐ | 5 − 4 | + | 4 − 5 | ⏐ ⏐ ⏐ b ⏐ ⏐ ⏐ 2 ×| 3 − 5 | + 2 ⏐ ⏐ ⏐ − 5 E.9329 Perform the following calculations : a | 2 − 3 | b | 3 − ı | c | ı − 4 | https://chingmath.fr chapExoCorrec/8632 sacados/8632 chapExoCorrec/2711 sacados/2711 chapExoCorrec/8546 sacados/8546 chapExoCorrec/859 sacados/859 chapExoCorrec/354 sacados/354 -5-4-3-2-1012345678 chapExoCorrec/321 sacados/321 chapExoCorrec/11409 sacados/11409 chapExoCorrec/11431 sacados/11431 chapExoCorrec/1595 sacados/1595 chapExoCorrec/1936 sacados/1936 chapExoCorrec/11290 sacados/11290 chapExoCorrec/4376 sacados/4376 chapExoCorrec/9329 sacados/9329
EntermededistanceEncadrementValeurabsolueIntervalleDroitegraduéeLadistancedexà2estinférieureouégaleà3−1x5|x−2|3x−15-153<x<7x−41|x1|<1 14. Center of an interval and equation E.9332 1 Which points are at a distance of 5 from the number 3 ? 2 Solve the equation : ⏐ ⏐ x − 3 ⏐ ⏐ = 5 E.332 Solve the following equations : a | x | = 3 b | x − 2 | = 3 c | x − 4 | = 7 d | x + 2 | = 3 e | x − 4 | = 0 f | x − 2 | = − 1 E.330 Translating the following equations into a distance problem, give the set of solutions to the equa- tions. Example: | x + 2 | = 3 translates to d ( x ; − 2) = 3 a | x − 4 | = 3 b | x + 2 | = 1.5 c | x − 5 | = ı d | x + 5 | = 2 e | x − 5 | = | x − 1 | f | x + 2 | = | x − 2 | E.11292 Give the two numbers that solve the equation : ⏐ ⏐ 3 × x − 5 ⏐ ⏐ = 4 15. Center of an interval and inequation E.9331 1 Solve the equation : ⏐ ⏐ x − 3 ⏐ ⏐ =5 2 a Express, in interval form, the set of numbers x veri-fying the relationship : ⏐ ⏐ x − 3 ⏐ ⏐ 5 b What relationship can be established between the num-ber 3 and the ends of the solution interval obtained in question a ? E.8287 1 Give the center of each of the intervals : a 5 ; 9 b − 2 ; 6 c 0 ; 4 2 Complete the blanks : a X ∈ 5 ; 9 = ⇒ d ( x; 7) ::: b X ∈ − 2 ; 6 = ⇒ d ( x;::: ) 4 c X ∈ 0 ; 4 = ⇒ d ( x;::: ) ::: E.349 Fill in the blanks : 1 equals 2 equals 3 equals 4 equals 5. E.9330 Complete the blanks below : a ⏐ ⏐ x − 3 ⏐ ⏐ 2 = ⇒ x ∈ 1 ; : : : b ⏐ ⏐ x − 5 ⏐ ⏐ 1 = ⇒ x ∈ : : : ; : : : c ⏐ ⏐ x + 1 ⏐ ⏐ 2 = ⇒ x ∈ : : : ; : : : E.352 Complete the following table line by line. 16. Unlimited development E.68 Determine the fractional expressions corresponding to the following infinite decimal expansions : a x = 0 ; 7 1 b y = 1 ; 2 17 https://chingmath.fr chapExoCorrec/9332 sacados/9332 chapExoCorrec/332 sacados/332 chapExoCorrec/330 sacados/330 chapExoCorrec/11292 sacados/11292 chapExoCorrec/9331 sacados/9331 chapExoCorrec/8287 sacados/8287 chapExoCorrec/349 sacados/349 chapExoCorrec/9330 sacados/9330 chapExoCorrec/352 sacados/352 EntermededistanceEncadrementValeurabsolueIntervalleDroitegraduéeLadistancedexà2estinférieureouégaleà3−1x5|x−2|3x−15-153<x<7x−41|x1|<1 chapExoCorrec/68 sacados/68
AB E.67 1 Give the fractional entries for the following decimal de-velopments : A = 0.1 7 ; B = 0.784 84 2 Justifying your approach, give the unlimited expansion of the fractional number B = 5 6 . 17. Share E.9531 Perform the following calculations and give the result in simplified form : a 1 2 + 1 2 × 5 3 b 1 4 + 3 10 × 5 2 c 1 9 + 2 6 16 3 Hint: the details of the calculation steps will be taken into account during the evaluation. 18. Unclassified exercises E.312 1 Translate the following equations in terms of distance and give their solutions : a | x +2 | =5 b | x − ı | = 2 c | x − 2 | = | x +2 2 | 2 Solve the following equations algebraically: a | x − 3 | = 1 b | x − 3 | = 3 c | 2 x + 1 | = | 3 x − 4 | 3 In each case, represent on a graduated line the solutions of the following inequations : a | x + 2 | > 2 b | x − 3 | 5 c | 2 x + 1 | > − 1 E.322 1 In terms of distance, what must x verify in the equation | x − 10 | =5 ? Give then the possible values of x . 2 In terms of distance, what must x verify in the equation | x − 10 | 5 ? Then give the possible values of x . E.308 We want to compare the following numbers : A = 513 6103515625 ; B =0 ; 000 000 084 049 9 ; C = 1 11897691 1 Compare these three numbers using your calculator. Make a guess. 2 a Using the calculator, find the value of 5 14 . b Show that : A = 8 404 992 10 14 c What is the nature of the number A ? 3 We assume the following statement : Proposition: An irreducible fraction a b is a decimal num-ber if and only if its denominator can be factored into prime factors of the form 2 m × 5 n , where m and n are natural num-bers. Justify that the number C is not a decimal number. 4 Revisit your conjecture from question 1 . E.9476 Let a , b , c , d and e be five numbers distinct in pairs. Consider the following sets : A = a ; c ; d ; B = c ; e ; C = a ; d Determine the expression of the following sets : a A ∩ B b A ∪ B c B ∩ C d A ∩ B ∩ C E.9477 Definition: we call the cardinal of a set A the number of elements making up this set. We note this number card ( A ) Consider Ω the set of outcomes of a random experiment and A and B two events in this universe. 1 Give the number of elementary events making up this random experiment. 2 a Determine the value of the following numbers : card ( A ) ; card ( B ) ; card ( A ∪ B ) ; card ( A ∩ B ) b Which formula is found? 3 Determine the value of the following numbers : card ( A ∩ B ) ; card ( A ∪ B ) ; card ( A ∩ B ) https://chingmath.fr chapExoCorrec/67 sacados/67 chapExoCorrec/9531 sacados/9531 chapExoCorrec/312 sacados/312 chapExoCorrec/322 sacados/322 chapExoCorrec/308 sacados/308 fichierPlus/308/ Cet exercice est fait pour une calculatrice avec un affichage de 10 chiffres. chapExoCorrec/9476 sacados/9476 chapExoCorrec/9477 sacados/9477 AB