Question

Prosze rozwiązać te zadania

Answer 1

1.

a)

licznik ---- (5 i 3/4 - 2 i 3/7) * 14/3 = (5 i 21/28 - 2 i 12/28) * 14/3 = 3 i 9/28 * 14/3 = 93/28 * 14/3 = 31/2

mianownik --- (4 i 1/5 + 3 i 2/9) : 167/15 = (4 i 9/45 + 3 i 10/45) * 15/167 = 7 19/45 * 15/167 = 334/45 * 15/167 = 2/3

 

licznik  / mianownik = (31/2) / (2/3) = 31/2 : 2/3 = 31/2 * 3/2 = 93/4 = 23,25

 

b)

licznik ---- (9/4 : 5/2)^2 + 1/6 = (9/4 * 2/5)^2 + 1/6 = (9/10)^2 + 1/6 = 81/100 + 1/6 = 486/600 + 100/600 = 586/600 = 293/300

mianownik --- 3/5 + 2/5 : 3/10 = 3/5 + 2/5 * 10/3 = 3/5 + 4/3 = 9/15 + 20/15 = 29/15

 

licznik / mianownik ---- 293/300 : 29/15 = 293/300 * 15/29 = 293/580

 

3.

a)    A= {0, 2, 4, 6, 6, 8, 10, 12}                  B= {0,4, 5, 6, 7, 8, 11, 12}

A u B = {0, 2, 4, 5, 6, 7, 10, 11, 12}

A ∧ B = {0, 4, 6, 8, 12}

B \ A = {7, 11}

A \ B = { 2, 10}

 

b)    A= { xC: x2  ≤ 36}  = <-6, 6>                    B= { x C: x2 > 16} = (-oo, -4) u (4, +oo)

A u B = R

A ∧ B = <-6, -4) u (4, 6>

A \ B =<-4, 4>

B \ A =(-oo, -6) u (6, +oo)

 

4.

a)

             | ----------\

----------o-------------o---------->

            -3               5             -3 zamalowane kołko, 5 nie zamalowane

 

b)

                  / -----------|

--------------o-------------o------------>

                 -2               4        -2 - nie zamalowane kołko, 4 - zamalowane

 

c)

                 /------------\

--------------o-------------o----------->

                2                 7            kołka nie zamalowane

 

d)

  <-----------------------|

--------------------------o--------------->

                                2               kółko zamalowane

 

e)

                 /---------------------->

-------------o----------------->

                 3           kółko nie zamalowane

f)

           |------------->

---------o------------->

          -4       kołko zamalowane

 

5.

a)

A= (1; 4> ,  B= <3, 6>

A u B = (1, 6>

A ∧ B = <3, 4>

A \ B = (1, 3)

B \ A = (4, 6>

 

b)

A = <-6, 2) , B= (0, 4)  

A u B = <-6, 4)

A ∧ B = (0, 2>

A \ B = <-6, 0>

B \ A = <2, 4)

 

c)

A= (-1, 2> u <4, 7>       B = <1, 3)

A u B = (-1, 2> u <4, 7>

A ∧ B = {Ф}

A \ B = A

B \ A = B

 

d)

A= <-1, 2)       B= <-3, 0> u <1, 4)

A u B = <-3, 4)

A ∧ B = (-1, 0> u <1, 2)

A \ B = (0, 1)

B \ A = <-3, -1> u <2, 4)

 

6.

a)

|x - 3| = 4  

x - 3 = 4  lub  x - 3 = -4

x = 4 + 3   lub   x = -4 + 3

x = 7   lub      x = -1

 

b)

|1/2x + 4| = 2

1/2x + 4 = 2   lub   1/2x + 4 = -2

1/2x = -2       lub    1/2x = -6

x = - 4          lub        x = -12

 

c)

|8 - x| = 1

8 - x = 1    lub   8 - x = -1

-x = 1 - 8    lub    -x = -1 - 8

-x = -7        lub       -x = -9

x = 7          lub        x = 9

 

d)

|x + 2| = -2

Równanie nie posuada rozwiazania, gdyż wartość bezwzględna zawsze w wyniku daje nam liczbe dodatnią, a tu mamy (-2).

 

e)

|x + 5| > 3

x + 5 > 3  lub   x + 5 < - 5

x > -2      lub      x < -10

x ∈ (-oo, -10) u (-2, +oo)

 

f)

|x - 3| ≤ 4

x - 3 ≤ 4   i x - 3 ≥ -4

x ≤ 7       i     x ≥ -1

x ∈ <-1, 7>

 

g)

|x + 2| < 1

x + 2 < 1    i x + 2 > - 1

x < -1       i     x > -3

x ∈ (-3, -1)

 

h)

|2x - 3| ≥ 5

2x - 3 ≥ 5    lub    2x - 3 ≤ - 5

2x ≥ 8         lub       2x ≤ -2

 x ≥ 4          lub        x ≤ -1

x∈ (-oo, -1> u <4, +oo)

 

7.

a) ... = √(2 * 8) = √16 = 4

b) ... = √25 * √49 = 5 * 7 = 35

c) ... = √(125 : 5) = √ 25 = 5

d) ... = ∛8 = 2

e) ... = ∛(24 : 3) = ∛8 = 2

f) ... = √100 - √64 = 10 - 8 = 2

g) ... = ∛(81 : 3) + √(27 * 3) = ∛27 + √81 = 3 + 9 = 12

h) ... = √[18∛(2 * 4)] = √[18∛8 ] = √(18 * 2) = √36 = 6

 

9.

15 000zł -------- 100%

x zł -------------- 9%

x = (15 000 * 9) / 100 = 150 * 9 = 1350zł  -----  odsetki

 

5 * 1350zł = 6 750zł

 

15 000zł + 6 750zł = 21 750zł  ------  odpowiedź

 

11.

a) ... = 5a * 5a = 25a²

b) ... = (1/2 * 4)^5 = 2^5 = 32

c) ... = [ 3 : 1/5] : 5]^4 = (3 * 5 * 1/5)^4 = 3^4 = 81

d) ... = (9/4 * 4)^5 : 3^5 = 9^5 : 3^5 = (9 : 3)^5 = 3^5 = 243

e) ... = [3^6 * (3^2)^5 ] : (3^3)^4 = [3^6 * 3^10) : 3^12 = 3^16 : 3612 = 3^4 = 81

f) ... = 1/2 + 3^2 = 1/2 + 9 = 9,5

g) ... = [1/3 * (-3)] : 9 = -1/9

 

12.

a) 3 / √6 = 3√6/6 = √6/2

b) 5 / (3√10) = 5√10 / (3 * 10) = 5√10 / 30 = √10 / 6

c) 3 / √3 = 3√3 / 3 = √3

d) 6√2 / √3 = (6√2 * √3) / 3 = 6√6/ 3 = 2√6