Question

Zadania są podane w załączniku.

Answer 1

11.

7 + x   ;   9 + 2x  ;   11 + 3x

  a1            a2              a3

 

a3 - a2 = a2 - a1

11 + 3x - (9 + 2x) = 9 + 2x - (7 + x)

11 + 3x - 9 - 2x = 9 + 2x - 7 - x

2 + x = 2 + x

x - x = 2 - 2

0 = 0

odp. Dla każdego x ∈ R ciąg ten bedzie ciagiem arytmetycznym.

 

10.

a6 = 6

a14 = 22

S42 = ?

 

a1 + 5r = 6

a1 + 13r = 22

 

-a1 - 5r = -6

a1 + 13r = 22

--------------------

          8r = 16

            r = 2

a1 + 5r = 6

a1 + 10 = 6

a1 = -4

 

a42 = a1 + 41r

a42 = -4 + 41 * 2

a42 = -4 + 82

a42 = 78

 

S42 = (a1 + a42) / 2 * n

S42 = (-4 + 78) / 2 * 42

S42 = 74/2 * 42

S42 = 74 * 21

S42 = 1 554  -------------  odpowiedź

 

 

9.

x  ;  x + 3  ;   x + 4

a1   a2            a3

a2 / a1 = a3/a2

(x + 3) / x = (x + 4) / (x + 3)

x(x + 4) = (x + 3)(x + 3)

x² + 4x = x² + 3x + 3x + 9

4x = 6x + 9

4x - 6x = 9

-2x = 9

x = -9/2

x = -4,5   ---------  odpowiedx

 

8.

a3 = 4

a4 = -2

a1 = ?

 

q = a4/a3

q = -2/4

q = -1/2

 

a3 = a1 * q^2

4 = a1 * (-1/2)^2

4 = a1 * 1/4       /*4

a1 = 16   -----------   odpowiedx

 

7.

a1 = -3

a145 = 2157

 

a145 = a1 + 144r

2157 = -3 + 144r

2157 + 3 = 144r

144r = 2160

r = 2160 : 144

r = 15   --------------   odpowiedź

 

 

6.

y = -3x + 1   ||     y = ax + b   ⇔  -3 = a

 

y = -3x + b     i    A = (-2, 1)

                                     x    y

1 = -3 * (-2) + b

1 = 6 + b

1 - 6 = b

b = -5

 

odp.  y = -3x - 5

 

5.

y = -6x + 1   _|_   y = ax + b   ⇔  -6 * a = -1

                                                           a = 1/6

y = 1/6x + b   i  A = (-6, 2)

                                  x   y

2 = 1/6 * (-6) + b

2 = -1 + b

2 + 1 = b

b = 3

 

odp.   y = 1/6x + 3

 

 

4.

A = (5, -1)   B = (-7, -3)

 

Xs = (5 - 7)/2 = -2/2 = -1

Ys = (-1 - 3)/2 = -4/2 = -2

 

S = (-1, -2)   ------------  srodek okregu

r = 8

 

(x+ 1)² + (y + 2)² = 8²

 

odp. (x+ 1)² + (y + 2)² = 64

 

3.

a)

x(x - 6)²(x + 1) = 0

x(x - 6)(x - 6)(x + 1) = 0

x = 0  ∨  x - 6 = 0   ∨  x - 6 = 0  ∨ x + 1 = 0

                x = 6                x = 6            x = -1

 

b)

x³ + 5x² - 4x - 20 = 0

x²(x + 5) - 4(x + 5) = 0

(x + 5)(x² - 4) = 0

(x + 5)(x - 2)(x + 2) = 0

x + 5 = 0 ∨  x - 2 = 0  ∨  x + 2 = 0

x = -5               x = 2              x = -2

 

2.

x^5 + 3x^4 - 4x^3 - 12x^2 =

x^4(x + 3) - 4x^2(x + 3) =

(x + 3)(x^4 - 4x^2) = x²(x + 3)(x² - 4) = x²(x + 3)(x - 2)(x + 2)