Odpowiedź:
zad 1
2x/(x + 1) = 1/(x - 1) + 1
założenie:
x +1 ≠ 0 ∧ x - 1 ≠ 0
x ≠ - 1 ∧ x ≠ 1
D: x ∈ R\ { - 1,1 }
2x/(x +1) = 1/(x - 1) + 1 | * (x + 1)(x - 1)
2x(x - 1) = x + 1 + (x + 1)(x - 1)
2x² - 2x = x + 1 + x² - 1
2x² - 2x = x² + x
2x² - 2x - x² - x = 0
x² - 3x = 0
x(x - 3) = 0
x = 0 ∨ x - 3 = 0
x = 0 ∨ x = 3
∧ - znaczy "i"
∨ - znaczy "lub"
zad 2
d - średnica koła = 16 cm
r - promień koła = d/2 = 16/2 cm = 8 cm
α - kąt środkowy = 75°
l = długość łuku = παr/180° = π * 75°/180° * 8 cm = π * 5/12 * 8 cm =
= π * 5/3 * 2 cm = 10π/3 cm
P - pole wycinka kołowego = παr²/360° = π * 75°/360° * 8² cm² =
= π * 5/25 * 64 cm² = 320π/25 cm² = 12,8π cm²
zad 3
(3ctg60° - 3cos30°)/(sin²30° + cos²45°) = (3 * √3/3 - 3 * √3/2)/[(1/2)² + (√2/2)²] =
= (√3 - 3√3/2)/(1/4 + 2/4) = (2√3 - 3√3)/2 : 3/4 = - √3/2 * 4/3 = - √3 * 2/3 =
= - 2√3/3
