Решите графически систему уравнений {y−p2=0 {y+2p=3 выбери правильный вариант ответа: 1. p1=−1, y1=1p2=3, y2=9 2. p=1, y=1 3. p1=0, y1=0p2=2, y2=4 4. нет решений 5. p1=−3, y1=9p2=1, y2=1 6. p=0, y=0

a) (2x - y)(x + 2y) = -3

1) 2x - y = 3 => y = 2x - 3

x + 2y = -1 => x + 2(2x - 3) = - 1 =>

x + 4x - 6 = -1 => 5x = 5 => x = 1

y = 2x - 3 = 2 - 3 = -1.

Первое решение (x, y) = (1, -1)

2) 2x - y = - 3 => y = 2x + 3

x + 2y = 1 => x +2(2x + 3) = 1 =>

x + 4x + 6 = 1 => 5x = - 5 => x = -1

y = 2x + 3 = -2 + 3 = 1.

Второе решение (x, y) = (-1, 1)

3) 2x - y = 1 => y = 2x - 1

x + 2y = -3 => x + 2(2x - 1) = -3 =>

x + 4x - 2 = -3 => 5x = -1 => x = - 1/5 - нецелое.

4) 2x - y = -1 => y = 2x + 1

x + 2y = 3 => x + 2(2x + 1) = 3 =>

x + 4x + 2 = 3 => 5x = 1 => x = 1/5 - нецелое.

Всего два решения (x, y) = (1, -1) и (x, y) = (-1, 1)

б) 2x² + xy - y² = -3

x² - y² + x² + xy = -3

(x - y)(x + y) + x(x + y) = -3

(x + y)(x - y + x) = -3

(x + y)(2x - y) = -3

1) x + y = 3 => y = 3 - x

2x - y = -1 => 2x - 3 + x = -1 =>

3x - 3 = -1 => 3x = 2 => x = 2/3 - нецелое

2) x + y = 1 => y = 1 - x

2x - y = -3 => 2x - 1 + x = -3 =>

3x -1 = -3 => 3x = -2 => x = -2/3 - нецелое

3) x + y = -3 => y = -3 - x

2x - y = 1 => 2x +3 + x = 1 =>

3x + 3 = 1 = 3x = -2 => x = -2/3 - нецелое

4) x + y = -1 = y = -1 - x

2x - y = 3 => 2x +1 + x = 3 =>

3x + 1 = 3 => 3x = 2 => x = 2/3 - нецелое.

Второе уравнение не имеет решений в целых числах.

а)sin 2x=√3 cos x
2sinxcosx-√3cosx=0
cosx(2sinx-√3)=0
cosx=0⇒x=π/2+πn,n∈Z
sinx=√3/2⇒x=(-1)^n*π/3+πk,k∈Z
б)sin 2x=√2 cos x
2sinxcosx-√2cosx=0
cosx(2sinx-√2)=0
cosx=0⇒x=π/2+πn,n∈Z
sinx=√2/2⇒x=(-1)^n*π/4+πk,k∈Z в)sin(0,5п+x)+ sin 2x=0
г)cos(0,5п+x)+ sin 2x=0
-sinx+2sinxcosx=0
-sinx(1-2cosx)=0
sinx=0⇒x=πn,n∈Z
cosx=1/2⇒x=+-π/3+2πk,k∈Z
д)sin 4x+√3 sin 3x+sin 2x=0
2sin3xcosx+√3sin3x=0
sin3x(2cosx+√3)=0
sin3x=0⇒3x=πn,n∈Z⇒x=πn/3,n∈Z
cosx=-√3/2⇒x=+-5π/6+2πk,k∈Z
е)cos 3x+sin 5x=sin x
cos3x+sin5x-sinx=0
cos3x+2sin2xcos3x=0
cos3x(1+2sin2x)=0
cos3x=0⇒3x=π/2+πn,n∈Z⇒x=π/6+πn/3,n∈Z
sin2x=-1/2⇒2x=(-1)^(k+1)*π/6+πk,k∈Z⇒x=(-1)^(n+1)*π/12+πk/2,k∈Z