Question

Please help

Answer 1

54).

Let, the points are A(a, - a), B(- a, a), and C(a√3, - a√3).

The lenth of AB = distance between A and B = √[(- a - a)² + (a + a)²] = √[(- 2a)² + (2a)²] = √[4a² + 4a²] = √[8a²] = 2√2a.

The lenth of BC = distance between B and C = √[(a√3 + a)² + (- a√3 - a)²] = √[2(a√3 + a)²]= √[2(a√3 + a)²] = (√3 + 1)√2a.

The lenth of AC = distance between A and C = √[(a√3 - a)² + (- a√3 + a)²] = √[2(a√3 + a)²]= √[2(a√3 + a)²] = (√3 + 1)√2a.

So, the length of two sides BC and AC are equal, i.e, (√3 + 1)√2a units.

Since, the two sides (BC and AC) have equal lengths, it is isosceles.

 

Option C is the correct choice.

55).

The roots of the quadratic equation are - 3 + 5i and - 3 - 5i.

Then the factors are (x - (- 3 + 5i)) and (x - (- 3 - 5i)).

If (x - a) and (x - b) are two factors, then the quadratic equation is (x - a)(x - b) = 0.

So, the required quadratic equation is (x - (- 3 + 5i))(x - (- 3 - 5i)) = 0

Using foil method : (a + b)(c + d) = ac + bc + ad + bd.

x²  - x(- 3 + 5i) - x(- 3 - 5i) + [( - 3 + 5i)(- 3 - 5i)] = 0

x²  + 3x - 5ix + 3x + 5ix + [( - 3)² - (5i)²)] = 0

x²  + 3x + 3x + 9 - 25i² = 0

Substitute i² = - 1.

x²  + 3x + 3x + 9 - 25(- 1) = 0

x²  + 6x + 9 + 25 = 0

x²  + 6x + 34 = 0.

The quadratic equation is x²  + 6x + 34 = 0.

Given, options are wrong.

Correct answer is x²  + 6x + 34 = 0.

56).

The trigonometric equation is 4cos² x = 1.

Divide each side by 4.

cos² x = 1/4

⇒ cos x = ± 1/2.

• cos (x) = - 1/2.

cos (x) = cos(2π/3)

The genaral solution of cos(θ) = cos(α) is θ = 2nπ ± α, where n is an integer.

⇒x = 2nπ ± (2π/3)

If n = 0, x = 2(0)π + (2π/3) and x = 2(0)π - (2π/3) = 2π/3 and - 2π/3,

If n = 1, x = 2(1)π + (2π/3) and x = 2(1)π - (2π/3) = 2π + 2π/3 and 2π - 2π/3 = 8π/3 and 4π/3.

⇒ x = 2π/3 and x = 4π/3 in the interval [0, 2π).

• cos (x) = 1/2.

cos (x) = cos(π/3)

The genaral solution of cos(θ) = cos(α) is θ = 2nπ ± α, where n is an integer.

⇒x = 2nπ ± (π/3)

If n = 0, x = 2(0)π + (π/3) and x = 2(0)π - (π/3) = π/3 and - π/3,

If n = 1, x = 2(1)π + (π/3) and x = 2(1)π - (π/3) = 2π + π/3 and 2π - π/3 = 7π/3 and 5π/3.

⇒ x = π/3 and x = 5π/3 in the interval [0, 2π).

Therefore, the solutions of the given equation are x = π/3, x = 2π/3, x = 4π/3 and x = 5π/3 in the interval [0, 2π).

 

Option a is the correct choice.

Answer 2

Contd..........

57).

Let the points are A(- 3, 4), B(2, - 6), and C(- 1, 0).

To find the ratio of the line divided by the point, we need to use the distance formula : d = √[(x₂ - x₁)² + (y₂ - y₁)²] between the points (x₁, y₁) and (x₂, y₂).

The distance from A to C : √[(- 1 + 3)² + (0 - 4)²] = √[(2)² + (- 4)²] = √[4 + 16] = √20 = 2√5.

The distance from B to C : √[(- 1 - 2)² + (0 + 6)²] = √[(- 3)² + (6)²] = √[9 + 36] = √45 = 3√5.

The ratio from AC to BC : 2√5 : 3√5 = 2 : 3.

 

 option d is the correct choice.

58).

Let, the two vertices of the triangle are A(5, 9),  B(- 4, 1), C(x₁, y₁) and the median is M(1,1 ).

Median : M(1, 1) = [(5 - 4 + x₁)/3, (9 + 1 + y₁)/3]

(1, 1) = [(1 + x₁)/3, (10 + y₁)/3]

Equating the x and y coordinates.

1 = (1 + x₁)/3 and 1 = (10 + y₁)/3

3 = 1 + x₁ and 3 = 10 + y₁

x₁ = 3 - 1 = 2 and  y₁ = 3 - 10 = - 7

Therefore, the third vertex of the triangle C(x₁, y₁) is (2, - 7).

 

Option b is the correct choice.