Explanation

Given: A : B = 3 : 1 and B : C = 1 : 2.

To find A : B : C, make the ratio of B the same in both expressions.

A : B = 3 : 1 means A = 3B.

B : C = 1 : 2 means C = 2B.

So, A : B : C = 3B : B : 2B, which simplifies to 3 : 1 : 2.

Now, multiply the entire ratio by 5 to make the B value in all ratios equal: 3 : 5 : 10.

Explanation

If two circles touch externally, the distance between their centers is equal to the sum of their radii.

This is because the circles are externally tangent, meaning the distance from one center to the other is the total length of both radii combined.

Explanation

x + 3y = 53

4x - 2y = 2

x = 8

y = 15

Explanation

The union of two sets A and B includes all elements that are in A, B, or both.

This is represented as: A ∪ B = {x | x ∈ A or x ∈ B}.

Explanation

Using the identity:

(1 - ω + ω²)ⁿ = (-1)ⁿ ω²ⁿ

Substituting n = 7:

(1 - ω + ω²)⁷ = (-1)⁷ ω²⁷

= -1 × ω²⁷ (since (-1)⁷ = -1)

= -128ω² (since ω²⁷ = (ω²)⁷ × ω⁰ = (ω²)⁷ = 2⁷ω² = 128ω², and multiplying by -1)

The correct answer is:

-128ω²

Explanation

When 5/3 is divided, it gives 1.666..., where 6 repeats infinitely.

Since the decimal never ends but follows a repeating pattern, it is non-terminating and recurring.

Explanation

To find (A ∪ B)':

First, find A ∪ B:

A ∪ B = {2, 3, 4} ∪ {1, 3, 5}

= {1, 2, 3, 4, 5}

Now, find the complement of A ∪ B:

(A ∪ B)' = U - (A ∪ B)

= {1, 2, 3, 4, 5} - {1, 2, 3, 4, 5}

= {}

Explanation

3x means three times of x.

2y means two times of y.

The expression 3x + 2y represents the sum of three times of x and two times of y.

Explanation

Supplementary angles add up to 180°, so set up an equation:

(3x + 5) + (9x - 5) = 180°

Combine like terms:

12x = 180°

Divide both sides by 12:

x = 15°

Explanation