Explanation

Everyone should bring their own lunch.

"Everyone": is a singular pronoun that refers to a group of people.

While grammatically "everyone" is singular, "their" is used as a gender-neutral pronoun to refer to the group.

Explanation

"brown" is an adjective that correctly describes the noun "dog."

The other options are incorrect because:

loudly – adverb (describes how the dog barked)

quickly – adverb

runs – verb

Explanation

Use "whom" when it's the object of the verb or preposition.

In this sentence, the person is receiving the book (object of the verb give), so "whom" is correct

Verb: give

Subject: You

Object: whom

Explanation

"but" is used to contrast intention with inability.

I would go, but I don’t have time.

Explanation

"I like both apples and oranges."

The phrase "both ... and ..." is a correlative conjunction pair used to show the inclusion of two items.

Other options like "but", "or", and "so" do not fit grammatically or logically here.

Explanation

1−sin2θ=cos2θ

Explanation

Factorial means multiplying all whole numbers down to 1:

$$n!=n\times (n-1)\times (n-2)\times \cdots \times 1$$

For consistency, we define:

$$0!=1$$

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Explanation

2x=1⇒x=0(since 20=1)

Explanation

The cube roots of unity are:  

1, ω, and ω², where:

- ω = -1/2 + (√3/2)i  

- ω² = -1/2 - (√3/2)i  

These are the complex cube roots of unity, and they satisfy:

1 + ω + ω² = 0  

and  

1 × ω × ω² = 1

Explanation

If $f(x) = 5$, then what is$f'(x)$ at $x = 1$?

This is a constant function, meaning:

$$f(x) = 5 quad text{for all } x$$

The derivative of a constant is always:

$$f'(x) = 0$$