Explanation

Denotative meaning is the literal, dictionary definition of a word.

It contrasts with connotative meaning, which includes emotional or cultural associations.

Explanation

We know the identity:

a³ - b³ = (a - b)(a² + ab + b²)

Here, 8x³ - 27y³ = (2x)³ - (3y)³

= (2x - 3y)((2x)² + (2x)(3y) + (3y)²)

Given 2x - 3y = 8 and xy = 2:

(2x)² + (2x)(3y) + (3y)² = 4x² + 6xy + 9y²

= 4x² + 6(2) + 9y²

= 4x² + 12 + 9y²

We need to express 4x² + 9y² in terms of 2x - 3y and xy:

(2x - 3y)² = 4x² - 12xy + 9y²

64 = 4x² - 12(2) + 9y²

64 = 4x² - 24 + 9y²

4x² + 9y² = 88

Now, 4x² + 6xy + 9y² = 88 + 12 = 100

So, 8x³ - 27y³ = (2x - 3y)((2x)² + (2x)(3y) + (3y)²)

= 8 × 100

= 800

Explanation

The Halting problem is about determining whether a given program will terminate (halt) or run indefinitely for a specific input.

It was proven by Alan Turing that there is no general algorithm to solve this problem for all possible programs.

Explanation

Over 6000 languages are spoken globally, including many with distinct dialects and regional variations.

This linguistic diversity is mainly concentrated in Asia and Africa, where countries like Papua New Guinea alone have over 800 languages.

Explanation

Expansion of metal on heating is a physical change because it does not alter the metal’s chemical composition.

When cooled, the metal returns to its original size, showing that the change is reversible.

Explanation

Explanation

"The" is used for a specific person or thing known to the speaker and listener.

"Boy in red shirt" is a specific boy, so "The" is appropriate.

Explanation

To find the fourth proportional, let's denote it as x. The proportion will be:

3 : 9 = 7 : x

We can cross-multiply:

3x = 9 × 7

3x = 63

x = 63 / 3

x = 21

Explanation

Given:

Side of square = 1200 cm

Width of strip = 10 cm

Side of larger square (including strip) = 1200 + 2(10)

= 1200 + 20

= 1220 cm

Area of larger square = 1220² = 1488400 cm²

Area of square = 1200² = 1440000 cm²

Area of strip = Area of larger square - Area of square

= 1488400 - 1440000

= 48400 cm²

Explanation

A transitive verb requires a direct object to complete its meaning — here, "closed" acts on "the door".

The verbs sat, smiled, arrived are intransitive, meaning they do not take a direct object.