Explanation

Given f(x) = 6x/(3x - 2)

The denominator cannot be zero:

3x - 2 ≠ 0

3x ≠ 2

x ≠ 2/3

So, the domain of f(x) is all real numbers except x = 2/3.

The answer is R - {2/3}.

Explanation

The Axiom of Choice is a fundamental principle in set theory that states you can select an element from each set in a collection of nonempty sets, even without a specific rule.

It is essential for many proofs in mathematics but is independent of other axioms of set theory.

Explanation

A rocket accelerates in space by expelling exhaust gases backward, which, according to Newton’s third law, produces an equal and opposite force pushing the rocket forward.

This force is called thrust and does not depend on external air or gravity.

Explanation

When Thorium (Th) emits a beta particle (β⁻), a neutron converts into a proton, increasing atomic number by 1.

So Thorium (Z=90) becomes Protactinium (Z=91).

Explanation

Given the conditions:

1. √x requires x ≥ 1 (from x - 1 ≥ 0)

2. Denominator x² - 2x ≠ 0 implies x ≠ 0 and x ≠ 2

The domain is indeed x ≥ 1, excluding x = 2.

So, the domain is (1, 2) ∪ (2, ∞).

Explanation

Oscillatory systems gradually lose energy due to frictional forces (like air resistance or internal damping).

This damping reduces amplitude over time, preventing indefinite oscillation.

Explanation

Given f(t) = 1/t and g(t) = t³

To find gof(t), substitute f(t) into g(t):

gof(t) = g(f(t))

= g(1/t)

= (1/t)³

= 1/t³

Explanation

The differential equation (1 - x²)y'' - 2xy' + n(n + 1)y = 0 is known as Legendre's equation.

It arises in problems with spherical symmetry, such as solutions to Laplace’s equation in spherical coordinates.

Explanation

Given a ≠ 0, b ≠ 0, and ab = 0:

In this case, a and b are called Zero divisors.

Explanation

Zakat is payable on a minimum of 7½ or 7.5 tolas (approximately 87.48 grams) of gold.

This is the nisab threshold for gold, above which zakat must be paid at 2.5%.