Explanation

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

Median is the middle value when data is arranged in order.

Ordered data: 63, 65, 66, 67, 69 → middle value is 66.

Explanation

Given a right-angled triangle with a base angle of 45°, it's an isosceles right triangle, meaning the base and perpendicular are equal.

Base length = 3 dm

Perpendicular length = 3 dm

Explanation

Step 1: Calculate the mean of the marks.

Mean = (6 + 8 + 10 + 12 + 4 + 20) / 6

= 60 / 6

= 10

Step 2: Calculate the mean deviation.

Mean Deviation = (|6-10| + |8-10| + |10-10| + |12-10| + |4-10| + |20-10|) / 6

= (4 + 2 + 0 + 2 + 6 + 10) / 6

= 24 / 6

= 4

Step 3: Calculate the coefficient of dispersion.

Coefficient of Dispersion = Mean Deviation / Mean

= 4 / 10

= 0.4

Explanation

A matrix has an inverse only if its determinant is non-zero, meaning it is non-singular.

Singular matrices (with determinant = 0) do not have an inverse.

Explanation

The Law of Excluded Middle states that for any proposition, either it is true or its negation is true—no third option exists.

It is a fundamental principle in classical logic.

Explanation

Chords that are equally distant from the center of a circle are always equal in length (congruent).

This is a fundamental property of chords in a circle.

Explanation

Specific death rate refers to the death rate calculated for a particular segment or subgroup of the population (e.g., by age, sex, cause).

It helps in understanding mortality patterns more precisely than the overall crude death rate.

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.