Explanation

A sequence is a function whose domain is the set of natural numbers (1, 2, 3, ...).

Each natural number input gives a specific term in the sequence.

Explanation

The ratio test and root test are inconclusive when the limit of the ratio (or root) of successive terms is exactly 1.

These tests fail to determine convergence or divergence in this case.

Explanation

A quadratic equation has the form ax² + bx + c = 0, and it typically has two roots. 

These roots can be real or complex, and they can be distinct or repeated (in the case of a double root).

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

Mean Value Theorem applies to functions that are continuous on a closed interval [a, b].

And differentiable on the open interval (a, b).

Explanation

A parabola is defined as the set of points equidistant from a fixed point (focus) and a fixed line (directrix).

Its eccentricity (e) = 1, which distinguishes it from ellipses (e < 1) and hyperbolas (e > 1).

Explanation

To find the center of the circle, rewrite the equation in standard form:

45x² - 60x + 45y² + 36y = -19

Divide by 45:

x² - (4/3)x + y² + (4/5)y = -19/45

Complete the square:

(x - 2/3)² + (y + 2/5)² = ...

The center is (2/3, -2/5).

Explanation

The rate of change of cost with respect to the number of units produced is called Marginal Cost.

It represents the additional cost of producing one more unit.

Explanation

Prime Minister Muhammad Ali Bogra presented the Bogra Formula in 1953 to resolve differences between East and West Pakistan over constitutional representation, particularly in the national legislature.

It aimed to provide parity between the two wings of the country.

Explanation

The resolution passed on April 19, 1946, at the Delhi convention, clarified doubts about the word 'states' in the original Lahore Resolution

It demanded `an unequivocal understanding be given to implementing the establishment of Pakistan without delay’.

NA23-7-2023