Theorem | MCQs

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

Basic assumptions (also called axioms or postulates) are statements accepted as true without proof.

Theorems, on the other hand, require proof based on these assumptions.

Explanation

An angle inscribed in a semicircle is always 90° due to the Thales' Theorem.

It is called a right angle because the angle subtended by a diameter is always right.

Explanation

The total sum of all exterior angles of any triangle (or any simple polygon) is always 360°, regardless of its shape or size.

Each exterior angle of an equilateral triangle is 120°.

Explanation

This is a statement of the Triangle Inequality Theorem, which states that the sum of any two sides of a triangle must be greater than the third side.

It ensures the sides form a valid triangle.

Explanation

Using the Pythagorean theorem:

a² + b² = c²

where a and b are the lengths of the two sides, and c is the length of the hypotenuse.

Given:

a = 5 cm

b = 12 cm

Substitute the values:

5² + 12² = c²

25 + 144 = c²

169 = c²

Take the square root:

c = √169

= 13 cm

Explanation

Lagrange's Mean Value Theorem is a special case of Cauchy's Mean Value Theorem when one function is f(x) and the other is g(x) = x.

Cauchy’s theorem is more general and applies to two functions.

Explanation

A semicircle is half of a circle.

A central angle which is subtended by a semicircle has a measure of 180°.

A circle is associated with a complete rotation, which is 360°.

A semi-circle is associated with half of a rotation which is 180°.

If an angle is inscribed in a semicircle, it will be half the measure of a semicircle (180 degrees), therefore measuring 90 degrees.

Explanation

The maximum power transfer theorem states that maximum power is delivered to the load when the load resistance 𝑅𝐿 is equal to the source (or internal) resistance Rs of the power source.

Mathematically, this can be expressed as:

𝑅𝐿 =𝑅𝑠

When the load resistance matches the source resistance, the power transferred to the load is maximized.