Explanation

The formula ∑x/n calculates the arithmetic mean of a set of numbers, where:

∑x = sum of all values

n = number of values

This formula provides the average value of the dataset.

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

Work is done every time a force moves something over a distance. By multiplying the force by the distance traveled in the direction of the force, you can calculate the energy transferred or the work done. Energy transmitted = work completed = force x distance traveled in the direction of  force.

AGM 25-12-2022

Explanation

Given dt/dx = xy^(1/2)

Separate variables:

dt/y^(1/2) = x dx

Integrate both sides:

∫y^(-1/2) dy = ∫x dx

2y^(1/2) = (x²)/2 + C

y^(1/2) = x²/4 + C'

Square both sides:

y = (x²/4 + C')²

For simplicity, let's assume C' = 0:

y = (x²/4)² = x⁴/16

Explanation

To find the 3rd term of the sequence 

𝑎𝑛 =n+3, substitute .n=3 into the formula:

𝑎3 = 3 + 3 = 6

Thus, the 3rd term of the sequence is 6.

Explanation

Given points A(3, 1) and B(7, -5), let's calculate the distance using the distance formula:

Distance = √((x2 - x1)² + (y2 - y1)²)

= √((7 - 3)² + (-5 - 1)²)

= √((4)² + (-6)²)

= √(16 + 36)

= √52

= √(4 × 13)

= 2√13

Explanation

Let's simplify the expression:

√(12 + 2√35)

This can be rewritten as:

√(7 + 5 + 2√(7 × 5))

= √(√7 + √5)²

= √7 + √5

Explanation

Given matrix A = [1 0, 0 1]:

This is a Unit matrix or Identity matrix, where all elements on the main diagonal are 1, and the rest are 0.

Explanation

To simplify 5/(2 - √3), rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator (2 + √3):

Step 1: Multiply by the conjugate

(5 / (2 - √3)) * ((2 + √3) / (2 + √3))

= (5(2 + √3)) / ((2 - √3)(2 + √3))

Step 2: Simplify the denominator

(2 - √3)(2 + √3) = 2² - (√3)²

= 4 - 3

= 1

Step 3: Simplify the expression

(5(2 + √3)) / 1

= 10 + 5√3

Explanation

Given:

a < b

b < c

Using transitive property, if a < b and b < c, then a < c.