Explanation

If |A| ≠ 0, then matrix A is Non-singular (also known as invertible).

Explanation

The notation n√x^m can be rewritten in power form as:

(x^m)^(1/n) = x^(m/n)

Explanation

Using the change of base formula:

log_b^a / log_b^d = log_d^a

Explanation

Let's expand both expressions:

(a + b)² = a² + 2ab + b²

(a - b)² = a² - 2ab + b²

Now, add them:

(a + b)² + (a - b)² = a² + 2ab + b² + a² - 2ab + b²

= 2a² + 2b²

Explanation

Let's use the Pythagorean theorem:

Height of pole (perpendicular) = 12m

Length of string (hypotenuse) = 20m

Distance between foot of pole and end point of string (base) = √(hypotenuse² - perpendicular²)

= √(20² - 12²)

= √(400 - 144)

= √256

= 16m

Explanation

Given:

x̄ (mean) = 12

∑x (sum) = 132

The formula for mean is:

x̄ = ∑x / n

Rearrange to find n:

n = ∑x / x̄

= 132 / 12

= 11

Explanation

Cost price of 1 apple = 34 ÷ 8 = Rs. 4.25

Cost price of 12 apples = 4.25 × 12 = Rs. 51

Selling price of 12 apples = Rs. 57

Profit = 57 − 51 = Rs. 6 per dozen.

Explanation

Work = Persons × Days = 3 × 4 = 12 person-days

With 4 persons: 12 ÷ 4 = 3 days.

Explanation

Explanation

10% of 5400 = 540

5400 - 540 = 4860