Explanation

The number of diagonals in a polygon can be calculated using the formula:

n(n-3)/2

where n is the number of sides.

For a 12-sided polygon (dodecagon):

n = 12

Number of diagonals = 12(12-3)/2

= 12(9)/2

= 108/2

= 54

Explanation

Given ∫12x⁻⁵ √(-3x⁻⁴ + 1) dx

Let u = -3x⁻⁴ + 1

du/dx = 12x⁻⁵

du = 12x⁻⁵ dx

∫√u du = ∫u^(1/2) du

= (2/3)u^(3/2) + C

Substituting back u = -3x⁻⁴ + 1:

= (2/3)(-3x⁻⁴ + 1)^(3/2) + C

Explanation

The function y = x + 100 is in the slope-intercept form y = mx + c, where:

m = 1 (slope)

c = 100 (y-intercept)

This confirms that the graph of the function is a straight line.

Explanation

- Magnetic Induction (B) is also called Magnetic Flux Density.  

- It represents the strength of the magnetic field over a given area.  

- Unit: Tesla (T) → 1 T = 1 Weber/m²  

- Formula: B = Φ / A  

  (Where Φ = magnetic flux, A = area)  

- It indicates how much magnetic field passes through a surface.  

- It's a vector quantity (has both magnitude and direction).  

- Higher B → stronger magnetic field at that point.

Explanation

Given y = 2/x

Let's find the first and second derivatives:

dy/dx = -2/x²

d²y/dx² = 4/x³

Now, let's evaluate y + 1/2 * d²y/dx²:

y + 1/2 * d²y/dx² = 2/x + 1/2 * 4/x³

= 2/x + 2/x³

= (2x² + 2)/x³

= 2(x² + 1)/x³

= 2(1 + x²)/x³

Explanation

The atomic number of silver (Ag) is 47. Its electronic configuration is [Kr] 4d¹⁰ 5s¹.

The valence electrons are in the outermost shell, which is the 5s orbital with 1 electron.

The number of valence electrons is 1.

Explanation

The center of gravity is the point where the entire weight of an object appears to act.

It can lie inside or outside the object, depending on its shape (e.g., outside for a ring or hollow sphere).

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

Given lim x → 1⁺ (2 + √x - 1)

As x approaches 1 from the right:

√x approaches √1 = 1

So, √x - 1 approaches 1 - 1 = 0

2 + 0 = 2

The limit is 2.

Explanation

Given f(x) = 6x/(3x - 2)

The denominator cannot be zero:

3x - 2 ≠ 0

3x ≠ 2

x ≠ 2/3

So, the domain of f(x) is all real numbers except x = 2/3.

The answer is R - {2/3}.