Explanation

Explanation

The set {x: x ∈ N, x ≤ 10} has 10 elements (1, 2, 3, 4, 5, 6, 7, 8, 9, 10).

The cardinality of the power set of a set with n elements is 2^n.

So, the cardinality of the power set is 2^10 = 1024.

Explanation

1. Arrange Data in Ascending Order:

   $$\mathrm{<b><font\; color="\#0000ff">2.5</font></b>}<b><font\; color="\#0000ff">,</font></b>\mathrm{<b><font\; color="\#0000ff">2.8</font></b>}<b><font\; color="\#0000ff">,</font></b>\mathrm{<b><font\; color="\#0000ff">2.8</font></b>}<b><font\; color="\#0000ff">,</font></b>\mathrm{<b><font\; color="\#0000ff">2.9</font></b>}<b><font\; color="\#0000ff">,</font></b>\mathrm{<b><font\; color="\#0000ff">3.0</font></b>}<b><font\; color="\#0000ff">,</font></b>\mathrm{<b><font\; color="\#0000ff">3.3</font></b>}<b><font\; color="\#0000ff">,</font></b>\mathrm{<b><font\; color="\#0000ff">3.4</font></b>}<b><font\; color="\#0000ff">,</font></b>\mathrm{<b><font\; color="\#0000ff">3.6</font></b>}<b><font\; color="\#0000ff">,</font></b>\mathrm{<b><font\; color="\#0000ff">3.7</font></b>}<b><font\; color="\#0000ff">,</font></b>\mathrm{<b><font\; color="\#0000ff">4.0</font></b>}<b><font\; color="\#0000ff">,</font></b>\mathrm{<b><font\; color="\#0000ff">4.4</font></b>}<b><font\; color="\#0000ff">,</font></b>\mathrm{<b><font\; color="\#0000ff">4.8</font></b>}<b><font\; color="\#0000ff">,</font></b>\mathrm{<b><font\; color="\#0000ff">4.8</font></b>}<b><font\; color="\#0000ff">,</font></b>\mathrm{<b><font\; color="\#0000ff">5.2</font></b>}<b><font\; color="\#0000ff">,</font></b>\mathrm{<b><font\; color="\#0000ff">5.6</font></b>}$$

2. Find Median:

   • There are 15 data points (odd number), so the median is the 8th value in the ordered list.

   • The 8th value is 3.6.

Explanation

The given partial differential equation:

∂²U/∂x² + ∂²U/∂y² = f(x,y)

is known as the Poisson equation.

If f(x,y) = 0, it would reduce to the Laplace equation.

Explanation

To integrate √(2x + 1), we can use the power rule of integration and the chain rule.

Let u = 2x + 1, then du/dx = 2, and du = 2dx.

∫√(2x + 1) dx = (1/2) ∫u^(1/2) du

= (1/2) (2/3) u^(3/2) + C

= (1/3) (2x + 1)^(3/2) + C

Explanation

Step 1: Evaluate the integral

∫(x² - x - 1) dx = (x³ / 3) - (x² / 2) - x

Step 2: Apply the limits of integration

[(1³ / 3) - (1² / 2) - 1] - [((-1)³ / 3) - ((-1)² / 2) - (-1)]

= [(1/3) - (1/2) - 1] - [(-1/3) - (1/2) + 1]

Step 3: Simplify the expression

= (1/3) - (1/2) - 1 + (1/3) + (1/2) - 1

= (2/3) - 2

= (2/3) - (6/3)

= -4/3

Explanation

Harmonic Mean (H.M): The reciprocal of the arithmetic mean of the reciprocals of the numbers.

Geometric Mean (G.M): The nth root of the product of the numbers.

Arithmetic Mean (A.M): The sum of the numbers divided by the count of numbers.

Explanation

A compound proposition that is always true, regardless of the truth values of its components, is called a tautology.

The statement "P ∨ ¬P" (P or not P) is a tautology because it is always true, whether P is true or false.

Explanation

Total ways to draw 3 balls = C(15, 3) = 455

Ways to draw 3 red balls = C(5, 3) = 10

Probability = 10/455 = 2/91

Explanation

1. Intersection of $D_f$ and $D_g$: The function $\frac{f}{g}$ is defined only where both $f$ and $g$ are defined.

2. Excluding zeros of $g$: The function $\frac{f}{g}$ is undefined where $g(x)=0$ because division by zero is not allowed.