Explanation

Given f(x) = x³ - (1/x³):

f(1/x) = (1/x)³ - (1/(1/x)³) = (1/x³) - x³

Now, let's calculate f(x) + f(1/x):

f(x) + f(1/x) = (x³ - 1/x³) + (1/x³ - x³)

= x³ - x³ - 1/x³ + 1/x³

= 0

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. Leibniz’s theorem gives a formula to find the nth derivative of the product of two functions.

2. It uses the binomial expansion:

Explanation

The hyperbolic tangent function tanh(y) is defined as:

tanh(y) = (e^y - e^(-y)) / (e^y + e^(-y))

Explanation

Given n(n + 1)(n + 5):

1. n(n + 1) is always even (multiple of 2) since it's the product of two consecutive integers.

2. For divisibility by 3:

    If n is a multiple of 3, the expression is divisible by 3.

    If n = 3k + 1, then n + 5 = 3k + 6 is divisible by 3.

    If n = 3k + 2, then n + 1 = 3k + 3 is divisible by 3.

Given these cases, n(n + 1)(n + 5) is divisible by 3.

Explanation

Given the operation a * b = a + b + 1:

Let's find the identity element 'e' such that a * e = a.

a * e = a + e + 1 = a

This implies e + 1 = 0, so e = -1.

Now, to find the inverse 'x' of 'a' such that a * x = e = -1:

a * x = a + x + 1 = -1

This implies a + x = -2.

x = -2 - a

x = -a - 2

Explanation

Given the function f(x) = 3x - 2

This is a linear function with a slope of 3. Since it's linear and defined for all real numbers x, its range is all real numbers.

As x varies from -∞ to ∞, f(x) also varies from -∞ to ∞.

Explanation

Given y = acos(x) + bsin(x) + 4

First derivative: dy/dx = -asin(x) + bcos(x)

Second derivative: d²y/dx² = -acos(x) - bsin(x)

Notice that d²y/dx² = -(acos(x) + bsin(x))

Given y = acos(x) + bsin(x) + 4, we rearrange to find acos(x) + bsin(x) = y - 4

Substituting into the second derivative equation gives:

d²y/dx² = -(y - 4)

d²y/dx² = -y + 4

d²y/dx² + y = 4

The answer is d²y/dx² + y = 4.

Explanation

Let's directly evaluate the system of equations:

1x + 3y = 1

3x + λy = 3

For infinitely many solutions, these equations must represent the same line. Multiplying the first equation by 3 gives:

3x + 9y = 3

Comparing this with the second equation 3x + λy = 3, we see that λ must equal 9 for the equations to be identical.

Explanation

The rank of a matrix is the number of non-zero rows in its row echelon form.

It represents the maximum number of linearly independent rows or columns.