Explanation

The given differential equation is a second-order linear homogeneous equation with constant coefficients. The characteristic equation is:

r^2 - 4r + 4 = 0

which has a repeated root r = 2. Therefore, the general solution is:

y = (c1 + c2x)e^2x

where c1 and c2 are arbitrary constants.

Explanation

• The correct answer is: a + (-b) ∈ H.
• A nonempty subset H of G is a subgroup of G if and only if it is closed under the group operation + and contains the inverse of each of its elements.

Explanation

The given equation y = Asin(x) + Bcos(x) is a general solution to the differential equation y'' + y = 0, where A and B are arbitrary constants.

Explanation

The correct answer is: n.

A cyclic permutation of length n is a permutation that shifts each element one position to the right (or left) in a cycle. For example, the cyclic permutation (1 2 3 4 5) maps 1 to 2, 2 to 3, 3 to 4, 4 to 5, and 5 to 1.

The order of a cyclic permutation of length n is n, because it takes n applications of the permutation to return to the original arrangement.

Explanation

Ext(A) denotes the exterior of the set A, which is defined as the interior of the complement of A, i.e., Int(A^c).

Explanation

The formula p ∨ (p ∧ q) can be simplified using the distributive property of ∨ over ∧, which states that p ∨ (q ∧ r) = (p ∨ q) ∧ (p ∨ r).

Applying this property, we get:

p ∨ (p ∧ q) = (p ∨ p) ∧ (p ∨ q) = p ∧ (p ∨ q) = p

Explanation

The group G = {-1, 1, -i, i} is a multiplicative group, and the order of an element a in G is the smallest positive integer n such that a^n = 1.

For -i, we have:

(-i)^2 = -1

(-i)^3 = i

(-i)^4 = 1

So, the order of -i is 4.

Explanation

The set of integers, denoted by Z, forms an abelian group under the operation of addition (+). This means that the following properties hold: 

• Closure: For any integers a, b, their sum a + b is also an integer.
• Associativity: (a + b) + c = a + (b + c) for any integers a, b, c.
• Commutativity: a + b = b + a for any integers a, b.
• Identity: There exists an identity element, which is 0, such that a + 0 = a for any integer a.
• Inverse: For each integer a, there exists an inverse element, which is -a, such that a + (-a) = 0.

The set of integers does not form a group under multiplication, because the inverse property does not hold (except for 1 and -1). Division is not even a binary operation on the set of integers, because it is not defined for all pairs of integers (e.g., 1/2 is not an integer).

Therefore, the correct answer is Addition.

Explanation

The given expression is: (x^3 + 2x^2y - xy - 2y^2) / (x + 2y)

To evaluate the limit, we can substitute x = 2 and y = 1: (2^3 + 2(2)^2(1) - 2(1) - 2(1)^2) / (2 + 2(1)) = (8 + 8 - 2 - 2) / 4 = 12 / 4 = 3.

Explanation

• The vector ax(bxc) is the cross product of vector a and the cross product of vectors b and c.
• This resulting vector is perpendicular to vector a and vector (bxc), which is parallel to vector c.
• Therefore, ax(bxc) lies in the plane containing vectors a and c.