An orthogonal matrix is defined as a square matrix whose transpose is equal to its inverse.

A skew-symmetric matrix satisfies the condition a_ij = -a_ji for all i and j.

A symmetric matrix satisfies the condition Aij = Aji for all i and j.

A symmetric matrix is defined by the property A = A^T, meaning it is equal to its transpose.

A matrix A is symmetric if it is equal to its transpose, i.e., A = A^T.

For a downward-opening parabola with vertex (h, k), the equation is y - k = -a(x - h)^2. Here, h = 3 and k = -2.

The standard form of a parabola that opens to the left is y^2 = -4px.

The general form of a parabola that opens to the right is y^2 = 4px.

Height = distance * tan(60) = 30 * √3 ≈ 30 * 1.732 = 51.96 m.

The perimeter P of a square is given by P = 4 * side. If P = 32 cm, then side = 32/4 = 8 cm.

Probability of selecting a boy = Number of boys / Total students = 6/15 = 2/5.

Probability of selecting a boy = Number of boys / Total students = 6/15 = 2/5.

Probability of selecting a boy = Number of boys / Total students = 15/40 = 3/8.

A transversal intersecting two parallel lines forms 4 pairs of corresponding angles.

When a transversal intersects two parallel lines, the same-side interior angles are supplementary, meaning they add up to 180 degrees.

The rank of matrix B is the number of non-zero rows in its row echelon form, which is 3.

The determinant of B is calculated as (2*7) - (3*5) = 14 - 15 = -1.

The determinant of C is 0 because it has a row of zeros.

Using the determinant formula for 3x3 matrices, det(C) = 1(3*0 - 1*1) - 0 + 2(-1*1 - 3*2) = 0 - 0 - 12 = -12.

The determinant of an upper triangular matrix is the product of its diagonal elements. Here, det(C) = 1 * 1 * 1 = 1.

C^2 = C * C = [[1*1 + 2*3, 1*2 + 2*5], [3*1 + 5*3, 3*2 + 5*5]] = [[11, 16], [18, 35]].

Using the identity tan A = sin A / cos A, we find sin A = √(1 - cos²A) = √(1 - (5/13)²) = 12/13. Thus, tan A = (12/13) / (5/13) = 12/5.

Using the Pythagorean identity, sin B = √(1 - cos²B) = √(1 - 0.6²) = √(1 - 0.36) = √(0.64) = 0.8.

Cosine of 60° is 1/2, hence angle B = 60°.

Using the Pythagorean identity, sin B = √(1 - cos²B) = √(1 - (5/13)²) = √(1 - 25/169) = √(144/169) = 12/13.

The angle C for which cos C = 0.5 is 60°.

Cotangent is 1 when the angle A is 45°, as cot A = cos A/sin A = 1 when both are equal.

cot A = cos A/sin A = 3/4 implies sin A = 4/5 using the identity sin²A + cos²A = 1.

cot D = 3/4 implies tan D = 4/3. Therefore, sin D = 4/5 (using the identity sin²D + cos²D = 1).

cot θ = 1 implies tan θ = 1, which occurs at θ = 45°.