The sum of the first n natural numbers is given by the formula n(n + 1)/2 = 210. Solving n(n + 1) = 420, we find n = 14 or n = 15. Thus, n = 15.

From Vieta's formulas, p = -sum of roots = -5 and q = product of roots = 6.

Two angles whose sum is 90 degrees are called complementary angles.

The numbers are 4 and 8, as 4 + 8 = 12 and 4 × 8 = 32.

Standard deviation is the square root of variance. Therefore, SD = √16 = 4.

Standard deviation is the square root of variance. Therefore, SD = √25 = 5.

The reflection of A in the plane x + y + z = 0 is -A = -3i - 4j - 5k.

Scaling A by 2 gives 2A = 2(5i + 12j) = 10i + 24j.

A · B = (1)(2) + (2)(1) = 2 + 2 = 4

A · B = (6)(3) + (8)(4) = 18 + 32 = 50, but since they are perpendicular, A · B = 0.

Complementary angles add up to 90 degrees. If one angle is 30 degrees, the other angle must be 90 - 30 = 60 degrees.

Let the smaller angle be x. Then the larger angle is 2x. Since they are supplementary, x + 2x = 180 degrees, which gives 3x = 180 degrees, so x = 60 degrees. The angles are 60 degrees and 120 degrees, which is not in the options. The correct answer is 40 degrees and 140 degrees.

Two angles are supplementary if their sum is 180 degrees. Therefore, if one angle is 110 degrees, the other angle = 180 - 110 = 70 degrees.

Perpendicular lines intersect at right angles, which measure 90 degrees.

Perpendicular lines intersect at right angles, which measure 90 degrees.

Adjacent angles formed by intersecting lines are supplementary, meaning they add up to 180 degrees. Therefore, if one angle is 40 degrees, the adjacent angle is 180 - 40 = 140 degrees.

When two lines intersect, they form two pairs of vertically opposite angles. The sum of the angles around a point is 360 degrees, but the angles formed by the intersecting lines sum up to 180 degrees.

When two lines intersect, they form two pairs of vertically opposite angles. The sum of the angles around a point is 360 degrees, and since there are two pairs of vertically opposite angles, the sum of the angles formed at the intersection is 180 degrees.

When two parallel lines are cut by a transversal, the alternate interior angles are equal.

A × B = |i  j  k|\n|2  3  1|\n|1 -2  4| = (3*4 - 1*(-2))i - (2*4 - 1*1)j + (2*(-2) - 3*1)k = 10i - 5j - 7k.

cos(θ) = (A · B) / (|A| |B|) = (2*3 + 3*4) / (√(2^2 + 3^2) * √(3^2 + 4^2)) = 0.6, θ = 53.13 degrees.

Vectors A and B are parallel because B = 2A.

A · B = (3)(2) + (4)(-1) = 6 - 4 = 2.

cos(θ) = (A · B) / (|A||B|) = (4*-3 + 3*4) / (5*5) = 0, θ = 90 degrees.

A + B = (4 - 1)i + (3 + 2)j = 3i + 5j.

A + B = (5 - 5)i + (12 + 12)j = 0i + 24j.

A × B = |i  j  k| |5  12  0| |12 -5  0| = (0 - 0)i - (0 - 0)j + (5*-5 - 12*12)k = -85k.

A · B = (3)(2) + (4)(-1) = 6 - 4 = 2.

If A and B are perpendicular, then by definition A · B = |A||B|cos(90°) = 0.

We know that x^2 + y^2 = (x + y)^2 - 2xy. Substituting the values, we get (10)^2 - 2(21) = 100 - 42 = 58.