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.

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, 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.

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

|-3| = 3, so |x| + x = 3 - 3 = 0.

2x - 3 = 2(4) - 3 = 8 - 3 = 5.

|z|² = (1)² + (1)² = 1 + 1 = 2.

z^3 = (1 + i)^3 = 1 + 3i + 3i^2 + i^3 = 1 + 3i - 3 - i = -2 + 2i.

The conjugate of a complex number z = a + bi is given by z* = a - bi. Thus, the conjugate of z = 1 + i√3 is 1 - i√3.

z^2 = (1 + i√3)^2 = 1^2 + 2(1)(i√3) + (i√3)^2 = 1 + 2i√3 - 3 = -2 + 2i√3.

z² = (2 + 2i)² = 4 + 8i - 4 = 8i.

The modulus of z is given by |z| = √(3^2 + 4^2) = √(9 + 16) = √25 = 5.

z1 * z2 = (2 + 2i)(2 - 2i) = 4 - 4i^2 = 4 + 4 = 8.

The number 60 appears most frequently (3 times), so the mode is 60.

Replacing 5 with 7 removes the most frequent number, so the new mode is 7.

The number 5 appears most frequently (3 times), so the mode is 5.

Both 2 and 3 appear 3 times, but since 2 is the smallest, the mode is 2.

The age 22 appears 3 times, which is more than any other age, so the mode is 22.

Probability of losing = 1 - P(winning) = 1 - 0.2 = 0.8.

Vertically opposite angles are equal. Therefore, if one angle measures 120 degrees, the other angle also measures 120 degrees.

In a parallelogram, opposite angles are equal. Therefore, if one angle is 70 degrees, the opposite angle is also 70 degrees.

Area = base * height = 8 * 5 = 40 cm²

In a 30-60-90 triangle, the sides opposite the 30 degrees and 60 degrees angles are in the ratio 1:√3.

Number of people preferring tea = 70% of 200 = 0.7 * 200 = 140.

The sum of the angles in a triangle is always 180 degrees. Therefore, if two angles are 45 degrees each, the third angle = 180 - (45 + 45) = 90 degrees.