Total sum = Mean * Number of elements = 30 * 6 = 180.

Mean = Sum / 5. Therefore, Sum = Mean * 5 = 6 * 5 = 30.

For the median to be 9, x must be 9. The ordered set is 3, 7, 9, 9, 15.

Removing 5, which is the mode, leaves 3 as the new mode since it appears twice.

The original mode is 4. Increasing it by 1 gives a new mode of 5.

Perimeter = 2(length + width). So, 50 = 2(15 + width). Solving gives width = 10 cm.

Substituting x = 0: 4(0) - 5y = 20 => -5y = 20 => y = -4.

Probability of not occurring = 1 - P(event) = 1 - 0.3 = 0.7.

By Vieta's formulas, the sum of the roots p + q = -b/a.

The equation can be factored as (x + 1)(x + 1) = 0, giving the root -1 with multiplicity 2.

Substituting x = -1 into the equation gives (-1)^2 + 2(-1) + k = 0, leading to 1 - 2 + k = 0, thus k = 1.

Using Vieta's formulas, p = -(3 + 4) = -7 and q = 3 * 4 = 12. Therefore, p + q = -7 + 12 = 5.

The area of a circle is given by A = πr². If the radius is doubled (r becomes 2r), the new area is A' = π(2r)² = 4πr², which is four times the original area.

Let the sides be 3x, 4x, and 5x. Then, 3x + 4x + 5x = 36. Thus, 12x = 36, giving x = 3. The longest side is 5x = 15 cm.

A triangle with sides in the ratio 3:4:5 is a right-angled triangle.

The discriminant is given by b^2 - 4ac. Here, b = 2, a = 1, c = 1, so the discriminant is 2^2 - 4*1*1 = 0.

The sum of the roots is -1 + -3 = -4, and the product is (-1)(-3) = 3. Thus, k = 3.

Using Vieta's formulas, k = (-1)(-2) = 2.

For the roots to be equal, the discriminant must be zero. Thus, 4^2 - 4*1*k = 0 leads to k = 4.

The product of the roots ab is given by c/a. Here, c = 6 and a = 1, so ab = 6.

The discriminant must be greater than zero for real and distinct roots: (-4)^2 - 4*1*k > 0, which simplifies to 16 - 4k > 0, or k < 4.

For real and distinct roots, the discriminant must be greater than zero: (-6)^2 - 4*1*k > 0, leading to k < 9.

A triangle with sides in the ratio 3:4:5 is a right triangle, as it satisfies the Pythagorean theorem.

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.

A · B = (1)(2) + (2)(1) = 2 + 2 = 4