The x-coordinate of point P is the coefficient of i in the position vector, which is 2.

Distance = |r| = √(3^2 + 4^2) = √(9 + 16) = √25 = 5.

Probability of not A = 1 - P(A) = 1 - 0.4 = 0.6.

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.

The discriminant is 0, indicating that the roots are real and equal.

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

For both roots to be negative, k must be greater than 0.

The product of the roots is given by c/a = 6/1 = 6.

For both roots to be positive, k must be greater than 12 (from Vieta's formulas). The minimum integer k is 13.

Using Vieta's formulas, p = -(3 + (-2)) = -1 and q = 3 * (-2) = -6. Therefore, p + q = -1 - 6 = -7.

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

Factoring gives (x - 3)(x - 5) = 0, hence the roots are 3 and 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.

Range = Maximum - Minimum. Therefore, Maximum = Range + Minimum = 20 + 10 = 30.

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.

If the shortest side is 6 cm, the sides are 6, 8, and 10 cm. Perimeter = 6 + 8 + 10 = 24 cm.

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

The condition for equal roots is given by the discriminant: b^2 - 4ac = 0.

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.

Using the product of the roots, c = 2 * 3 = 6.

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

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.

The equation can be expressed as (x - 3)^2 = 0, which expands to x^2 - 6x + 9 = 0. Thus, k = 9.

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