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.

The product of the roots is (-2)(-3) = 6, so k = 6.

The discriminant must be greater than zero: 5^2 - 4(1)(k) > 0, leading to k < 25.

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

Using Vieta's formulas, n = 3 * 4 = 12.

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

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.

By Vieta's formulas, ab = 10, which is the constant term of the polynomial.

By Vieta's formulas, the sum of the roots a + b + c = -(-3) = 3.

Using the sum of roots, b = - (4 + (-1)) = -3.

For the roots to be equal, the discriminant must be zero, which means b^2 = 4ac.

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

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

The sum of the roots is -(-2) + -(-3) = 5, hence p = 5.

Using Vieta's formulas, p = -(-2 - 3) = 5 and q = (-2)(-3) = 6. Therefore, p + q = 5 + 6 = 11.

For the roots to be equal, the discriminant must be zero. Thus, (-5)² - 4(1)(k) = 0, giving k = 25/4 = 6.25.

If A · B = 0, then A and B are orthogonal (perpendicular) to each other.

A · B = |A||B|cos(θ) => 15 = 5*3*cos(θ) => cos(θ) = 1, θ = 0°.

If A · B = |A||B|, then the angle between them is 0°, meaning they are parallel.

If A · B = |A||B|, then cos(θ) = 1, which means θ = 0°.

Increased sediment load generally decreases the velocity of the river due to increased friction.

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

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

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

Vertical drop = Distance × Slope = 2000 m × (1/1000) = 2 m.

Soil eroded = 2 tons/hectare/year × 50 hectares × 5 years = 500 tons.

Soil eroded = Erosion rate × Area × Time = 2 tons/hectare/year × 50 hectares × 5 years = 500 tons.