Increasing one number by 5 increases the total sum by 5. If there are n numbers, the mean increases by 5/n.

The original sum of the numbers is 30 * 10 = 300. Adding 50 gives a new sum of 300 + 50 = 350. The new mean is 350 / 11 = 31.82, which rounds to 32.

Let the integers be x, x+1, x+2, x+3, x+4. Mean = (5x + 10) / 5 = 12. Solving gives x = 10.

Let the sum of the five numbers be S. Then, S/5 = 20, so S = 100. Let the removed number be x. The new sum is S - x, and the new mean is (S - x)/4 = 18. Therefore, 100 - x = 72, which gives x = 28.

The first four prime numbers are 2, 3, 5, and 7. Mean = (2 + 3 + 5 + 7) / 4 = 17 / 4 = 4.25.

Mean = (4 + 8 + 10 + 12 + x) / 5 = 10. Solving gives x = 14.

Mean = (4 + 8 + 12 + x) / 4 = 10. Therefore, 4 + 8 + 12 + x = 40. So, x = 40 - 24 = 16.

Let the integers be x, x+1, x+2. Mean = (x + (x+1) + (x+2)) / 3 = 20. Thus, 3x + 3 = 60, so x = 19.

Total of three numbers = 3 * 12 = 36. Sum of the other two numbers = 36 - 8 = 28. Mean of the other two = 28 / 2 = 14.

New mean = (15*3 + 5) / 3 = 20/3 = 16.67.

Let the other two numbers be a and b. Then, (10 + a + b) / 3 = 15. Solving gives (a + b) = 35, so Mean of a and b = 35 / 2 = 17.5.

Let the three numbers be a, b, c. Then, (a + b + 20) / 3 = 30. Thus, a + b = 90, and the mean of a and b = 90 / 2 = 45.

For the median to be 8, x must be less than or equal to 8. The only value that satisfies this is x = 6.

Median = 9 (middle value of the ordered data set)

Arranging the data: 3, 7, 9, 15, 20. Median = 9 (middle value).

The vertex of the parabola occurs at x = -b/(2a) = 4/2 = 2. The minimum value is f(2) = 2^2 - 4*2 + 6 = 2.

The equation y^2 = 12x indicates that the parabola opens to the right since it is in the form y^2 = 4px.

Perimeter of a square = 4 × side. Therefore, side = 64 cm / 4 = 16 cm.

In an equilateral triangle, all sides are equal. Therefore, each side = Perimeter / 3 = 36 / 3 = 12 cm.

The product of the roots is given by c/a. Here, c = 1 and a = 2, so the product is 1/2.

The product of the roots of the equation ax^2 + bx + c = 0 is given by c/a. Here, c = 2 and a = 2, so the product is 2/2 = 1.

The product of the roots of the equation ax^2 + bx + c = 0 is given by c/a. Here, c = 6 and a = 2, so the product is 6/2 = 3.

The product of the roots is given by c/a, which is 9/3 = 3.

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

The discriminant must be less than zero: (-4)^2 - 4*2*k < 0 leads to k > 8.

The discriminant must be less than zero: (-4)^2 - 4*2*k < 0 leads to k > 8.

Dividing the entire equation by 3 gives x^2 + 4x + 4 = 0.

The equation can be factored as (2x - 3)(2x - 3) = 0, indicating a perfect square.

The discriminant is 3^2 - 4*5*(-2) = 9 + 40 = 49, which is a perfect square, hence the roots are rational.

The equation can be factored as (x + 3)^2 = 0, hence a = 3.