Determinant of E = 1(1*0 - 4*6) - 2(0*0 - 4*5) + 3(0*6 - 1*5) = 1(0 - 24) - 2(0 - 20) + 3(0 - 5) = -24 + 40 - 15 = 1.

Determinant of E = (1*4) - (2*2) = 4 - 4 = 0.

Calculating J^2 gives [[0, 1], [1, 0]] * [[0, 1], [1, 0]] = [[1, 0], [0, 1]], which is the identity matrix.

Using the determinant formula, det(J) = 1*(1*0 - 4*6) - 2*(0*0 - 4*5) + 3*(0*6 - 1*5) = 1*(-24) - 2*(-20) + 3*(-5) = -24 + 40 - 15 = 1.

The determinant is 0 because the rows are linearly dependent.

Det(F) = (2*4) - (1*3) = 8 - 3 = 5.

The equation x^2 = -12y indicates that the parabola opens downwards.

In the equation x^2 = 4py, we have 4p = 16, thus p = 4. The distance from the vertex to the focus is 4.

The length of the latus rectum for the parabola x^2 = 4py is 4p. Here, p = 4, so the length is 8.

To find the y-intercept, set x = 0. The equation becomes y = -0^2 + 4(0) - 3 = -3.

The vertex of the parabola y^2 = 4px is at (0, 0).

To find the vertex, use x = -b/(2a). Here, a = 1, b = -4, so x = 2. Substitute x = 2 into the equation to find y = -1. Thus, the vertex is (2, -1).

The vertex of the parabola y^2 = 4px is at (0, 0).

The vertex of the parabola y^2 = 4px is at (0, 0). Here, p = 5, but the vertex remains at (0, 0).

The polynomial can be factored as (x-1)^3, indicating that it has one real root with multiplicity 3, hence all roots are real and equal.

The sum of the roots is given by -b/a = 3/1 = 3.

The polynomial can be factored as (x - 1)^3, indicating that all roots are real and equal.

The discriminant D = b^2 - 4ac = 4^2 - 4(2)(2) = 16 - 16 = 0.

For equal roots, the discriminant must be zero: b^2 - 4ac = 0. Here, 4^2 - 4(2)(k) = 0 leads to k = 4.

For real and equal roots, the discriminant must be zero. Here, b^2 - 4ac = 0 gives 16 - 8k = 0, thus k = 8.

The discriminant must be non-negative: 4^2 - 4*2*k >= 0, which simplifies to k <= 4.

The discriminant D = b^2 - 4ac = 4^2 - 4(2)(-6) = 16 + 48 = 64.

For equal roots, the discriminant must be zero: (-4)^2 - 4*2*k = 0. Solving gives k = 4.

The discriminant must be non-negative: (-4)^2 - 4*2*k >= 0, which simplifies to k <= 2.

Using the quadratic formula x = [-b ± √(b^2 - 4ac)] / 2a, we find the roots to be -1 and 2/5.

The roots can be found using the quadratic formula: x = (3 ± √(9-8))/2 = 1 and 2.

The discriminant must be non-negative: (2p)^2 - 4(1)(p^2 - 4) >= 0 leads to p >= 2.

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

The vertex can be found using the formula (-b/2a, f(-b/2a)). Here, vertex is (-1, 0).

The discriminant must be negative: 2^2 - 4*1*k < 0 => 4 < 4k => k > 1.