The distance from the focus to the directrix is 6, so p = -3. The equation is x^2 = 4py, which gives x^2 = -12y.

The vertex is at (0, 0) and p = 2. The equation is y^2 = 4px, which gives y^2 = 8x.

The vertex form of a parabola is given by (x - h)^2 = 4p(y - k). Here, h = 2, k = 3, and p = 1 (distance from vertex to focus). Thus, the equation is (x - 2)^2 = 4(1)(y - 3) or y = (1/4)(x - 2)^2 + 3.

f'(x) = 2x + 2. At x = 1, f'(1) = 4. The point is (1, 3). The tangent line is y - 3 = 4(x - 1) => y = 4x - 1.

'Each' is singular, so it should be 'has' instead of 'have'.

The sentence is correct as 'is' agrees with the nearest subject 'students' (plural) when 'either/or' is used.

The correct form is 'is' instead of 'are' because 'Neither' is singular.

When 'neither/nor' is used, the verb should agree with the nearest subject, which is 'employees' (plural), so it should be 'were'.

The correct form is 'goes' instead of 'go' for the subject 'She'.

The correct form is 'The team is winning the match.' 'Team' is a collective noun and takes a singular verb.

The correct verb should be 'has' instead of 'have' because 'each' is singular.

The pronoun 'their' is incorrect; it should be 'his or her' to match the singular 'Each student'.

'more' is unnecessary; it should just be 'taller'.

The correct form is 'sing' instead of 'sings' after 'can'.

'Don't' should be 'doesn't' because 'she' is third person singular. The correct sentence is 'She doesn't like coffee.'

The correct form is 'She doesn't like to play tennis.' The contraction 'don't' should be 'doesn't' for third person singular.

The correct verb should be 'has' instead of 'have' because 'everyone' is treated as singular.

The correct verb should be 'is' instead of 'are' because 'neither' is singular.

The equation y = mx + c represents straight lines where m is the slope and c is the y-intercept.

Figure Z can be obtained by folding the paper along the dotted lines.

Figure S has a different number of sides compared to the others.

All options are directional arrows except ▲ and ▼ which are triangles.

Option D has the same angles and proportions as the given figure.

The equation x^2 = 12y can be rewritten as (y - 0) = (1/3)(x - 0)^2, indicating the focus is at (0, 3).

The standard form of a parabola is y^2 = 4px. Here, 4p = 12, so p = 3. The focus is at (p, 0) = (3, 0).

Integrating 3x^2 gives y = x^3 + C.

This is a separable equation. Integrating gives ln|y| = 2x + C, hence y = Ce^(2x).

Integrating both sides gives y = (3/3)x^3 + C = x^3 + C.

This is a separable differential equation. Integrating gives y = Ce^(4x), where C is the constant.

This is a separable equation. Integrating gives ln|y| = x + C, hence y = Ce^x.