The derivative y' = 2x. At x = 3, y' = 2(3) = 6.

The sum of the coefficients in the expansion of (x + 1)^n is given by (1 + 1)^n = 2^n. Here, n=8, so the sum is 2^8 = 256.

Sum of interior angles = (n-2) * 180° = (6-2) * 180° = 720 degrees

The sum of the roots is given by -b/a = 5/1 = 5.

The trace of a square matrix is defined as the sum of the elements on its main diagonal.

The trace of a matrix is the sum of its diagonal elements. Here, trace(H) = 2 + 3 + 4 = 9.

(2x + 3)² = 4x² + 12x + 9 by expanding the square of a binomial.

(a + b)² = a² + 2ab + b² by the expansion of the square of a binomial.

(x + 1)(x - 1) = x² - 1, which is the difference of squares.

Multiplying three negative ones gives −1.

(−1)^3 = −1.

According to BODMAS, 4 × 2 = 8, so 3 + 8 = 11.

The discriminant must be less than zero. Thus, k^2 - 4*1*16 < 0 leads to k^2 < 64, giving k < 8 and k > -8.

For equal roots, the discriminant must be zero. Thus, k^2 - 4*1*4 = 0, which gives k^2 = 16, so k = ±4.

For real and distinct roots, the discriminant must be positive: k^2 - 4(1)(16) > 0. Thus, k^2 > 64, leading to k < -8 or k > 8.

For the roots to be equal, the discriminant must be zero. Thus, (-5)^2 - 4(1)(k) = 0. Solving gives k = 6.25.

log_10(0.01) = log_10(10^-2) = -2.

log_2(1/8) = log_2(2^-3) = -3.

log_5(1) = 0 because 5^0 = 1.

In the equation y^2 = 4px, we have 4p = 20, thus p = 5.

According to the Pythagorean identity, sin² θ + cos² θ = 1 for any angle θ.

Rearranging gives 3x - 2x = 5 + 7, thus x = 12.

Adding 7 to both sides gives 4x = 16. Dividing by 4 gives x = 4.

Mean = (1 + 2 + 3 + 4 + 5 + 6) / 6 = 3.5. Variance = [(1-3.5)² + (2-3.5)² + (3-3.5)² + (4-3.5)² + (5-3.5)² + (6-3.5)²] / 6 = 2.92 (approx 2.5).

Rearranging to slope-intercept form: -2y = -5x + 10, thus y = (5/2)x - 5. The y-intercept is -5.

A matrix where all elements are zero is called a zero matrix.

(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca.

A square matrix is one where the number of rows is equal to the number of columns. Among the options, only 3x3 has equal rows and columns.

An identity matrix is a square matrix with ones on the diagonal and zeros elsewhere. The matrix [[1, 0], [0, 1]] fits this definition.

√4 = 2, 0.25 = 1/4, and 1 is rational. √2 is irrational.