The characteristic polynomial is det(A - λI) = det([[1-λ, 2], [3, 4-λ]]) = (1-λ)(4-λ) - 6 = λ^2 - 5λ + 2.

The characteristic polynomial is given by det(A - λI) = det([[2-λ, 1], [1, 2-λ]]) = (2-λ)(2-λ) - 1 = λ^2 - 3λ + 3.

The coefficient of x^0 is (-4)^5 = -1024.

The coefficient of x^0 is C(3,0) * 5^3 = 1 * 125 = 125.

The coefficient of x^0 is C(5,0) * 5^5 = 1 * 3125 = 3125.

The coefficient of x^2 is C(4,2) * (3)^2 * (-4)^2 = 6 * 9 * 16 = 864.

Using the binomial theorem, the coefficient of x^2 in (a + b)^n is given by nCk * a^(n-k) * b^k. Here, n=6, a=x, b=-4, and k=2. Thus, the coefficient is 6C2 * (1)^4 * (-4)^2 = 15 * 16 = 240.

Using the binomial theorem, the coefficient of x^3 is given by C(5,3) * (2)^3 * (3)^(5-3) = 10 * 8 * 9 = 720.

The coefficient of x^4 is C(6,4) * (2)^4 * (3)^(6-4) = 15 * 16 * 9 = 2160.

The coefficient of x^4 is C(5, 4) * (2)^4 * (-3)^1 = 5 * 16 * (-3) = -240.

The coefficient of x^4 is C(6,4) = 15.

The coefficient of x^5 is C(10,5) * (1/2)^5 = 252 * 1/32 = 7.875.

The coefficient of x^5 is C(7,5)(2)^2 = 21 * 4 = 84.

The coefficient of x^5 is C(8,5) * (2)^3 = 56 * 8 = 448.

Using the binomial theorem, the coefficient of x^5 in (x + 3)^8 is given by 8C5 * (3)^3 = 56 * 27 = 1512.

The coefficient of x^6 is C(8,6) * (5)^2 = 28 * 25 = 700.

The common ratio r = 12/4 = 3.

The conjugate of z = 2 + 5i is z̅ = 2 - 5i.

The conjugate of z = 5 - 2i is 5 + 2i.

The conjugate of z = 5 - 6i is 5 + 6i.

The derivative of sin^(-1)(x) is 1/√(1-x^2)

The derivative of y = sin^(-1)(x) is 1/√(1-x^2)

The determinant is calculated as (0*0) - (1*1) = 0 - 1 = -1.

The determinant is calculated as \( 1*4 - 2*3 = 4 - 6 = -2 \).

The determinant is 0 because the rows are linearly dependent.

The determinant of the identity matrix is 1.

The determinant is calculated as (1*1) - (1*1) = 1 - 1 = 0.

The determinant is calculated as \( 1(1*1 - 0*3) - 2(0*1 - 0*2) + 1(0*3 - 1*2) = 1 - 0 - 2 = -1 \).

The determinant is 0 because the rows are linearly dependent.

The determinant is calculated as \( 1*4 - 2*3 = 4 - 6 = -2 \).