cot(45°) = 1/tan(45°) = 1/1 = 1

cot(90°) = 1/tan(90°) = 1/0 = undefined.

Determinant of H = (3*5) - (2*1) = 15 - 2 = 13.

f''(x) = 24x - 12. At x = 1, f''(1) = 24(1) - 12 = 12.

The product of the roots is k = 2 * 4 = 8.

For no real roots, the discriminant must be less than zero: k² - 4*1*16 < 0, thus k < -8 or k > 8.

For no real roots, the discriminant must be less than zero: (-4)² - 4*1*k < 0, hence k > 4.

Since the roots are equal, k = 4 + 4 = 8.

sec(0°) = 1/cos(0°) = 1/1 = 1

sec(30°) = 1/cos(30°) = 1/(√3/2) = 2/√3 = √3

sec(45°) = 1/cos(45°) = 1/(√2/2) = √2.

sin(45°) = 1/√2

tan(0°) = 0

tan(90°) is undefined.

The 3rd term is C(5,2) * (3x)^3 * (-4)^2 = 10 * 27x^3 * 16 = -240.

The 5th term is given by C(6,4) * (3x)^4 * (-2)^2 = 15 * 81 * 4 = -4860.

The azimuthal quantum number l for d orbitals is 2.

The azimuthal quantum number (l) for a p-orbital is 1.

The coefficient of x^4 is C(6,4) * (-2)^2 = 15 * 4 = -60.

The coefficient of x^5 is given by 8C5 * (1/2)^3 = 56 * 1/8 = 7.

The coefficient of x^5 is given by 7C5 * (2)^2 = 21 * 4 = 84.

Determinant of B = (2*7) - (3*5) = 14 - 15 = -1.

Determinant of B = (5*3) - (1*2) = 15 - 2 = 13.

The discriminant D = b² - 4ac = 6² - 4*3*2 = 36 - 24 = 12.

The discriminant D = b² - 4ac = (-12)² - 4*3*12 = 144 - 144 = 0.

The discriminant is b² - 4ac = (-2)² - 4*1*1 = 0.

The discriminant is b² - 4ac = (-4)² - 4*1*4 = 0.

The gravitational constant (G) is 6.67 x 10^-11 N m²/kg².

The term containing x^4 is C(8,4) * (1/2)^4 = 70 * 1/16 = 4.375.

The term containing x^5 is C(8,5)(1/2)^3 = 56 * 1/8 = 7.