The value of \[ \left( \frac{1 + \cos\left(\frac{\pi}{12}\right) + i\sin\left(\frac{\pi}{12}\right)} {1 + \cos\left(\frac{\pi}{12}\right) - i\sin\left(\frac{\pi}{12}\right)} \right)^{72} \] is equal to:
If \[ f(9) = 0 \quad \text{and} \quad f'(9) = 0, \] then \[ \lim_{x \to 9} \frac{\sqrt{f(x)} - 3}{\sqrt{x} - 3} \] is equal to:
If \[ f(x) = \sqrt{2x + \frac{4}{\sqrt{2x}}} \] then \[ f'(2) \] is equal to:
The value of \[ \frac{15^3 + 6^3 + 3 \cdot 6 \cdot 15 \cdot 21}{1 + 4(6) + 6(36) + 4(216) + 1296} \] is equal to:
The eccentricity of the ellipse \[ \frac{(x - 1)^2}{2} + \left(y + \frac{3}{4}\right)^2 = \frac{1}{16} \] is:
If \[ f(x) = \begin{vmatrix} x & x^2 & x^3 \\ 1 & 2x & 3x^2 \\ 0 & 2 & 6x \end{vmatrix}, \] then \[ f'(x) \] is equal to:
Evaluate: \[ \int \frac{x^2}{1 + (x^3)^2} \, dx \]
Let \( f_n(x) \) be the \(n^{\text{th}}\) derivative of \( f(x) \). The least value of \( n \) such that \[ f_n(x) = f_{n+1}(x) \] where \[ f(x) = x^2 + e^x \] is:
The difference between the maximum and minimum value of the function \[ f(x) = \int_{0}^{x} (t^2 + t + 1)\, dt \] on the interval \[ [2, 3] \] is:
If the \(7^{\text{th}}\) and \(8^{\text{th}}\) terms of the binomial expansion \[ (2a - 3b)^n \] are equal, then the value of \[ \frac{2a + 3b}{2a - 3b} \] is equal to:
If \[ f(x) = \frac{1}{x^2 + 4x + 4} - \frac{4}{x^4 + 4x^3 + 4x^2} + \frac{4}{x^3 + 2x^2}, \] then the value of \[ f\left(\frac{1}{2}\right) \] is equal to: