A binary operation * is defined on the set of real numbers, by \(a * b = \frac{a}{b} + \frac{b}{a}\). If \((\sqrt{x} + 1) * (\sqrt{x} - 1) = 4\), find the v...

If \(\log_{3}a - 2 = 3\log_{3}b\), express a in terms of b.

Given that \(f(x) = 3x^{2} - 12x + 12\) and \(f(x) = 3\), find the values of x.

If \(\alpha\) and \(\beta\) are the roots of \(2x^{2} - 5x + 6 = 0\), find the equation whose roots are \((\alpha + 1)\) and \((\beta + 1)\).

Simplify \(\frac{\sqrt{3}}{\sqrt{3} -1} + \frac{\sqrt{3}}{\sqrt{3} + 1}\)

If the polynomial \(f(x) = 3x^{3} - 2x^{2} + 7x + 5\) is divided by (x - 1), find the remainder.

Find the domain of \(g(x) = \frac{4x^{2} - 1}{\sqrt{9x^{2} + 1}}\)

\(P = {1, 3, 5, 7, 9}, Q = {2, 4, 6, 8, 10, 12}, R = {2, 3, 5, 7, 11}\) are subsets of \(U = {1, 2, 3, ... , 12}\). Which of the following statements is true?

If \(4x^{2} + 5kx + 10\) is a perfect square, find the value of k.