A binary operation * is defined on the set of real numbers, R, by \(x * y = x + y - xy\). If the identity element under the operation * is 0, find the inver...

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

Express \(\log \frac{1}{8} + \log \frac{1}{2}\) in terms of \(\log 2\).

Given that \(a^{\frac{5}{6}} \times a^{\frac{-1}{n}} = 1\), solve for n.

Find the third term in the expansion of \((a - b)^{6}\) in ascending powers of b.

If \(f(x) = x^{2}\) and \(g(x) = \sin x\), find g o f.

Given that \(f(x) = 2x^{3} - 3x^{2} - 11x + 6\) and \(f(3) = 0\), factorize f(x).

Solve: \(\sin \theta = \tan \theta\)

If \(\sqrt{x} + \sqrt{x + 1} = \sqrt{2x + 1}\), find the possible values of x.