The binary operation \( \oplus \) defined on the set of real numbers is such that \( x \oplus y = \frac{xy}{6} \) for all \( x, y \in R \). Find the inverse...

If \( \tan \theta = \frac{5}{4} \) find \( \sin^2 \theta - \cos^2 \theta \)

A binary operation \( \oplus \) on the set of rational numbers is defined as \(x \oplus y = \frac{x^2-y^2}{2xy}\). Find \(-5 \oplus 3\)

If \(x = \begin{vmatrix} 1 & 0 & 1 \\ 2 & -1 & 0 \\ -1 & 0 & 1 \end{vmatrix}\) and \(y = \begin{vmatrix} -1 & 1 & 2 \\ 0 & -1 & -1 \\ 2 & 0 & 1 \end{vmatrix}...

The sum of the first n positive integers is

Find p, q for which \(\begin{pmatrix}2p & 8 \\ 3 & -5q\end{pmatrix}\) \(\begin{pmatrix}1 \\ 2\end{pmatrix}\) \(\begin{pmatrix}24 \\ -17\end{pmatrix}\)

Find the roots of x3 - 2x2 - 5x + 6 = 0

If \(T = 2\pi\sqrt{\frac{L}{g}}\), make \(g\) the subject of the formula

If the locus of the points which are equidistant from points P and Q meets line PQ at point N, then PN equals