For independent events A_1, A_2, A_3, ..., A_n if P(A_i) = 1 i+1, i = 1, 2, 3, ..., n, then the probability that none of the events occur is:

If f(x) = 2x + 3, then f⁻¹(x) is:

If $\vec{a}$ is a unit vector perpendicular to both the vectors $\vec{b} = \hat{j} + \hat{2k}$ and $\vec{c} = \hat{i} + 2\hat{j}$, then $\hat{a}$ is equal to

If the roots of the equation x² - 5x + k = 0 are in the ratio 2:3, then the value of k is:

A function $f: \mathbb{R} \rightarrow \{x \in \mathbb{R}: -1 < x < 1\}$ is defined as $f(x) = \frac{x}{1+|x|}$, then $f$ is:

If A is a square matrix such that $A^2 = A$ and I is the identity matrix of same order as A, then the value of $(A-2I)^2 - (2A + I)^2 + 11A$ is: