Let f: $\mathrm{X} \to \mathrm{X} \mathrm{be} \mathrm{such} \mathrm{that} \mathrm{f}\left[\mathrm{f}\right(\mathrm{x}\left)\right] = \mathrm{x}, \mathrm{for} \mathrm{all} \mathrm{x} \in \mathrm{X} \mathrm{and} \mathrm{X} \subseteq \mathrm{R}$ then

• f is one - to - one

• f is onto

• f is one - to - one but not onto

• f is onto but not one - to - one

If A and B are two matrices such that AB = B and BA = A, then A² + B² equals

• 2AB

• 2BA

• A + B

• AB

A relation p on the set of real number R is defined as {x$\mathrm{\rho }$y: xy > 0}. Then, which of the following is/are true?

• $\mathrm{\rho }$ is reflexive and symmetric

• $\mathrm{\rho }$ is symmetric but not reflexive

• $\mathrm{\rho }$ is symmetric and transitive

• $\mathrm{\rho }$ is an equivalence relation

The function f(x) = x² + bx + c, where b and c real constants, describes

• one - to - one mapping

• onto mapping

• not one-to-one but onto mapping

• neither one-to-one nor onto mapping

For any two real numbers $\mathrm{\theta } \mathrm{and} \mathrm{\varphi }$ we define $\mathrm{\theta R\varphi }$, if and only if ${\sec }^2\left(\mathrm{\theta }\right) - {\tan }^2\left(\mathrm{\varphi }\right)$ = 1. The relation R is

• reflexive but not transitive

• symmetric but not reflexive

• both reflexive and symmetric but not transitive

• an equivalence relation

We define a binary relation $~$ on the set of all 3 x 3 real matrices as A $~$ B,if and only if there exist invertible matrices P and Q such that B = PAQ^-1 .The binary relation $~$ is

• neither reflexive nor symmetric

• reflexive and symmetric but not transitive

• symmetric and transitive but not reflexive

• an equivalence relation

In the set of all 3 x 3 real matrices a relation is defined as follows. A matrix A is related to a matrix B, if and only if there is a non-singular 3 x 3 matrix P, such that B = P^-1AP. This relation is

• reflexive, symmetric but not transitive

• reflexive, transitive but not symmetric

• symmetric, transitive but not reflexive

• an equivalence relation

458.

For any two real numbers a and b, we define a R b if and only if sin²(a) + cos²(b) = 1. The relation R is

• reflexive but not symmetric

• symmetric but not transitive

• transitive but not reflexive

• an equivalence relation

The total number of injections (one-one into mappings) from {a₁, a₂, a₃, a₄} to {b₁, b₂, b₃, b₄, b₅, b_6, b₇} is

• 400

• 420

• 800

• 840

Let IR be the set of real numbers and f : IR ➔ IR be such that for all x, y ∈ IR, . Prove that f is a constant function.