Mean of n observations x₁, x₂, ..., x_n, is $\overline{\mathrm{x}}$. If an observation ${\mathrm{x}}_{\mathrm{q}}$, is replaced by x_q^', then the new mean is

• $\overline{\mathrm{x}} - {\mathrm{x}}_{\mathrm{q}} + {\mathrm{x}}_{\mathrm{q}}\text{'}$

• $\frac{\left(\mathrm{n} - 1\right)\overline{\mathrm{x}} + {\mathrm{x}}_{\mathrm{q}}\text{'}}{\mathrm{n}}$

• $\frac{\left(\mathrm{n} - 1\right)\overline{\mathrm{x}} - {\mathrm{x}}_{\mathrm{q}}\text{'}}{\mathrm{n}}$

• $\frac{\mathrm{n}\overline{\mathrm{x}} - {\mathrm{x}}_{\mathrm{q}} + {\mathrm{x}}_{\mathrm{q}}\text{'}}{\mathrm{n}}$

On set A = {1, 2, 3}, relations R and S are given by

R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)},

S = {(1, 19, (2, 2), (3, 3), (1, 3), (3, 1)}.

Then,

• R $\cup$ S is an equivalence relation

• R $\cup$ S is reflexive and transitive but not symmetric

• R $\cup$ S is reflexive and symmetric but not transitive

• R $\cup$ S is symmetric and transitive but not reflexive

173.

If the function f : $\mathrm{R} \text{→ R is defined by f(x) = }{\left({\mathrm{x}}^2 + 1\right)}^{35}, \forall \mathrm{x} \in \mathrm{R}$, then f is

• one-one but not onto

• onto but not one-one

• neither one-one nor onto

• both one-one and onto

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