In the group G = {1, 5, 7, 11} under multiplication modulo 12, the solution of $7^{- 1} {\otimes }_{12} \left(\mathrm{x} {\otimes }_{12} 11\right) = 5$ is equals :

• 5

• 1

• 7

• 11

If f : R $\to$ R is defined by f(x) = lxl, then

• f^-1(x) = - x

• f^-1 = $\frac{1}{\left|\mathrm{x}\right|}$

• the function f^-1(x) does not exist

• ${\mathrm{f}}^{-1}\left(\mathrm{x}\right) = \frac{1}{\mathrm{x}}$

On the set of all natural numbers N, which one of the following * is a binary operation?

• $\mathrm{a} * \mathrm{b} = \sqrt{\mathrm{ab}}$

• $\mathrm{a} * \mathrm{b} = \frac{\mathrm{a} - \mathrm{b}}{\mathrm{a} + \mathrm{b}}$

• a * b = a + 3b

• a * b = 3a - 4b

On the set of integers Z, define f : Z $\to$ Z as f(n) = $$\begin{gathered} \left\{\frac{\mathrm{n}}{2}, \mathrm{n} \mathrm{is} \mathrm{even} \\ 0, \mathrm{n} \mathrm{is} \mathrm{odd}\right\} \end{gathered}$$, then 'f' is

• injective but not surjective

• neither injective nor surjective

• surjective but not injective

• bijective

The inverse of 2010 in the group Q^* of all positive rational under the binary operation * defined by a * b = $\frac{\mathrm{ab}}{2010}, \forall \mathrm{a}, \mathrm{b} \in {\mathrm{Q}}^{+}$ is

• 2009

• 2011

• 1

• 2010

Define a relation R on A = {1, 2, 3, 4} as xRy if x divides y. R is

• reflexive and transitive

• reflexive and symmetric

• symmetric and transitive

• equivalence

On the set of all non-zero reals, an operation * is defined as a * b = $\frac{3\mathrm{ab}}{2}$. In this group, a solution of (2 * x) * 3^-1 = 4^-1 is

• 6

• 1

• 1/6

• 3/2

If A and B have n elements in common, then the numberofelements common to A x B and B x A is

• n

• 2n

• n²

• 0

Which of the following is false ?

• (N, *) is a group
  
  

• (N, +) is a semi-group

• (Z, +) is a group

• Set of even integers is a group under usual addition

300.

Let S be the set of all real numbers. A relation R has been defined on S by aRb $\iff \left|\mathrm{a} - \mathrm{b}\right| \le 1$, then R is

• symmetric and transitive but not reflexive

• reflexive and transitive but not symmetrIc

• reflexive and symmetric but not transitive

• an equivalence relation