In Z, the set of all integers, the inverse of - 7 w.r.t. * defined by ab = a + b + 7 for all a,b $\in$ Z is

• - 14

• 7

• 14

• - 7

In three element group {e, a, b} where e is the identity a⁵b⁴ is equal to

• a

• e

• ab

• b

The relation R = {(1, 1), (2, 2), (3, 3)} on the set { 1, 2, 3} is

• symmetric only

• reflexive only

• an equivalence relation

• transitrve only

Let M be the set of all 2 x 2 matrices with entries from the set ofreal number R. Then, the function f : M R defined by f(A) = $\left|\mathrm{A}\right|$ for ever A $\in$ M, is

• one-one and onto

• neither one-one nor onto

• one-one but not onto

• onto but not one-one

In the group (Q⁺, *) of positive rational numbers w.r.t. the binary operation * defined by a * b = $\frac{\mathrm{ab}}{3}, \forall \mathrm{a}, \mathrm{b} \in {\mathrm{Q}}^{+}$, the solution of the 3 equation 5 * 4 = 4^-1 in Q⁺ is

• $\frac{27}{20}$

• $\frac{20}{27}$

• $\frac{1}{20}$

• 20

On the set Q of all rational numbers the operation * which is both associative and commutative is given by a * b, is :

• a + b + ab

• a² + b²

• ab + 1

• 2a + 3b

The function f : X $\to$ Y defined by f(x) = sin(x) is one-one but not onto, if X and Y are respectively equal to :

• R and R

• $\left[0, \mathrm{\pi }\right] \mathrm{and} \left[0, 1\right]$

• $\left[0, \frac{\mathrm{\pi }}{2}\right] \mathrm{and} \left[- 1, 1\right]$

• $\left[\frac{- \mathrm{\pi }}{2}, \frac{\mathrm{\pi }}{2}\right] \mathrm{and} \left[- 1, 1\right]$

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