If the binary operation * on the set of integers Z is defined by a*b=a+3b² then find the value of 2 * 4.

Let * be a binary operation on N given by a*b=HCF(a,b), \(a,b\in N\). Write the value of 22*4.

If f:\(R\to R\) and g:\(R\to R\)are given by f(x) = sin x and g(x) = 5x² find gof(x).

State the reason for the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} not to be transitive.

Let f:\(R\rightarrow R\) is defined by f(x) = x². Is f one-one?

Prove that the greatest integer function f : R \(\rightarrow\)R, given by f(x) = [x] is neither one-one nor onto. Where [x] denotes the greatest integer less than or equal to x

Show that the absolute value function : R\(\rightarrow\)R given by f(x) = |x| is neither one-one nor onto.

State the reason why  the Relation R = [(a, b) : \(a\le b^{ 2 }\) on the set R of the real numbers is not reflexive

State the reason for the following Binary operation * defined on the set Z of integers to be not commutative a*b = ab²

If the binary operation * on the set of integers Z is defined by a*b = a + 3b² then find the value of 8*3

Define Reflexive. Give one example.

Define Transitive Relation. Give one example.

Let f and g be real function be \(f(x)=\sqrt { x+4 } ,x\ge 4\) find the function fg, \(\frac { f }{ g } \)

Which of the following binary operations are commutative?
(i) * On Z defined by a*b = a²+ b²
(ii) * On Q  defined by a*b = a²+2b

Let * be a binary operation On the set of all non-zero real numbers given by a*b = \(\frac { ab }{ 5 } \)
For all a, b \(\in R-[0]\)
Find the value of x, given that
(i) 2 * (x*5) = 6
(ii) 3 (x*3) = 9

Show that the relation R in the set Z of integers given by:
R = {(a, b) : 2 divides a-b} is an equivalence relation

Let f: \({ \{ 2,3,4,5 }\} \rightarrow \{ 3,4,5,9\} \) and \(g=\{ 3,4,5,9\} \rightarrow \{ 7,11,15\} \) be functions defined as 
f(2) = 3, f(3) = 4, f(4) = f(5) = 5 and g(3) = g(4) = 7, and g (5) = g(11) = 11. Find gof.

Show that if \(f:A\rightarrow B\) and \(g:B\rightarrow C\) are onto, then \(gof:A\rightarrow C\) is also onto.

Show that subtraction and division are not binary operations on N.

Show that \(*:R\times R\rightarrow R\) given by \(a*b\rightarrow a+2b\) is not associative.

In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.
(i) \(f:R\rightarrow R\) defined by f(x) = 3 - 4x
(ii) \(f:R\rightarrow R\) defined by \(f(x)=1+x^{ 2 }\) .

Determine whether or not each of the definition of '*' given below gives a binary operation. In the event that '*' is not a binary operation, give justification for this:
(i) On \(Z^{ + }\), define '*' by a *b=a-b
(ii) On \(Z^{ + }\), define '*' by a *b=ab
(iii) On R, define '*' by a * b=\(ab^{ 2 }\)
(iv) On \(Z^{ + }\), define '*' by a *b=\(\left| a-b \right| \)
(v) On \(Z^{ + }\), define '*' by a *b=a