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 the binary operation * defined on Q is defined as a*b=2a+b-ab, for all \(a,b\in Q,\) find the value of 3*4.

Let f : \(R\rightarrow R\) is defined by f(x) = |x|. Is function f onto? Give reasons.

Let R be a relation in the set of natural numbers N defined by R={(a,b)\(\in\) N x N;a < b} s relation R reflexive? Give a reason.

Let A be any non-empty set and P(A) be the power set of A. A relation R defined on P(A) by \(X\ R\ Y\Leftrightarrow X\ \cap Y=X,X,Y\in P(A)\)  . Examine whether R is symmetric.

Show that division is not a binary operation on N.

Let the function f : R\(\rightarrow\)R to be defined by f(x) = cos x \(\forall \) x \(\in\)R. Show that  is neither one-one nor onto.

For the set A = {1, 2, 3} define a relation R in the set A is follows:
R = {(1,1), (2, 2), (3, 3), (1, 3)}. Write the ordered pairs to be added to R to make it the smallest equivalence relation.

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 2*(x* 5) = 10

Given an example of a relation which is
(i) Reflexive, Symmetric and transitive
(ii) Reflexive, Symmetric and not transitive.

Let A be the set of all human beings in a town at a particular time. Determine whether the relation R = {(x, y) : x is wife of y ; x, Y\(\in \)A} is reflexive, symmetric and transitive.

Consider the relation perpendicular on a set  of lines in a plane. Show that this relation is symmetric and neither reflexive and nor transitive.

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

Show that the function \(f\ :\ N\rightarrow N\) given by f (x) = 2x  is one-one but not onto.

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

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

Let S = {1,2,3}. Determine whether the functions \(f:S\rightarrow S\) defined below has inverse. Find \(f^{ -1 }\) , if it exists:
(a) f = {(1, 1), (2, 2), (3, 3)}
(b) f = {(1, 2), (2, 1), (3, 1)}
(c) f = {(1, 3), (3, 2), (2, 1)}

Show that the relation S in the set R of real numbers, defined as
S = {(a, b): a,b \(\in\)R and a \(\le\)b³} is neither reflexive, nor symmetric nor transitive.

Consider \(f:R_+\rightarrow[-5,\infty) \) given by f(x) = 9x² + 6x - 5. Show that f is invertible with \(f^{-1}(y)={(\sqrt{(y+6)}-1)\over3}\)

Consider the binary operation * on the set {1, 2, 3, 4, 5} defined by a*b = min{a, b}. Write the operation table of the operation *.

Let R be a relation defined on the set of natural numbers Nas follow:
R = {(x, y) : x \(\in \) N, y \(\in \)N and 2x + y = 24}
Find the domain and range of the relation R. Also, find if R is an equivalence relation or not.

Write the following functions in the simplest form:
\({ tan }^{ -1 }\left[ \frac { 3{ a }^{ 2 }x-{ x }^{ 3 } }{ { a }^{ 3 }-3{ a }^{ 2 } } \right] ,\ a>0;\ -\frac { a }{ \sqrt { 3 } } \le x\le \frac { a }{ \sqrt { 3 } } \)

Show that the binary operation * on A = R-{-1} defined as a*b = a + b for all a, b, c A is  commutative and associative on A. Also find the identity element of * in A and prove that every element of A is invertible

Determine whether the operation * define below on Q is binary operation or not.
a*b = ab+1
If yes, check the commutative and the associative properties. Also check the existence
of identity element and the inverse of all elements is Q.

If \(A=\left( \begin{matrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3 \end{matrix} \right) \) and A³ - 6A² + 7A + kI³ = 0, find k.

Using elementary transformation, find the inverse of the matrix \(A=\left[ \begin{matrix} 8 & 4 & 3 \\ 2 & 1 & 1 \\ 1 & 2 & 2 \end{matrix} \right] \) and use it to solve the following system of lines equations :
8x + 4y + 3z = 19
2x + y + z = 5
x + 2y + 2z = 7

Using elementary column operations, find the inverse of the following matrix :
\(\left[ \begin{matrix} -1 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1 \end{matrix} \right] \)