List of top Mathematics Questions asked in CBSE CLASS XII

Show that the relation R defined in the set A of all polygons as R = {(P₁, P₂): P₁ and P₂ have same number of sides}, is an equivalence relation. What is the set of all elements in A related to the right angle triangle T with sides 3, 4 and 5?

Let f : R \(\to\) R be the Signum Function defined as \(f(x) =   \begin{cases}     1,     & \quad \text x>0 \\     0,  & \quad x=0 \\  -1, &\quad x<0 \end{cases}\)

and \(g: R \to R\) be the Greatest Integer Function given by g (x)= [x], where [x] is greatest integer less than or equal to x. Then does fog and gof coincide in (0,1]?

Define a binary operation *on the set {0,1,2,3,4,5} as
\(a*b =   \begin{cases}     a+b       & \quad \text{if } a+b<6 \\     a+b-6  & \quad \text{if } a+b\geq6    \end{cases}\)
Show that zero is the identity for this operation and each element a≠0 of the set is invertible with 6−a being the inverse of a.

Consider the binary operations*: R ×R → and o: R×R →R defined as \(a*b=\mid a-b \mid \)and a o b = a,∀a,b∈R. 
Show that * is commutative but not associative, o is associative but not commutative. 
Further, show that ∀a,b,c∈R, a * (b o c)= (a * b) o (a * c). [If it is so, we say that the operation * distributes over the operation o]. Does o distribute over *? Justify your answer.

Let A={1,2,3}. Then number of relations containing (1,2) and (1,3) which are reflexive and symmetric but not transitive is

Let A={1,2,3}. Then number of equivalence relations containing (1,2) is

Given a non empty set X, consider P(X) which is the set of all subsets of X.
Define the relation R in P(X) as follows:
For subsets A,B in P(X),ARB if and only if A⊂B. 
Is R an equivalence relation on P(X)? Justify you answer:

Find the number of all onto functions from the set {1,2,3,… ,n) to itself.

Given a non-empty set X, consider the binary operation * : P (X)×P (X)→P (X) given by A * B= A∩B ∀A,B in P (X) is the power set of X. Show that X is the identity element for this operation and X is the only invertible element in P (X) with respect to the operation*.

If \(f:R\to R\) is defined by \(f(x)=x^2-3x+2,find \,f(f(x)).\)

Let f : W \(\to\) W be defined as f(n)=n−1, if is odd and f(n)=n+1, if n is even. Show that f is invertible. Find the inverse of f. Here, W is the set of all whole numbers.

For each binary operation * defined below, determine whether * is commutative or associative. 
(i) On Z, define a * b=a−b 
(ii) On Q, define a * b=ab+1 
(iii) On Q, define a * b= \(\frac {ab}{2}\).
(iv) On Z⁺, define a * b=2ab 
(v) On Z⁺, define a * b=ab 
(vi) On R−{−1},define a * b= \(\frac {a}{b+1}\)

Show that the relation R in the set A of all the books in a library of a college, given by R = {(x, y): x and y have same number of pages} is an equivalence relation.

Find which of the operations given above has identity.

Let * be a binary operation on the set Q of rational numbers as follows: 
(i) a * b=a−b 
(ii) a * b=a²+b²
(iii) a * b=a+ab 
(iv) a * b= (a−b)² 
(v) a * b= \(\frac {ab} {4}\)
(vi) a * b=ab²

Find which of the binary operations are commutative and which are associative. 

Let * be the binary operation on N defined by a * b=H.C.F. of a and b. Is * commutative? Is * associative? Does there exist identity for this binary operation on N?

Is * defined on the set {1,2,3,4,5} by a*b=L.C.M. of a and b a binary operation? Justify your answer.

Let * be the binary operation on N given by a*b=L.C.M. of a and b. Find 
(i) 5*7, 20*16 
(ii) Is * commutative? 
(iii) Is * associative? 
(iv) Find the identity of * in N 
(v) Which elements of N are invertible for the operation *?

Let*′ be the binary operation on the set {1,2,3,4,5} defined by a*′b=H.C.F. of a and b. Is the operation *′ same as the operation * defined in Exercise 4 above? Justify your answer.

Consider a binary operation * on the set {1,2,3,4,5} given by the following multiplication table. 
(i) Compute (2 * 3)*4 and 2 *(3 * 4) 
(ii)Is * commutative? 
(iii)Compute (2 * 3)*(4 * 5).

(Hint: use the following table)

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∨.

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 = ab²
(iv) On Z⁺, define * by a * b = |a − b|
(v) On Z⁺, define * by a * b = a

Let f : X → Y be an invertible function. Show that the inverse of f^-1 is f, i.e., (f^-1)^-1 = f.