List of top Questions asked in CBSE CLASS XII

Assume X,Y,Z,W and P are matrices of order 2 x n, 3 x k, 2 x p, n x 3, and p x k respectively.
The restriction on n, k and p so that PY+WY will be defined are: 
 A. k = 3,p = n 
 B. k is arbitrary, p = 2 
 C. p is arbitrary, k=3 
 D. k=2,p=3

Assume X,Y,Z,W and P are matrices of order 2 x n, 3 x k, 2 x p, n x 3, and p x k respectively. If n = p, then the order of the matrix 7X-5Zis 

The number of all possible matrices of order 3 × 3 with each entry 0 or 1 is:

Which of the given values of x and y make the following pair of matrices equal \(\begin{bmatrix}3x+y&5\\y+1&2-3x\end{bmatrix}=\begin{bmatrix}0&y-2\\8&4\end{bmatrix}\)

A=\([a_{ji}]_{m*n}\) is a square matrix, if

Find the value of a,b,c, and d from the equation: \(\begin{bmatrix}a-b&2a+c\\2a-b&3c+d\end{bmatrix}=\begin{bmatrix}-1&5\\0&13\end{bmatrix}\)

Find the value of x, y, and z from the following equation:
 I.\(\begin{bmatrix} 4&3&\\x&5\end{bmatrix}=\begin{bmatrix}y&z\\1&5\end{bmatrix}\)

  II. \(\begin{bmatrix}x+y&2\\5+z&xy\end{bmatrix}=\begin{bmatrix}6&2\\5&8\end{bmatrix}\)

   III. \(\begin{bmatrix}x+y+z\\x+z\\y+z\end{bmatrix}=\begin{bmatrix}9\\5\\7\end{bmatrix}\)

Check the injectivity and surjectivity of the following functions:

1. f: N \(\to\) N given by f(x) = x²
2. f: Z \(\to\) Z given by f(x) = x²
3. f: R \(\to\) R given by f(x) = x²
4. f: N \(\to\) N given by f(x) = x³
5. f: Z \(\to\) Z given by f(x) = x³

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:

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?