List of top Mathematics Questions asked in CBSE CLASS XII

If f : R→R be given by f(x)= \((3-x^3)^\frac {1} {3}\),,then fof(x) is 

Determine whether each of the following relations are reflexive, symmetric, and transitive.

  1. Relation R in the set A={1,2,3,...13,14} defined as R={(x,y): 3x-y=0}.
  2. Relation R in the set of N natural numbers defined as R={(x,y): y=x+5 and x<4}.
  3. Relation R in the set A={1,2,3,4,5,6} defined as R={(x,y): y is divisible by x}.
  4. Relation R in the set of Z integers defined as R={(x,y): x-y is an integer}
  5. Relation R in the set of human beings in a town at a particular time given by
    1. R={(x,y): x and y work at same place}
    2. R={(x,y): x and y live in the same locality}
    3. R={(x,y): x is exactly 7 am taller that y}
    4. R={(x,y): x is wife of y}
    5. R={(x,y):x is father of y}

For the matrices A and B, verify that (AB)′= B'A' where
(i)A=\(\begin{bmatrix}1\\-4\\3\end{bmatrix},\,B=\begin{bmatrix}-1&2&1\end{bmatrix}\)

(ii)A=\(\begin{bmatrix}0\\1\\2\end{bmatrix},\,B=\begin{bmatrix}1&5&7\end{bmatrix}\)

For the matrix A=\(\begin{bmatrix}1&5\\6&7\end{bmatrix}\),verify that 
  I. (A+A') is a symmetric matrix
  II. (A-A') is a skew symmetric matrix

  1. Show that the matrix A=\(\begin{bmatrix}1&-1&5\\-1&2&1\\5&1&3\end{bmatrix}\) is a symmetric matrix.
  2. Show that the matrix A=\(\begin{bmatrix}0&1&-1\\-1&0&1\\1&-1&0\end{bmatrix}\) is a skew symmetric matrix

Find the inverse of each of the matrices, if it exists. \(\begin{bmatrix}1&-1\\2&3\end{bmatrix}\)

If A'=\(\begin{bmatrix}-2&3\\1&2\end{bmatrix}\)and B=\(\begin{bmatrix}-1&0\\1&2\end{bmatrix}\),then find (A+2B)'

If A=\(\begin{bmatrix}-1&2&3\\5&7&9\\-2&1&1\end{bmatrix}\)and B=\(\begin{bmatrix}-4&1&-5\\1&2&0\\1&3&1\end{bmatrix}\),then verify that
 (i)(A+B)'=A'+B'
 (ii)(A-B)'=A'-B'

Let A= {−1,0,1,2}, B={−4,−2,0,2} and f,g: A→B be functions defined by \(f(x)=x^2-x, \,x\in A\, and \,g(x)=2\mid\frac{ x-1}{2}\mid-1,x\in A.\). Are f and g equal? Justify your answer.    (Hint: One may note that two function \(f:A\to B \,and \: g:A\to B\) such that \(f(a)=g(a) \forall \,a \in\,A,\) are called equal functions). 

Evaluate the determinants in Exercises 1 and 2. 
(i) \(\begin{vmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{vmatrix}\) 

(ii) \(\begin{vmatrix}x^2&-x+1&x-1\\& x+1&x+1\end{vmatrix}\)

Evaluate the determinants in Exercises 1 and 2. 

 \(\begin{vmatrix}2&4\\-5&-1\end{vmatrix}\)

Let f :X\(\to\) Y be an invertible function. Show that f has unique inverse.
(Hint: suppose g1 and g2 are two inverses of f. Then for all y∈ Y, fog1(y) = IY (y) = fog2 (y). Use one-one ness of f).

Show that f :[−1,1]→R,given by f(x)= \(\frac {x}{x+2}\) is one-one. Find the inverse of the function f :[−1,1] \(\to\) Range f.
(Hint: For y ∈Range f, y = f(x)= \(\frac {x}{x+2}\) , for some x in [−1, 1], i.e.,x= \(\frac {2y}{1-y}\)

Find the inverse of each of the matrices, if it exists. \(\begin{bmatrix}2&1\\1&1\end{bmatrix}\)

Find the transpose of each of the following matrices:
  I.\(\begin{bmatrix}5\\\frac{1}{2}\\-1\end {bmatrix}\)

  II.\(\begin{bmatrix}1&-1\\2&3\end{bmatrix}\)

  III.\(\begin{bmatrix}-1&5&6\\\sqrt3&5&6\\2&3&-1\end{bmatrix}\)

The bookshop of a particular school has 10 dozen chemistry books, 8 dozen physics books, 10 dozen economics books. Their selling prices are Rs 80, Rs 60 and Rs 40 each respectively. Find the total amount the bookshop will receive from selling all the books using matrix algebra.

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

Let L be the set of all lines in XY plane and R be the relation in L defined as R = {(L1, L2): L1 is parallel to L2 }. Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4.