MULTIPLE CHOICE QUESTIONS FOR CLASS XII

METRIX & DETERMINANT

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VERY SHORT ANSWER QUESTIONS

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                        Total 131 questions

                        ..----------------------------..

1) If A is and m x n  matrix and B is n x p matrix does AB exists? If yes, write its order.         Yes,.mxp

 2) If A= 2    1    4 and B=  3     -1 

               4    1    5               2      2

                                             1      3 , write the order of AB and BA.      2x2 and 3x3

3) If A=  4      3  and      B= - 4

               1      2                      3 

write AB.                                         -7

                                                          2

4) If A= 1

              2

              3 write AA'.             1    2     3

                                               2   4     6

                                               3   6     9

5) Give an example of two non-zero  2 x 2 matrices A and B such that AB= O.             A= 2    0 &B= 0     0

                              3     0         2    -1

6) If A= 2      3 

              5      7 find A+ A'.       4     8

                                                  8    14

7) If A= i    0 

             0     i write A².              -1      0

                                                    0     -1

8) If A= cosx        sinx 

            - sinx         cosx                    find x satisfying 0< x < π/2 when A+ A'= I.                                             π/3

 

9) If A= cosx         sinx 

            - sinx          cosx                  find AA'.                                     1      0

                                                   0       1

10) If A= 1    0 and B= x      0

                y     5             1      -2 and the relation A+ 2B= I,  where I is 2 x 2 unit matrix. find x and y.        0,-2

11) If A= 1      -1

               -1       1  satisfies the matrix equation A²=kA,write the value of k.                                     2

12) If A= 1       1

                1       1 satisfies A⁴= kA, then  write the value of k.              8

13)      -1        0         0

If A=     0       -1        0

             0        0       -1                        find A².                                  -A or I₃

14)     -1       0        0

If A=    0      -1        0

            0.      0       -1 find A³.         A

15) If A= -3       0

                 0      -3 find A⁴.         81     0

                                                    0    81

16) If A= x      2 and B= 3

                                        4 and the relation AB= 2 find x.                -2

17) If A= [aᵢⱼ] is a 2 x 2 matrix such that aᵢⱼ= i+ 2j, write A.            3      5

                                                 4       6

18) write matrix A satisfying A+ B= C where B= 2     3  & C = 3      -6

                    -1     4           -3       8

                                                  1     -9

                                                 -2      4

19) if A= [aᵢⱼ] is a square Matrix that aᵢⱼ = i² - j², then write whether A is symmetric or skew symmetric.     sk

20) For any square matrix write whether AA' is symmetric or skew-symmetric.                               symm

21) If  A= [aᵢⱼ] is a skew symmetric matrix, then write the value of ᵢ∑aᵢⱼ.  0 

22) If  A= [aᵢⱼ] is a skew symmetric matrix, then write the value of ᵢ∑ ⱼ∑aᵢⱼ .                                                 0

23) If A and B are symmetric matrices, then write the condition for which AB is also symmetric.       AB= BA

24) If B is a skew-symmetric matrix, write whether the matrix AB A' is symmetric or Skew-symmetric.   Sk

25) If B is a symmetric matrix, write whether the matrix AB A' is  symmetric or Skew-symmetric.   sy

26) If A is a skew-symmetric and n ∈ ℕ such that (Aⁿ)ᵀ = K Aⁿ, write the value of K.                                    (-1)ⁿ

27) If A is a symmetric matrix and n∈ ℕ, write whether Aⁿ is symmetric or Skew-symmetric or neither of these two.                   sy

28) if A is a symmetric matrix and n is an even natural number, write whether Aⁿ is symmetric or Skew-symmetric or neither of these two.   sy

29) If A is a skew-symmetric matrix and n is an odd natural number, write whether Aⁿ is symmetric or Skew-symmetric or neither of these two.                                                 sk

30) If A and B are symmetric matrices of the same order, write whether AB - BA is symmetric or Skew-symmetric or neither of these two.                           sk

31) write a square matrix which is both symmetric as well as skew symmetric.                      null matrix

32) Find the values of x and y, if 2A+ B= C where A= 1   3  B= y    0 

                                   0   x       1    2

And C= 5       6

              1       8                              3,3

33) If  x+ 3         4       =  5        4

           y-4          x+ y        3        9 then find x and y.                               2,7

34) If A is a singular Matrix, then write the value of |A|.                      0

35) For what value of x, the following matrix is singular ?

 5 - x     x+1

   2         4                                         3

35) write the value of the determinant of 2      3        4 

                          2x    3x       4x 

                          5      6         8           0

36) State whether the matrix 2     3

                                                   6     4 is singular or non-singular.           ns

37) find the value of the determinant 4200       4201

                       4202       4203         -2

38) find the value of the determinant    101   102    103

                         104   105    106

                         107   108    109      0

39) Write the value of the determinant a      1       b+c

                       b      1       c+a 

                       c      1       a+b           0

40) If A= 0      i    &  B= 0        1 

                 i      1             1        0 find the value of |A|+| B|.                       0

41) if A= 1         2   & B= 1     0

                3        -1           -1     0 find |AB|.                                                   0

42) Evaluate 4785      4787 

                       4789      4791            8

43) if w an imaginary cube root of unity, find the value of 1    w    w²

                                        w   w²    1 

                                        w²   1     w   0

44) If A= 1     2 and B= 1      -4 

                 3   -1                3     -2 find |AB|.                                             -70

45) If A=[aᵢⱼ] is a 3x3 diagonal matrix such that a₁₁=11, a₂₂= 2 and a₃₃ = 3, then find |A|.                      6

46) If A=[aᵢⱼ] is a 3x3 scalar Matrix such that a₁₁=2, then write the value of |A|.                                             8

47) If I₃ denotes Identity matrix of order 3x3, write the value of its determinant.                                     1

48) A matrix A of order 3x3 has determinant 5. What is the value of |3A| ?                                     135

49) On spending by first row, the value of the determinants of 3x3 square Matrix A=[aᵢⱼ] is a₁₁ C₁₁+ a₁₂C₁₂+ a₁₃ C₁₃, where Cᵢⱼ is the cofactor of aᵢⱼ in A. write the expression for its value on expanding by the second column.               a₁₂ C₁₂+ a₂₂C₂₂+ a₃₂ C₃₂

50) let A= [aᵢⱼ] be a square matrix of order 3x3 and Cᵢⱼ denote cofactor of aᵢⱼ in A. If |A|= 5, write the value of a₃₁C₃₁+ a₃₂C₃₂+ a₃₃Ca₃₃.                  5

51) In question 18, write the value of write the value of a₁₁C₂₁+a₁₂C₂₂ + a₁₃ C₂₃ .                    0

52) Write the value of 

sin 20    - cos 20 

sin 70      cos 70                            1

53) If a square Matrix satisfying A'. A = I, write the value of |A|.       ±1

54) If A and B are square matrices of the same order such that |A| = 3 and AB= I, then write the value |B|.   1/3

55) A is a skew-symmetric of order 3, write the value of |A|.              0

56) If A is a square matrix of order 3 with the determinants 4, then write the value of |-A|.                -4

57) if A is square Matrix such that |A|= 2, Write the value of |AA'|.     4

58) find the value of the determinant 243    156      300 

                         81      52      100

                         -3        0          4       0

59) write the value of the determinant of 2        -3      5

                            4       -6      10

                            6       -9      15       0

60) If the matrix 5x       2

                            -10       1 is singular, find the value of x.                    -4

61) If A is a square matrix of order n x n such that |A| = K, then write the value of |-A|.                        (-1)ⁿ

62) find the value of the determinant 2²      2³       2⁴

                       2³      2⁴       2⁵

                       2⁴      2⁵       2⁶           0

63) if A and B are nonsingular matrices of the same order, write whether AB is singular or non-singular.                                 ns

64) A metrix of order 3 x 3 has determinant 2. What is the value of |A(3I)|, where I is the identity matrix of order 3 x 3.                               54

65) If A and B are square matrices of order 3 such that |A| = -1, |B| = 3, then find the value of |3AB|.     -81

66) Write the value of a+ ib   c+id

                                      -c+id    a+ib      a²+b²+c²+d²

67) write the cofactor of a₁₂ in the following metrix 2    -3       5

                              6     0       4

                              1     5      -7         46

68) If 2x+5       3

          5x+2       9 = 0, find x.         -13

69) Write the adjoint of the matrix A= -3       4                               -2    7

       7      -2                               4    -3

70) If A is square matrix such that A(adj A)= 5I, where I denotes the identity matrix of the same order. Then, find the value of |A|.            5

71) If A is a square Matrix of order 3 such that |A|= 5, write the value of |adj A|.                                          25

72) If A is a square matrix of order 3 such that |adj A|= 64, find |A|.   ±8

73) if A is a non-singular Square matrix such that |A|=10, find |A⁻¹|.   1/10

74) If A, B, C are three non-null square matrices of the same order, write the condition on A such that AB= AC => B= C.           A must be invertible

75) If A is a non-singular square Matrix such that A⁻¹= 5       3

                                      -2      -1 then find (A')⁻¹.                            5      -2 

                                              3      -1

76) if adj A= 2       3

                      4       -1 and adjoint of B= 1         -2

     -3          1 find adj of AB.      -6     5

                                                    -2  -10

77) If A is a symmetric matrix, write whether A' is symmetric or Skew-symmetric.            Sy

78) If A is a square matrix of order 3 such that |A| = 2, then write the value of adj(adj A).            2A

79) If A is a square matrix of order 3 such |A|=3, then find the value of |adj (adj A)|.                                  81

80) If A is a square matrix of order 3 such that adj(2A)= k adj(A), then write the value of k.                   4

81) If A is a square matrix, then write the matrix adj(A') - (adj)'.   Nm

82) Let A be a 3 x 3 square Matrix such that A(adj A)= 2I, where I is the identity Matrix, write the value of |adj A|.                                             4

83) if A is a nonsingular symmetric matrix, write whether A⁻¹ is Symmetric or Skew-symmetric.   Sy

84) If A= cosx       Sinx

              - sinx       cos x and A(adjoint A)= k      0

                        0       k then find the value of k.                              1

85) If A is an invertible Matrix such that |A⁻¹|= 2, find the value of |A|.  1/2

86) If A is a square Matrix such that A(adj A)= 5      0      0

                  0      5      0

                  0      0      5 then write the value of |adj A|.                           25

87) If A= 2      3

                5     -2 be such that A⁻¹= kA, then find the value of k.       1/19

88) Let A be a square matrix such that A² - A + I= O, then write A⁻¹ in terms of A.                A⁻¹=(I - A)

89) using cramer's rule write the solution of the system of equations 3x+4y=7; 7x - y= 6.                  1,1

90) find the inverse of the matrix

 3      -2.                               5       2

-7       5                                7       3

91) find the inverse of cosx     sinx 

                                        -sinx    cosx

                           Cox     - sinx 

                           Sinx      cosx

92) If A= 1      -3

                2       0 write adj A.     0   3 

                                                    -2   1

93) If A= a      b   & B = 1      0

                c      d              0      1 find adj(AB).                                d       - b 

                                             - c        a

94) If A= 1   0  0 & B= x   & C= 1

                0   1  0          y           -1

                0   0  1          z            0 with the relation AB= C, then find x,y, z.    1, -1,0

95)If A= 1   0    0 & B= x   & C= 1

                0  -1   0          y            0

                0   0  -1          z            0      with the relation AB= C, then find x,y and z.                   1,0, -1

96)If A= 1   0    0 & B= x   & C= 1

                0   y   0          -1           0

                0   0   1           z            1 with the relation AB= C, then find x,y and z.              1,0,1

97) If A= 3   -4  0 & B= x  

                9   2  0           y    with the relation AB= C, then find x,y.   2/3,-2

98) If A= 1   0  0 & B= x   & C= 2

                0   0  1          y           -1

                0   1  0          z            3 with the relation AB= C, then find x,y, z.   2,3, -1

99) If A= 2   4   & B= n   & C= 8

                4   3            1           11 with the relation AC= B, then find n.      2

100) ) A Matrix has 12 elements. Find the possible orders of the matrix.           1x12 or 12x1 or 2x6 or 6x2 or 3x4 or 4x3.

101) Construct a 2x2 matrix whose (i,j)th element as= {(i+j)²/2}. 

                     1/2        9/2

                       0            2

102) If 3A= 3      -4

                   15      5 find 4A.  4  -16/3

                                               20   20/3

103) If A+  1    3      4    -1

                  -1     2  = 6     2                    find A.                                 3       -4

                                            7        0

104) If 2(x     y) + (1     2)=(3    8), find the values of x and y.      1,3

105)  if A= 3    and B= x & C= 3

                   4.                 y         11 with the relation 2A + 3B= C, find the value of x and y.                           -1,1

106) Under what condition the product of two matrices AB of two metrices A and B can be determined ?      

107) If A= 1    2    3  & B=  7     8

                  4    5    6            0     9  find which one of the product AB and BA cannot be determined.        AB

108) Find the if A and B are two square matrix order of the matrix B if  [3   4     2]B=[2  1  0  1  2]      3x5

109) If A= 1  2  3 

&    B= 4   5    6       & C= 3

            5   6    7                2

            6   7  8                  1 with the relation ABC then find the order of ABC.                                             1x1

110) If A= 3  0  & B= a  b  & C= a  b

                  0   3           c  d           c  d Prove AB= 3C.

111) If A= x    2x & B= 2  & C= 16

                 2y    y            3            21 find the value of x and y.           2,3

112) If A= 1    1

                   0    1 find A² and A³.

                            1    2  & 1       3

                            0    1      0       1

113) If A= -3  2

                  -6   4 show A²= A

114)If A= cosx  -sinx  B= cosx   sinx

                  sinx   cosx     - sinx  cosx show that AB= I

115) If A= cosx  sinx       

                - sinx  cosx      show that

A²= cos2x     sin2x

       - sin2x    cos2x

116) Find the value of AB when 

A= a        & B= a    b        c

      b

      c.                       a²     ab    ac 

                               ab      b²    bc 

                               ac      bc    c²

117) In the case of two matrices A and B if AB= BA, show that (A+B)(A-B)= A² - B².            

118) IF A= 7  1  2 & B= 3  & C= 4

                   9  2  1          4           2

                                        5                      find AB+ 2C.                                  43

                                                        44

119) If A and B are two square matrices of the same order then what condition (A+B)²= A²+2AB+B² ?                                            BA= AB

120) for what value of k, the inverse of the matrix A= 2.   k

                              3    5 does not exist?                                          10/3

121) find the value of the determinant of the square matrix A= 4         -3

      9          7                                  55

122) If A= sin 70      cos 70

                   Sin 20    - cos 20 

find |A|.                                         -1

123) value of       logₓy     1 

                                 1     logₓy          0

124) find the number of elements of a determinants of order n.     n²

125) find the cofactor of the element a in the determinant 

7    -6      5

1     2      a 

3    -2     1                                      -4

126) show that  a-b   b-c   c-a

                             b-c  c-a    a-b = 0

                             c -a   a- b  b-c 

127) Evaluate    91   92     93

                            94   95     96

                            97    98    99          0

128) Solve: x+ a    b

                       a    x+ b = 0.   0, -(a+b)

129) Solve: 1     1     1

                     p     x     p  = 0

                     q    q      x                 p,q

130) If A=2   3  &B= 0   4 & C= 2   3

                -1   6          4    0        -1   6

Is the relation AB= 4C true? 

131) If A= 1      -1 

                  -1      1 show that A²= 2A

MULTIPLE CHOICE QUESTIONS

----------*******------------*********--------

                         Total questions - 80

1) If A= 1    0    0

              0    1    0

              a     b  -1 then A² is equal to

A) null matrix     B) a unit matrix.

C) - A      D) A

2) If A= i     0

             0      i, n∈ ℕ, then A⁴ⁿ equal 

A) 0   i      B) 0   0  C). 1  0  D) 0   i

     i   0           0   0        0  1      i    0

 

3) If A and B are two Matrices such that AB= A and BA= B, then B² is equals to 

A) B.       B) A       C) 1         D) 0

4) If AB= A and BA= B, where A and B are square matrices, then 

A) B²= B and A²= A. 

B) B²≠ B and A²= A 

C) A²≠ A and B²= B 

D) A²≠ A and B²≠ B 

5) If A and B are two Matrices such that AB= B and BA= A, then A²+ B² is equal to 

A) 2AB  B) 2BA C) A+B. D) AB 

6)

7) if the matrix AB is zero, then 

A) it is not necessary that either A= O or B= O.

B) A= O or B= O  C) A= O and B= O 

D) all the above Statements are wrong. 

8) 

9) If A and B are square matrices of order 3, A is non-singular and AB= O, then B is a

A) null matrix.  B) singular Matrix 

C) unit Matrix 

D) non singular matrix 

10) If A= n   0   0 and B= a₁  a₂  a₃

                0   n   0              b₁  b₂  b₃

                0   0   n              c₁  c₂  c₃ then AB is equals to 

A) B  B) nB.     C) Bⁿ   D) A+ B 

11) If A= 1       a 

                0       1 then Aⁿ (where n ∈ ℕ) equals 

A) 1  na. B) 1  n²a  C) 1  na  D) n  na 

     0   1        0    1        0   0        0    n 

12) If A= 1   2   x  and B= 1    -2    y 

                0   1   0               0     1    0

                0   0   1               0     0    1 and AB= I, then x+ y is 

A) 0.       B) -1 C) 2    D) none

13) If A= 1      -1      &  B=a     1 

                2      -1               b    -1 and (A+ B)² = A²+ B², values of a and b 

A) a= 4, b= 1         B) a= 1, b= 4. 

C) a= 0, b= 4         D) a= 2, b= 4 

14) If A= a       b 

                c      -a  is such that A²= I, then 

A) I+ a² + bc= 0    B) I- a² + bc= 0 

C)  I- a² - bc= 0.     D) I+ a² - bc= 0  

15)  If S= [Sᵢⱼ] is a scalar matrix such that sᵢⱼ = k and A is a square matrix of the same order, then AS= SA= ? 

A) Aᵏ B) k+ A C) kA. D) kS 

16) If A is a square Matrix such that A² = A, then (I+A)³ - 7A is equals to 

A) A       B) I - A C) I.    D) 3A 

17)  if a  matrix A is both symmetric and skew symmetric, then 

A) A is a diagonal matrix 

B) A is zero matrix. 

C) A is scalar matrix 

D) A is a square matrix  

18) the matrix 0      5      -7 

                        -5       0      11

                         7     -11      0 is 

A) a  skew-symmetric matrix 

B) a  symmetric matrix 

C) a diagonal matrix.

D) an upper triangular matrix 

19) If a square matrix, then AA is a 

A) skew symmetric matrix 

B) symmetric matrix 

C) diagonal Matrix 

D) none.

20) If A and B are symmetric matrices, then ABA is 

A) symmetric matrix.

B) skew-symmetric matrix 

C) diagonal Matrix 

D) scalar matrix 

21) If A= 5      x 

                y      0     and A= A', then 

A) x= 0, y= 5            B) x+y= 5 

C) x= y.                     D) none 

22) If A= 3 x 4 matrix and B is a matrix such that A'B and BA' are both defined. then, B is of the type 

A) 3x4.  B) 3x3 C) 4x4 D) 4x3  

23) If A= [aᵢⱼ] is a square matrix of even order such that aᵢⱼ = i² - j³, then 

A) A is a skew-symmetric matrix and |A|= 0 

B) A is symmetric metrix and |A| is a square. 

C) A is a symmetric Matrix and |A|= 0        D) none. 

24) If cosx         - sinx 

          sin x           cosx then A'+ A= I, if 

A) x=nπ, n ∈ Z 

B)x=(2n+1)π/2 , n ∈ Z 

C) x= 2nπ +π/3,  n ∈ Z.  D) none 

25) If A= 2       0       -3

                4       3        1

               -5       7        2 is expressed as the sum of a symmetric and skew symmetric matrix, then the symmetric matrix is 

A) 2     2      -4            B) 2    4      -5

     2     3       4                 0    3        7

    -4     4       2.                -3    1        2 

C) 4     4      -8           C) 1     0       0 

     4     6       8                0     1       0 

    -8     8       4                0     0       1 

26) Iut of the given matrices, choose that matrix which is a scalar Matrix: 

A) 0   0  B) 0   0   0 C) 0   0  D) 0 

     0   0.      0   0   0     0    0       0 

27) the number of all possible matrices of order 3x3 with entry 0 or 1 is 

A) 27   B) 18    C) 81      D) 512 

28) Which of the given values of x and y make the following pairs of matrices equal ?

    3x+7   5         =  0    y-2 

     y+1  2 - 3x        8      4 

A) x= -1/3, y= 7 

B) x= 7, y= 2/3 

C) x= -1/3, y= -2/5 

D) not possible to find. 

29) If A= 0       2 and kA= 0       3a

                3      -4               2b      24

then the values of k, a, b are respectively 

A) -6,-12,-18 B) -6,4,9 

C) -6,-4,-9.     D) -6,12,18 

30) If I= 1   0   and J= 0      1 

               0   1              -1      0 and    B= cosx       sinx 

     -sinx       cosx  then B equals 

A) I cosx + J sinx.

B) I sinx + J cosx  

C) I cosx - J sinx 

D) - I cosx + J sinx 

31) The trace of the matrix 

A= 1    -5       7

      0     7       9

     11    8       9 is 

A) 17.     B) 25      C) 3      D) 12 

32) If A=[aᵢⱼ] is a scalar matrix of order n x n such that aᵢⱼ = k, for all i, then trace of A equal to

A) nk.  B) n+ k C) n/k D) none 

33) If A and B are square matrices of order 2, then det(A+B)= 0 is possible only when

A) det(A+B)= 0 or  det(B)= 0 

B) det(A)= 0 + det(B)= 0 

C) det(A)= 0 and  det(B)= 0 

D) A'+ B= O.

34) which of the following is not correct in a given determinant of A, where A=[aᵢⱼ]₃ ₓ ₃.

A) Order of minor is less than order of the det (A) 

B) minor of an element can never be equal to cofactor of the same element.

C) Value of a is determinant is obtained by multiplying elements of a row or column by corresponding cofactors 

D) order of minors and cofactors of the elements of A is same 

35) Let x    2     x 

             x²   x     6  

             x     x    6 

Then the value ofax⁴+bx³+cx²+dx+e is equal to 

A) 0    B) -16    C) 16 D) none.

36) the value of the determinant.               a²               a                 1

     cos nx      cos(n+1)x    cos(n+2)x 

    sin nx        sin(n+1)x     sin(n+2)x is independent of 

A) n.       B) a       C) x        D) none 

37) If ∆₁= 1  1   1 & ∆₂= 1   bc    a 

                  a  b   c            1  ca     b 

                  a² b² c²           1 ab     c then A) ∆₁ +∆₂= 0.     B) ∆₁+2∆ = 0 C) ∆₁+ ∆₂ D) none 

38) Dₖ = 1           n             n 

               2k    n²+n+2      n²+n 

              2k-1     n²          n²+n+2 and ⁿₖ₌₁∑ Dₖ = 48, then n equals 

A) 4.       B) 6        C) 8     D) none 

39) Let x²+ 3x     x-1       x+3 

              x+1         -2x       x-4 

              x-3         x+4        3x 

= ax⁴ + bx³+ cx² + dx+ e be an identity in x, where a, b, c, d, e are independent of x. Then the value of e is 

A) 4      B) 0.    C) 1      D) none 

40) using the factor theorem it is found that a+b, b+c and c+a are three factors of the determinant 

  -2a        a+ b       a+ c 

 b+a         -2b       b+c 

 c+a        c+b      -2c  The other factor in the value of the determinant is 

A) 4.      B) 2     C) a+b+c    D) none 

41) if a,b,c are distinct, then the value of x satisfying 

   0      x²-a       x³- b

 x²+a    0         x²+ c = 0 is

x⁴+ b   x- c         0   

A) c       B) a       C) b       D) 0 

42) 

43) 

44) 

45)

46) The value of 5²     5³        5⁴

                             5³     5⁴        5⁵ 

                             5⁴     5⁵        5⁶ is 

A) 5²     B) 0 .      C) 5¹³            D) 5⁹ 

47) 

48) 

49) 

50) If A is an invertible matrix, then which of the following is not true

A) (A²)⁻¹= (A⁻¹)² B) |A⁻¹|= |A⁻¹|.  

C) (A')⁻¹= (A⁻¹)' D) |A| ≠ 0 

51) If A is an invertible matrix of order 3, then which of the following is not true

A) |adj A|= |A|² B) (A⁻¹)⁻¹= A

C) If BA= CA, then B= C, where B and C are square matrices of order 3.

D) (AB)⁻¹= B⁻¹A⁻¹, where B= [bᵢⱼ]₃ ₓ ₃ and |B| ≠ 0

52) If A= 3      4        & B= -2      -2

                2      4                  0      -1 then (A+B)⁻¹=

A) is a skew-symmetric matrix

B) A⁻¹+ B⁻¹ C) does not exist D) n.

53) If S= a      b 

                c     d then adj A is 

A) -d  -b     B)d    -b     C) d  b D) d   c

     -c   a        -c     a.         c  a       b   a

54) If A is a singular matrix, then adj A is

A) non-singular B) singular.

C) symmetric D) not defined

55) If A, B are two n x n non-singular matrices, then

A) AB is non-singular.

B) AB is singular

C) (AB)⁻¹= A⁻¹B⁻¹

D) (AB)⁻¹ does not exist

56) If A= a    0     0 

                0    a     0 

                0    0     a then the value of |adj A| is 

A) a²⁷ B) a⁹ C) a⁶. D) a² 

57)     1      2       -1

If A= -1      1        2

           2    -1        1 then det(adj(adj A)) is

A) 14⁴. B) 14³ C) 14² D) 14

58) If B is a non-singular matrix and A is a square matrix, then det(B⁻¹ AB) is equal to 

A) Det (A⁻¹) B) Det (B⁻¹)

C) Det (A). D) Det (B) 

59) For any 2 x 2 matrix, if A(adj A) = 10      0

     0      10 then |A| is equal to

A) 20 B) 100 C) 10. D) 0 

60) If A⁵ = O such that Aⁿ for 1≤n ≤ 4, then (I - A)⁻¹ equals 

A) A⁴ B) A³ C) I+ A D) none.

61) If A satisfies the equation x³ - 5x³ + 4x + K= 0, then A⁻¹exists if

A) K≠ 1 B) K≠2 C) K≠-1 D) none.

62) If for the matrix A, A³ = I, then A⁻¹= 

A) A². B) A³ C) A D) none

63) If A and B are square matrices such that B= - A⁻¹BA, then (A+ B)² 

A) O. B) A²+ B². C) A²+2AB+B² D) A+B

64) 2     0        0

       0     2        0

       0     0        2 Then A⁵ is

A) 5A B) 10A C) 16A. D) 32A 

65) For non-singular square matrix A, B and C of the same order (AB⁻¹C)= 

A) A⁻¹BC⁻¹ B) C⁻¹B⁻¹A⁻¹

C) CBA⁻¹ D) C⁻¹BA⁻¹.

66)                5     10      3

the matrix   -2     -4       6

                    -1     -2        b is a singular matrix, if the value of b= 

A) -3 B) 3 C) 0 D) non-existent.

67) If d is the determinant of a square matrix A of order n, then the determinant of its adjoint is

A) dⁿ B) dⁿ⁻¹. C) dⁿ⁺¹ D) d 

68) if A is a matrix of order 3 and |A| = 8, then |adj A|=

A) 1 B) 2 C) 2³ D) 2⁶.

69) If A² - A + I= 0, then the inverse of A is.

A) A⁻² B) A+ I C) I - A. D) A - I 

70) if A and B are invertible matrices, which of the following statement is not a correct.

A) adj A= |A| A⁻¹  

B) det(A⁻¹)= (A)⁻¹

C) (A+B)⁻¹ = A⁻¹+ B⁻¹.

D) (AB)⁻¹ = B⁻¹A⁻¹

71) If A is a square Matrix such that A² = I, then A⁻¹ is equal to

A) A+ I B) A. C) 0 D) 2A 

72) Let A= 1     2   & B= 1      0 

                    3    -5            0      2 and X be a matrix such that A= BX, then X is equal to

A) 1    2    B) -1    2   C) 2    4 

     3   -5.         3    5        3   -5 D) n 

73) If A= 2       3

                5      -2 be such that A⁻¹ = kA, then k equal to

A) 19 B) 1/19. C) -19 D) -1/19

74) If A= 1/3     1/3      2/3 

                  2         1         -2 

                   x         2          y is orthogonal, then x+y= 

A) 3 B) 0 C) -3. D) 1

75) if A= 1       0      1

                0       0      1

                a       b      2 then aI + bA + 2A² is 

A) A B) -A C) ab A D) none .

76) If A= 1       - tan x &  B= 1    tanx 

           tanx              1          -tanx     1 and C= a        - b 

              b          a and relation of A B⁻¹= C then

A) a=1, b= 1

B) a=cos 2x , b= sin 2x.

C) a=sin 2x, b= cos 2x D) n 

77) If a matrix A is such that 3A³ + 2A² + 5A + I= 0, then A⁻¹ is 

A) -(3A² + 2A + 5)

B) (3A² + 2A + 5)

C) (3A² - 2A - 5) D) none .

78) If A is an invertible Matrix, then det A⁻¹ is equal to

A) det(A) B) 1/det(A). C)1 D) n

79) If A= 2       -1

                3       -2 then Aⁿ= 

A) A= 1       0

           0       1 if n is an even natural number.

B) A= 1       0

           0       1 if n is an odd natural number

C) A= -1       0

            0       1 if n belongs to N

D) none

80) If ∆= a      b

                c      d then K∆ is equal to

A) Ka    Kb            B) Ka         b

     Kc    Kd                 c        Kd

C) Ka      b           D)  Ka          b

    Kc       d.                 c           d 

                      RELATION 

                  **************

                   FUNCTION

                   ***********

INVERSE TRIGONOMETRIC FUNCTION

*************************************

                   Total questions: 74

      

1) Write the value of sin⁻¹(-√3/2) + cos⁻¹(-1/2).                           π/3

2) write the difference between maximum and minimum values of sin⁻¹x for x ∈ [-1,1].                   π

3) if sin⁻¹x + sin⁻¹y+ sin⁻¹z= 3π/2, then write the value of x+ y +z.      3

4) if x> 1, then write the value of sin⁻¹{2x/(1+x²)} in terms of tan⁻¹x.        π - 2 tan⁻¹x.

5) If x < 0, then write the value of cos⁻¹{(1-x²)/(1+x²) in terms of tan⁻¹x.        - 2 tan⁻¹x

6) write the value of tan⁻¹x + tan⁻¹(1/x) for x> 0.                    π/2

7) write the value of tan⁻¹x + tan⁻¹(1/x) for x< 0.                  -π/2

8) what is the value of cos⁻¹(cos 2π/3) + sin⁻¹(sin 2π/3)?               π

9) if - 1< x <0, then write the value of sin⁻¹{2x/(1+x²)} + cos⁻¹{(1-x²)/(1+x²)}.                      0

10) write the value of sin(cot⁻¹x).   1/√(1+x²) 

11) Write the value of cos⁻¹(1/2) + 2 sin⁻¹(1/2).                    2π/3

12) write the range of tan⁻¹x.           (-π/2,π/2)

13) value of cos⁻¹(cos1540).    100°

14) value of sin⁻¹(sin(-600°)).    60°

15) value of cos⁻¹(sin(-600°)).   60° 

16) value of sin⁻¹(sin(1550°)).   70°

17) Evaluate sin(1/2 cos⁻¹ 4/5).   1/√10            

18) Evaluate sin(tan⁻¹ 3/4).        3/5   

19) write the value of cos⁻¹(tan 3π/4).                  π

20) write the value of cos(2sin⁻¹ 1/2).                 1/2

21) write the value of cos⁻¹(cos350) - sin⁻¹(sin(350°)).                   20°

22) write the value of cos²(1/2 cos⁻¹3/5.                  4/5

23) If tan⁻¹x+ tan⁻¹y=π/4, then write the value of x+ y+ xy.           1

24) write the value of cos⁻¹(cos 6).                             2π - 6

25) write the value of sin⁻¹(cos π/9).                    7π/18

26) value: sin{π/3 - sin⁻¹(sin (-1/2))}.             1

27) value tan⁻¹{tan(15π/4))}.     -π/4

28) value of cos⁻¹1/2 + 2 sin⁻¹ 1/2.                2π/3

29) value of tan⁻¹a/b - tan⁻¹{(a-b)/(a+b)}.               π/4

30) value of cos⁻¹(cos 5π/4).   3π/4

31) show: sin⁻¹{2x √(1- x²)}=2sin⁻¹x.

32) value:cos⁻¹(1/√2 secπ/4).   π/4

33) value of tan⁻¹(cot330°).      -π/3            

34) value of sin⁻¹(2sin 150).     π/2      

35)value of cos⁻¹sin cot⁻¹√3.    π/3                

36)value of sin(1/2cos⁻¹1/2).    1/2                 

37) value of sec² cot⁻¹1/√3 + tan² cosec⁻¹√2.                 5

38) value: sin(π/2 - sin⁻¹3/5).    4/5

39) State with arguments which of the following functions are not defined. cos⁻¹(3), sec⁻¹(√2), cot⁻¹(1/4), cosec⁻¹(-1/√5). 

               cos⁻¹(3), cosec⁻¹(-1/√5).

40) Indicate the correct one out of the following two statements:

     a) cos⁻¹3 + cod⁻¹2=π/2   

     b) sin⁻¹(1/2)+ cos⁻¹(1/2)=π/2.

41) show sin(cos⁻¹x)= cos(sin⁻¹x) state with reasons whether the equality is valid for all values of x.

42) value of cos⁻¹(cos 4π/3).   2π/3

43) value of cot⁻¹(-√3)+ tan⁻¹(1/√3). 0.33   

44) If cos⁻¹(1/√5)= x, then value of cosec⁻¹(√5).              π/2- x

45) If sin⁻¹x= k, find the value of cosec⁻¹(1/√(1-x²).       π/2 - k

46) value of tan⁻¹3+ tan⁻¹1/3.   π/2

47) value of sin⁻¹3/5 + cosec⁻¹5/4.      π/2

48) value of tan⁻¹(-2)+ tan⁻¹(-1/2).            -π/2

49) if sin⁻¹x +, sin y=sin⁻¹z, then express z in terms of x and y.    z= x√(1-y²)+ y√(1-x²).

50) If tan⁻¹x+ tan⁻¹y=π/2, show that, xy = 1.

51) Prove, cos(2 sin⁻¹(x/√2)= 1- x².

52) value of cos(2cos⁻¹3/4).     1/8

53) value of cos(2 sin⁻¹3/√5). 7/25

54) value of tan(2 tan⁻¹1/2).      4/5

55) value of cos(2 tan⁻¹1/3).     4/5

56) value of sin(2 tan⁻¹2/3).   12/13

57) value of sin(2 sin⁻¹4/5).   24/25

58) prove: sin⁻¹3/√5=1/2 cos⁻¹1/3

59) prove: tan⁻¹1/5=1/2 sin⁻¹5/13

60) value of sin(1/2 cos⁻¹4/5).     1/√10

61) If sin⁻¹x= cos⁻¹x, x is.          1/√2

62) If cot⁻¹x+ cot⁻¹2=π/2, x is.    1/2

63) If sin⁻¹x+2 cos⁻¹x= 2π/3, x is. √3/2

64) If cos(2 sin⁻¹x)=-1, x is.        ±1

65) The sum of two acute angles tan⁻¹x and tan⁻¹1/2 is 45°, find the value of x.           1/3

66) Find the value of tan⁻¹sin cos⁻¹ √(2/3).            π/6

67) value of sin⁻¹sin(sin 5π/6)ᶜ.  1/2 radian

68) value of cos⁻¹x+ cos⁻¹(-x), when 0< x < 1.              π

69) value of tan(cos⁻¹4/5 + tan⁻¹2/3).           17/6

70) value of tan{1/2(tan⁻¹x+ tan⁻¹1/x)}.         1

71) If A+ B+ C=π and A= tan⁻¹2, B= tan⁻¹3, show that C=π/4.

72) If sin⁻¹x + sin⁻¹y= 2π/3, find the value of cos⁻¹x + cos⁻¹y.   

73) show tan(1/2 cos⁻¹a)= √{(1-a)/(1+a)}.

74) show cos⁻¹√(3/5)= 1/2 cos⁻¹1/5 

MULTIPLE CHOICE QUESTIONS

_____________()______()__________

1) value of cos[π/3+ cos⁻¹(-1/2) is   

A) 0       B) -1.     C) 1          D) n   

2) value of sin{tan⁻¹(7π/6) + cos⁻¹(cos 7π/3) is

A) 0       B) -1      C) 1.     D) none 

3)

RELATION XII

 EXERCISE-A

               

1) Prove the following:

a) the relation '>' is not symmetric.

b) The relation '≤' is transitive.

c) The relation 'is similar to' on the set T of all Triangles in a plane is transitive.

d) The relation 'is a square of' on the set of natural numbers is not Reflexive.

2) give an example of a relation which is:

a) only symmetric.                    

b) only Reflexive.                    

c) only transitive.             

d) symmetrical and Reflexive and not transitive.       

e) symmetric and transitive but not reflexive.        

3) Decide in each of the following cases whether the relation is a) symmetric b) transitive c) reflexive. Justify by giving examples.

A) 'is greater than' on W.

B) 'is perpendicular to' on a set of lines in a plane.

C) 'is divisible by' on the set of real numbers.

D) 'is less than' on N.

E) ' is parallel to' on a set of lines.

F) ' is a multiple of'

3) Prove that every identity relation on a set is reflexive, but the converse is not necessarily true.

4) Three relation R₁, R₂ and R₃are defined on a set A={a,b,c} as follows:

R₁= {(a,a),(a,b),(a,c),(b,b),(b,c),(c,a),(c,b),(c,c)}.            

R₂={(a,a)}.                   

R₃={(b,c)}.          

R₄={(a,b),(b,c),(c,a)}.        

find whether or not each other relation R₁, R₂, R₃, R₄ on A is a)reflexive b) symmetric c) transitive.

                 

5) if A={1,2,3,4} define relations on A which have properties of being

a) reflexive, transitive but not symmetric.        

b) symmetric but neither reflexive nor transitive.

c) reflexive, symmetric and transitive.

     

6) Let R be a relation defined on the set of natural numbers N as R={(x,y): x,y ∈ N, 2x+ y= 41}.

 Find the domain and range of R. Also, verify whether R is

a) Reflexive

b) Symmetric

c) Transitive.

      

7) Is it true that relation which is symmetric and transitive is also reflexive ? give reasons.

8) Test whether the following relations R₁, R₂, and R₃ are

A) Reflexive B) Symmetric C) Transitive 

i) R₁ on Q₀ defined defined by (a,b)∈R₁ <=> a= 1/b.

ii) R₂ on Z defined by (a,b)∈ R₂ <=> |a- b| ≤ 5.

iii) R₃ on R defined by (a,b) ∈ R₃ <=> a² - 4ab + 3b²= 0.

9) An integer m is said to be related to another n if m is a multiple of n. check if the relation is symmetric, reflexive and Transitive.

10) Given the relation R={(1,2),(2,3)} on the set A={1,2,3}, add a minimum number of ordered pairs so that the enlarged relation is symmetric, transitive and Reflexive.

11) let A= {1,2,3} and R={(1,2),(1,1),(2,3)} be a relation on A. What minimum number of ordered pairs may be added to R so that it may become a transitive relation on A.

12) give an example of a relation which is

A) reflexive and symmetric but not transitive,

B) reflexive and transitive but not symmetric.

C) symmetric and transitive but not reflexive.

D) symmetric but neither Reflexive nor Transitive.

E) transitive but neither Reflexive nor symmetric.

13) Show that the relation '≥' on the set R of all real numbers is reflexive and transitive but not symmetric.

14) the following relation are defined on the set of real numbers.

a) aRb if a- b > 0

b) aRb iff 1+ ab > 0

c) aRb if |a| ≤ b.

 Find weather these relations are Reflexive, symmetric or Transitive.

15) Let A={1,2,3} and  

R₁={(1,1),(1,3), (3,1),(2,2),(2,1),(3,3)}; 

R₂={(2,2),(3,1),(1,3)},

R₃={(1,3),(3,3)}. Find whether or not each of the relations R₁, R₂ R₃ on A is

A) reflexive

B) symmetric

C) transitive.

16) Let A be the set of all human beings in a town at a particular time. Determine whether each of the following relations are reflexive, symmetric and transitive.

A) R={(x,y): x and y work at the same place}.

B) R={(x,y): x and y live in the same locality}.

C) R={(x,y): x is wife of y}.

D) R={(x,y): x is father of y}

17) Check whether the relation R defined on the set A={1,2,3,4,5,6} as R={(a,b): b= a+1} is Reflexive, symmetric or Transitive.

18) Check whether the relation R on R defined by R={(a,b): a ≤ b³} is Reflexive, symmetric or transitive.

              EXERCISE -2

            ****************

     EQUIVALENCE RELATION

       --------------------------

1) the relation 'is sister of' on the set of the members of the family is not an equivalence relation.

2) Write the word 'yes' if the following relations are equivalence relation on the given set. write the word 'no' if it is not.

a) 'is equals to' on N.                Yes

b) 'is a descendant' on a non- empty set of people.                               No

c) ' lives within 10 kilometres of' on a set or people who live in Delhi. No

d) " is the brother of" on the set of members of a family.             No

e) 'is stronger than' on the set of children of class IX in a particular school.                                        No

f) 'is divisible by' on the set of whole number(W).                    No

3) let M be the set of male members of a family and R means 'is brother of', prove that the relation R, over the set M, is an equivalence relation.

4) Let T be set of all triangles in a plane and the relation R means 'is similar to'. prove that R is an equivalence relation.

5) Show that for the set of all points in a plane, the relation 'at the same distance from the origin' is an equivalence relation.

6) Let I be the set of Integers and R be a relation on I x I defined as: R:{(a,b)∈ I x I : (a - b) is divisible by 7}.

Show that R is an equivalence relation.

7) a) Is the relation 'divides' on the set of positive integers an equivalence relation ? Prove it.     

        No, the relation is not symmetric.

b) Is the relation 'reciprocal of' on the set of non-zero real numbers an equivalence relation ? Prove it.

          No, the relation is only symmetric

8) Prove that the relation ">" on the set of real numbers is not an equivalence relation.

9) show that the relation ' is subset of ' with respect to sets is not an equivalence relation.

10) Show that the relation '≤' on the set of Integers is not an equivalence relation.

11) Show that " is the father of" is not an equivalence relation.

12) In the set of all triangles in a plane, show that the relation " is congruent to" is an equivalence relation.

13) Show that the relation "≥" on the set of real numbers is not equivalence relation.

14) prove that "is parallel to" for a set of straight lines in a plane is an equivalence relation.

15) State which of these is an equivalence relation.

A) 'is greater than' on W.

B) 'is perpendicular to' on a set of lines in a plane.

C) 'is divisible by' on the set of real numbers.

16) A={real numbers}. On A, a relation k is defined by; for all a, b belongs to A, aRb holds if and only if the difference between a and b is less than 2. Is R an equivalence relation. Justify your answer.

17) state which of these is an equivalence relation.

a) 'is less than' on N.

b) ' is parallel to' on a set of lines.

c) ' is a multiple of' 

__

18) Prove that the relation R on Z defined by (a,b)∈ R <=> a - b is divisible by 5 is an equivalence relation on Z.                              1

19) let n be a fixed positive integer. Define a relation R on Z as follows: (a,b)∈ R <=> a - b is divisible by n. Show that R is an equivalence relation on Z.

20) let S be a relation on the R of all real numbers defined by

S= {(a,b)∈ R x R: a²+ b²=1} prove that S is not an equivalence relation on R.

21) Let O be the origin. We define a relation between two points P and Q in a plane if OP= OQ. Show that the relation, so defined is an equivalence relation.

22) m is said to be related to n if m and n are integers and m - n is divisible by 13. does this define an equivalence relation ? 

23) Let Z be the set of Integers. show that the relation R={(a,b): a,b ∈ Z and a + b is even} is an equivalence relation on Z.

24) Let Z be the set of all integers and Z₀ be the set of all non zero integers. Let a relation R on Z x Z₀ be defined as a follows:

 (a,b)R(c,d) <=> ad = bc for all (a,b), (c,d) ∈ Z x Z₀, Prove that R is an equivalence relation on Z x Z₀

25) If R and S are relation on a set A, then Prove that

a) R and S are symmetric => R ∩ S and R U S are symmetric.

b) R is reflexive and S is any relation => R U S is reflexive.

26) R and S are transitive relation on a set A, then Prove that R U S may not be a transitive relation on A.

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

28) Let R be the relation defined on the set both A={1,2,3,4,5,6,7} by R={a,b} : both a and b are either odd or even}. Show that R is an equivalence relation. Further, show that all the elements of the subset {1,3,5,7} are related to each other and all the elements of the subset {2,4,6}.

29) Show that the relation R on the set A={ x∈ Z ; 0≤ x ≤ 12}, given by R{(a,b): a= b}, is an equivalence relation. Find the set of all elements related to 1.

30) Show that the relation R, on the set A of all polygon 

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 angled triangle T with sides 3, 4 and 5 ?        

                             Set of all triangles

31) 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.

                                {y=2x+c: c ∈ R}

32) Show that the relation R defined by R={(a,b): a-b is divisible by 3 ; a,b ∈ N} is an equivalence relation.

              EXERCISE ---3

              **************

VERY SHORT ANSWER QUESTIONS

__________________________________

1) Write the domain of the relation R defined on the set Z of integers as follows: (a,b)∈ R <=> a²+b²=25.

                            {0,±3,±4,±5}

2) If R={(x,y): x²+ y² ≤ 4, x,y ∈ Z} is a relation on Z, write the domain of R.

              {0,±1,±2}

3) write the Identity relation on set A={a,b,c}.               {a,a),(b,b),(c,c)} 

4) write the smallest reflexive relation on set A{1,2,3,4}.        {(1,1),(2,2),(3,3),(4,4)}

5) If R={(x,y): x+ 2y= 8} is a relation on N by, then write the range of if R.         {1,2,3}

6) If R is symmetric relation on a set A, then write a relation between R and R⁻¹.                       R =R⁻¹ 

7) let R={(x,y): |x² - y²|<1} be a relation on the set {1,2,3,4,5}. Write R as a set of ordered pairs. 

          {(1,1),(2,2),(3,3),(4,4),(5,5)}. 

8) If A={2,3,4}, B {1,3,7} and R={(x,y): x∈A, y ∈B and x< y} is relation from A to B, then write R⁻¹.           {(3,2),(7,2),(7,3),(7,4)}

9) let A={3,5,7}, B{2,6,10} and R be a relation from A to B defined by R={(x,y): x and y are relatively prime}. Then, write R and R⁻¹.       R= {(3,2),(3,10),(5,2),(5,6),(7,2),(7,6),(7,10)}

      R⁻¹={(2,3),(10,3),(2,5),(6,5),(2,7), (6,7),(10,7) 

10) If A={3,5,7} and B{2,4,9} and R is a relation given by " is less than", write R as a set ordered pairs.   

         {(3,4),(3,9),(5,9),(7,9)} 

11) A{1,2,3,4,5,6,7,8} and if R={(x,y): y is one half of x; x,y ∈A} is a relation on A, then write R as a set of ordered pairs.

 {(2,1),(4,2),(6,3),(8,4)}

12) let A={2,3,4,5} and B{1,3,4}. If R is the relation from A to B given by a R b iff " a is a divisor of b". write R as a set of ordered pairs.      {(2,4),(4,4),(3,3)}

              EXERCISE --4

                  **************

1) " is greater than" for the set of real numbers is

A) symmetrica relation

B) Transitive relation

C) Reflexive relation

D) equivalence relation  D) none

2) is the cube of" for the set of real numbers is

A) symmetrica relation

B) Transitive relation

C) Reflexive relation

D) equivalence relation D) none

3) 'is the sister of" for the set of real numbers is

A) symmetrica relation

B) Transitive relation

C) Reflexive relation

D) equivalence relation  D) none

4) 'is similar to" for the set of real numbers is

A) symmetrica relation

B) Transitive relation

C) Reflexive relation

D) equivalence relation  D) none

5) is perpendicular to" for the set of real numbers is

A) symmetrica relation

B) Transitive relation

C) Reflexive relation

D) equivalence relation  D) none

6) Write down a relation which is

only Transitive.

A) is greater than

B) is perpendicular to

C) is a multiple of

D) is a friend of

7) Write down a relation which is

only symmetric.

A) is greater than

B) is perpendicular to

C) is a multiple of

D) is a friend of

8) Write down a relation which is

only Reflexive and Transitive.

A) is greater than

B) is perpendicular to

C) is a multiple of

D) is a friend of

9) Write down a relation which is

only symmetric and reflexive

A) is greater than

B) is perpendicular to

C) is a multiple of

D) is a friend of

10) If A is the set of the members of a family and R means "is brother of" then it is

A) symmetrica relation

B) Transitive relation

C) Reflexive relation

D) equivalence relation D) none

11) The relation R in the set {1,2,3} given by R={(1,2),(2,1) is

A) symmetrica relation

B) Transitive relation

C) Reflexive relation

D) equivalence relation  D) none

12) The relation R in the set {1,2,3,4} given by R={(1,2),(2,2),(1,1), (4,4), (1,3), (3,3),(3,2)} is

A) R is reflexive and symmetric but not Transitive 

B) R is reflexive and transitive but not symmetric

C) R is symmetric and Transitive but not reflexive

D) R is an equivalence relation

13) The relation R in the set A of real numbers defined as R= {(a,b): a≤ b} is

A) R is reflexive and symmetric but not Transitive 

B) R is reflexive and transitive but not symmetric

C) R is symmetric and Transitive but not reflexive

D) R is an equivalence relation

14) The relation R in the set of real numbers defined as R={(a,b): a≤ b²} is

R is reflexive and symmetric but not Transitive 

B) R is neither reflexive nor transitive nor symmetric

C) R is symmetric and Transitive but not reflexive

D) R is an equivalence relation

15) In the set of all triangles in a plane, the relation of similarity is 

A) symmetric

B) transitive

C) reflexive

D) equivalence relation

16) Is the the relation "is the square of" for the set of natural numbers N is 

A) symmetric

B) transitive

C) reflexive

D) equivalence relation

E) none

17) "is smaller than"

A) symmetric

B) transitive

C) reflexive

D) equivalence relation E) none

18) "is the father of"

A) symmetric

B) transitive

C) reflexive

D) equivalence relation E) none

19) "is parallel to" for set of straight lines.

A) symmetric

B) transitive

C) reflexive

D) equivalence relation E) none

20) "is a multiple of" for a set of positive integers

A) symmetric

B) transitive

C) reflexive

D) equivalence relation E) none

21) "is congruent to"

A) symmetric

B) transitive

C) reflexive

D) equivalence relation E) none

22) "is congruent to" in a set of triangles is

A) symmetric

B) transitive

C) reflexive

D) equivalence relation

23) If R is a relation in N x N defined by (a,b) R (c,d) if only of a+ d = b+c, then R is

A) symmetric

B) transitive

C) reflexive

D) equivalence relation

24) The relation R in the set A={1, 2, 3, 4,5} given by R={(a,b): | a - b | is even} is 

A) symmetric

B) transitive

C) reflexive

D) equivalence relation

25) Let I be the set of all integers and R be the relation on I defined by a R b iff (a+b) is an even integer for all a, b belongs to I. Then it is

A) symmetric

B) transitive

C) reflexive

D) equivalence relation

26) Let I be the set of all integers and R be the relation  on I defined by R={(x,y): x,y belongs to I, x- y is divisible by 11} is 

A) symmetric

B) transitive

C) reflexive

D) equivalence relation

27) The relation R in the set {1,2,3} given by R={(1,2),(2,1) is

A) symmetrica and Transitive

B) symmetric and reflexive

C) symmetric, Reflexive and Transitive.

D) symmetric but neither reflexive nor Transitive.