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1.4 Refractive index of a solid is observed to have the same value along all directions. Comment on the nature of this solid. Would it show cleavage property?

We know that amorphous solid are isotropic in nature i.e., their properties are same in all directions.According to given description, the nature of given solid is amorphous. Clevage property :- When we cut the solid with a sharp edged tool, they cut into two pieces with irregular surfaces.

Q. 13 Consider a binary operation ∗ on N defined as a * b = a^3 + b^3 . Choose the
correct answer.

(A) Is ∗ both associative and commutative?
(B) Is ∗ commutative but not associative?
(C) Is ∗ associative but not commutative?
(D) Is ∗ neither commutative nor associative?

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Q. 12State whether the following statements are true or false. Justify. 

(ii) If ∗ is a commutative binary operation on N, then a *(b*c) =( c*b)*a

(ii) If ∗ is a commutative binary operation on N, then R.H.S                        (* is commutative)                          (  as * is commutative)              =    L.H.S             Hence, statement  (ii) is true.

Q.12 State whether the following statements are true or false. Justify.

(i) For an arbitrary binary operation ∗ on a set N,a*a = a,\; \forall a\in N

(i) For an arbitrary binary operation ∗ on a set N, An operation * on a set N as  Then , for  b=a=2 Hence, statement (i) is false.    

Q. 11 Let A = N \times N and ∗ be the binary operation on A defined by
      (a, b) * (c, d) = (a + c, b + d)

Show that ∗ is commutative and associative. Find the identity element for ∗ on
A, if any.

 and ∗ be the binary operation on A defined by Let    Then,  We have        Thus it is commutative. Let    Then,  Thus, it is associative. Let   will be a element for operation *  if  for all . i.e.  This is not possible for any element in A . Hence, it does not have any identity.              

Q. 10 Find which of the operations given above has identity.

An element   will be identity element for operation * if    for all           when    . Hence,   has identity as 4. However, there is no such element    which satisfies above condition for all rest five operations. Hence, only (v) operations have identity.

Q. 9 Let ∗ be a binary operation on the set Q of rational numbers as follows:

(vi) a* b = ab^2

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

On the set Q ,the operation * is defines as   .It is observed that: For            for   Hence, the * operation is not  commutative. It can be observed that     for all  The operation * is  not associative.

Q. 9 Let ∗ be a binary operation on the set Q of rational numbers as follows:

(v) 

a * b = \frac{ab}{4}

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

On the set Q ,the operation * is defines as   .It is observed that: For            for   Hence, the * operation is commutative. It can be observed that     for all  The operation * is associative.

Q.9 Let ∗ be a binary operation on the set Q of rational numbers as follows:

(iv) a * b = (a-b)^2

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

On the set Q ,the operation * is defined as   .It is observed that: For            for   Hence, the * operation is commutative. It can be observed that     for all  The operation * is not associative.

Q.9 Let ∗ be a binary operation on the set Q of rational numbers as follows:

(iii) a * b = a + ab

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

On the set Q ,the operation * is defines as   .It is observed that: For          for   Hence, the * operation is not  commutative. It can be observed that     for all  The operation * is not associative.

Q. 9 Let ∗ be a binary operation on the set Q of rational numbers as follows:

(ii) a*b = a^2 + b^2

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

On the set Q ,the operation * is defines as  .It is observed that: For          Hence, the * operation is commutative. It can be observed that      for all    The operation * is not associative.                           

Q.9 Let ∗ be a binary operation on the set Q of rational numbers as follows:

(i)  a * b = a - b

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

On the set Q ,the operation * is defines as  .It is observed that:              here  Hence, the * operation is not commutative. It can be observed that      for all    The operation * is not associative.                           

Q. 8 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?

a ∗ b = H.C.F. of a and b for all  H.C.F. of a and b = H.C.F of b and a for all  Hence, operation  ∗  is commutative. For   ,                      Hence,  ∗ is associative. An element  will be identity for operation *  if   for . Hence, the operation * does not have any identity in N.    

Q. 7 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.

A =  Operation table is as shown below: 1 2 3 4 5 1 1 2 3 4 5 2 2 2 6 4 10 3 3 6 3 12 15 4 4 4 12 4 20 5 5 10 15 20 5 From the table, we can observe that Hence, the operation is not a binary operation.          

Q.6 Let ∗ be the binary operation on N given by a* b = L.C.M.\; of \;a\; and \;b. Find 

(iv) the identity of ∗ in N

(iv) the identity of ∗ in N We know that                       for  Hence, 1 is the identity of ∗ in N.  

Q.6 Let ∗ be the binary operation on N given by a ∗ b = L.C.M. of a and b. Find

(iii) Is ∗ associative?

a  b = L.C.M. of a and b (iii)                Hence, the operation is associative.  

Q.6 Let ∗ be the binary operation on N given by a * b = L.C.M. \;of \;a\; and \;b. Find 

(ii) Is ∗ commutative?

(ii)     for all    Hence, it is commutative.  

Q.6 Let ∗ be the binary operation on N given by a * b = L.C.M. \;of\; a\; and \;b. Find

(i)  5 ∗ 7, 20 ∗ 16

a*b=LCM of a and b (i)  5 ∗ 7, 20 ∗ 16

Q.5 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.

   for   The operation table is as shown below: 1 2 3 4 5 1 1 1 1 1 1 2 1 2 1 2 1 3 1 1 3 1 1 4 1 2 1 4 1 5 1 1 1 1 5 The operation ∗′ same as the operation ∗ defined in Exercise 4 above.

Q.4 Consider a binary operation ∗ on the set {\{1, 2, 3, 4, 5\} given by the following
multiplication table (Table 1.2).

(iii) Compute (2 ∗ 3) ∗ (4 ∗ 5).

(Hint: use the following table)

(iii) (2 ∗ 3) ∗ (4 ∗ 5). from the above table  
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