**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 . 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?

14953/

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

(ii) If ∗ is a commutative binary operation on N, then

(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,

(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 and ∗ be the binary operation on A defined by

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)

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)

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)

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)

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

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

(ii)

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)

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 by 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 . Find

(iv) 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?

Q.6 Let ∗ be the binary operation on N given by . Find

(ii) Is ∗ commutative?

Q.6 Let ∗ be the binary operation on N given by . Find

(i) 5 ∗ 7, 20 ∗ 16

Q.5 Let ∗′ be the binary operation on the set defined by

. 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 { 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

- Previous
- Next

Exams

Articles

Questions