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1. a * b = 2ab = 2ba = b * a

Therefore commutative

a * (b * c) = a * (2bc) = 4abc

(a * b) * c = (2ab) * c = 4 abc

Therefore is associative.

2. a * e = a ⇒ 2ae = a ⇒ e = \(\frac{1}{2}\)

e * a = a ⇒ 2ea = a ⇒ e = \(\frac{1}{2}\)

Therefore identity element is \(\frac{1}{2}\).

3. a * b = \(\frac{1}{2}\) ⇒ 2ab = \(\frac{1}{2}\) ⇒ b = \(\frac{1}{4}\) a, a ≠ 0.

Answer is (B)

(i) {(1, 1), (1, 2), (2, 1), (2,2), (2, 3)} 

(ii) {(1,1), (2,2), (3,3), (1,2), (2,1), (1,3), (3,1)} (2,3), (3,2)} 

There are two equivalence relation

(D) 5

Given, set A = {1, 2, 3}

Now, the number of equivalence relations as follows

R₁ = {(1, 1), (2, 2), (3, 3)}

R₂ = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)}

R₃ = {(1, 1), (2, 2), (3, 3), (1, 3), (3, 1)}

R₄ = {(1, 1), (2, 2), (3, 3), (2, 3), (3, 2)}

R₅ = {(1, 2, 3) ⇔ A x A = A²}

Thus, maximum number of equivalence relation is ‘5’.

Answer is (2) associative

Given that * is a binary operation defined on Q.

(i) a * b = a – b, ∀ a, b ∈Q and b * a = b – a

So, a * b ≠ b * a

Thus, * is not commutative.

(ii) a * b = a² + b²

b * a = b² + a²

Thus, * is commutative.

(iii) a * b = a + ab

b * a = b + ab

So clearly, a + ab ≠ b + ab

Thus, * is not commutative.

(iv) a * b = (a – b)², ∀ a, b ∈Q

b * a = (b –a)²

Since, (a – b)² = (b – a)²

Thus, * is commutative.

For Commutativity,

Let (a, b), (c, d) ∈ R × R

(a, b) * (c, d) = (a + c, b + d)

and (c, d) * (a, b) = (c + a, d + b)

= (a + c, b + d) [∵ Commutative law holds for real number]

⇒ (a, b) * (c, d) = (c, d) * (a, b)

Hence, * is commutative.

For Associativity,

Let (a, b), (c, d) and (e, f) ∈ R × R

((a, b) * (c, d)) * (e, f) = (a + c, b + d) * (e, f) = (a + c + e, b + d + f)

(a, b) * ((c, d) * (e, f)) = (a, b) * (c + e, d + f) = (a + c + e, b + d + f)

((a, b) * (c, d)) * (e, f)) = (a, b) * ((c . d) * (e, f ))

∴ * is associative,

Let (e₁,e₂) be identity

⇒ (a, b) * (e₁, e₂) = (a, b)

⇒ (a + e₁ , b + e₂) = (a, b)

⇒ a + e₁ = a and b + e₂ = b

⇒ e₁ = 0, e₂ = 0

(0, 0) ∈ R × R is the identity element.

True

In order to prove composition of functions is associative we need to show

[fo(goh)](x) = [(fog)oh](x)

Let us suppose, f(x) = x, g(x) = 2x, h(x) = x + 2

Now,

[fo(goh)](x) = f(g(h(x))) = f(g(x+2)) [∵ h(x) = x + 2 ]

= f(2(x+2)) = f(2x+4)

= 2x+4 (i)

[(fog)oh](x) = (fog)oh(x) = (fog)(h(x))

= (fog)(x+2) = f(g(x+2))

= f(2(x+2)) = f(2x+4)

= 2x+4 (ii)

From (i) and (ii), we observe that

[fo(goh)](x) = [(fog)oh](x)

Thus, composition of functions is associative.

False

Given that, f : R → R be the function defined by

f (x) = sin (3x+2) ∀ x ∈R

f is invertible if it is bijective that is f should be one-one and onto.

Now, we know that sin x lies between -1 and 1.

So, the range of f(x) = sin (3x+2) is [-1,1] which is not equal to its co-domain.

i.e., range of f ≠ R (co-domain)

In other words, range of f is less than co-domain, i.e there are elements in co-domain which does not have any pre-image in domain.

so, f is not onto.

Hence, f is not invertible.

a ≤ a² is not true for all a. So (a, a) ∉ R hence not Reflexive.

(a, b) ∈ R ⇒ a ≤ b² does not imply that b ≤ a². So (b, a) ∉ R, hence not Symmetric. (a, b),(b, c) ∈ R ⇒ a ≤ b² and b ≤ c² does not imply that a ≤ c². So (a, c) ∉ R, hence not Transitive.

(1, 1) ∉ R, hence not reflexive (1, 2) ∈ R 

⇒ (2, 1) ∈ R, also (2, 1) ∈ R 

⇒ (1,2) ∈ R . 

(a, b) ∈ R = 9 (b, a) ∈ R, hence symmetric 

(1,2) ∈ R (2, 1) ∈ R but (1, 1) ∈ R hence not transitive.

(x, x) ∈ R as x and x have same number of pages. ∀ x ∈ A, hence 

Reflexive 

(x, y) ∈ R ⇒ x and y have same number of pages 

⇒ y and x have same number of pages 

⇒ 9 (y, x) ∈ R hence Symmetric (x, y) ∈ R and (y, z) ∈ R 

⇒ x and y and y and z have same number of pages 

⇒ x and z have same number of pages 

⇒ 9 (x, z) ∈ R, hence Transitive. 

Since R is reflexive, symmetric and transitive, R is an equivalence Relation.

False

In order to prove composition of functions is commutative we need to show

(fog)(x) = (gof)(x)

Let us suppose, f(x) = 2x, g(x) = 1 + x²

Now,

(fog)(x) = f(g(x)) = f(1+x²)

= 2(1+x²) = 2+2x² (i)

(gof)(x) = g(f(x)) = g(2x)

= 1+(2x)² = 1+4x² (ii)

From (i) and (ii), we observe that

(fog)(x) ≠ (gof)(x)

Thus, composition of functions is not commutative.

Let A = {1, 2, 3}.
A relation R on A is defined as R = {(1, 2), (2, 1)}.
It is clear that (1, 1), (2, 2), (3, 3) ∉ R.
∴ R is not reflexive.
Now, as (1, 2) ∈ R and (2, 1) ∈ R, then R is symmetric.
Now, (1, 2) and (2, 1) ∈ R
However, (1, 1) ∉ R
∴ R is not transitive.
Hence, R is symmetric but neither reflexive nor transitive

1. (-1, -1) ∉ R, R is not reflexive

(-1,1) ∈ R but (1, -1) ∉ R, R is not symmetric

(-1,1) ∈ R, (1, 1) ∈ R and (-1, 1) ∈ R,

R is transitive.

2. R is not reflexive, not symmetric and not transitive. So R is not an equivalence relation.

False

Given that, A = {1, 2, 3} and R = {(1, 1), (1, 2), (2, 1), (3, 3)}

Now,

R is not reflexive ∵ (2,2) ∉ R

(1, 1) ∉ R, hence not reflexive (1, 2) ∈ R

⇒ (2, 1) ∈ R, also (2, 1) ∈ R

⇒ (1,2) ∈ R . (a, b) ∈ R = 9 (b, a) ∈ R,

hence symmetric

(1,2) ∈ R (2, 1) ∈ R but (1, 1) ∈ R

hence not transitive.

False

Given that, R = {(3, 1), (1, 3), (3, 3)} be a relation defined on the set A = {1, 2, 3}

Now,

R is not reflexive ∵ (1,1),(2,2) ∉ R.

R is symmetric ∵ (3,1) ∈ R ⇒ (1,3) ∈ R

R is not transitive ∵ (1,3) ∈ R and (3,1) ∈ R but (1,1) ∉ R.

Given a * b = 2a + b -3

3 * 4 = 2(3) + 4 -3

= 6 + 4 - 3 = 7

(C) 2

Given that, a * b = ab/2 ∀ a, b ∈ Q ~ {0}

Let e be the identity element for * such that

a*e = e*a = a

Now, a * e = ae/2

a = ae/2

⇒ 2a = ae

⇒ e = 2

R is not transitive as (1, 2) ∈ R, (2, 1) R But (1, 1) ∉ R.

[Note : A relation R in a set A is said to be transitive if (a, b) ∈ R, (b, c) ∈ R ⇒ (a, c) ∈ R ∀ a, b, c ∈ R]

Given binary operation is a*b = 2a + b – ab 

∴ 3* 4 = 2 x 3 + 4 – 3 

⇒ 4 x 3* 4 = –2 

3 * 4 = 2 × 3 + 4 – 3 × 4 = –2

Let E ∈ P (X) be an identity elements, then 

A * E = E * A = A ∀ A ∈ P(X) 

⇒ A ∩ E = E ∩ A = A ∀ A ∈ P(X) 

⇒ X ∩ E = X as X ∈ P (X) 

⇒ X C E Also E C X as E ∈ P (X) 

∴ E = X 

Thus, X is the identity element. 

Let A ∈ P(X) be invertible, then there exists 

B ∈ P(X) such that A * B = B * A = X, the identity element. 

⇒ A ∩ B = B ∩ A = X X C A & X C B 

Also A, B, ⊂ X as A, B ∈ P (X) 

A = X = B 

X is the only invertible element and – X₁ = B = X.

f(x) = ax + 3 ; g(x) = 4x -3

fog = f[g(x)]

= f(4x – 3)

= a(4x – 3) + 3

= 4ax – 3a + 3

gof = g[f(x)]

= g(ax + 3)

= 4(ax + 3) – 3

= 4 ax + 12 – 3

= 4ax + 9

But fog = gof

4ax – 3a + 3 = 4ax + 9

-3a + 3 = 9

– 3a = 6

a = – 2

The value of a = – 2

∵ gof = fog = IR 

⇒ fog = IR 

⇒ fog(x) = I (x) 

⇒ f (g(x)) = x [∵ I(x) = x being identity function] 

⇒ 10(g(x)) + 7 = x [∵ f(x) = 10x + 7]

⇒ g(x) = (x - 7)/10

i.e., g : R → R is a function defined as g(x) = (x - 7)/10

gof = fog = I_R
=> fog = I_R
=> fog(x) = I (x)
=>f (g(x)) = x       [I(x) = x being identity function]
=>10 (g(x)) + 7 = x       [f(x) = 10x + 7]
=>g(x)=x-7/10

i.e., g : R→R is a function defined as g(x)=x-7/10.
 

fog (x) = f(g(x))
= f(|5x - 2| )
= ||5x - 2||
= |5x -2|

(2 * 3) *4 = (4 + 7) *4 

= 11 * 4 = 44.

Correct Answer is: (a) British

(c) both (a) and (b)

Correct Answer is: (a) Chabahar

India is one of the biggest consumers of crude oil.

Correct Answer is: (a) JIM

India’s latest project with its neighbouring countries: The Government of India gives high importance to strengthen the friendly relations and to promote understanding with neighbouring countries. As part of this policy, the Government has undertaken several development projects with our neighbouring countries. Details of projects (a sample) is given below.

Note: Students should find and collect information (or) pictures related to the project of Gol (Government of India) with its neighbouring countries taking the sample few projects given here (or) other projects also. Students should paste the pictures in a note book underlining the project, prepare it as an album, submit to the subject teacher.

Projects

Bhutan: Projects completed in Bhutan with GOI assistance include Paro airport, Bhutan broadcasting station, major highways, and construction of Mini – Hydel projects. Penden cement plant with 300 tonnes capacity was completed in 1982 with GOI assistance.

Bangladesh: India is gifting 10 ambulances to Bangladesh.

Maldives: The 200 – bed Indira Gandhi Memorial Hospital (IGMH) was set up in Maldives in 1995 at a cost of about Rs 40 crores.

Nepal: The recently completed projects include 22 bridges on the East – West Highway. Setting up of trauma – center at Bir hospital in Kathmandu.

Sri Lanka: GOI has committed US $ 7.5 million to set up an India cancer centre in Colombo.

The above mentioned are only guidelines that could help a student to find what type of projects are generally undertaken.

(b) (i), (iii),(iv)

Correct Answer is: (c) (ii) only

The three big elements in our eastern policy are stronger emphasis on physical connectivity commercial and security related.

Correct Answer is: (d) 2 and 3

(c) both (a) and (b)

“Broudly, non – alignment means not tying yourself off with military blocks. It means trying to view things, as far as possible, not form the military point of view, though that has to come in sometimes, but independently, and trying to maintain friendly relations with all countries”.

– Jawaharlal Nehru

NAM means Non – Aligned Movement.

1. The NAM is meant for mutual assistance among nations for peace and progress. 

2. It aimed to maintain national independence in foreign affairs by not , jointing any military alliance.

Basic Determinants of a Foreign Policy:

• Geographical position and size of the territory
• Nation’s history, traditions and philosophical basis
• Natural resources
• The compulsion of economic development
• Political stability and structure of government
• The necessity of peace, disarmament and nonproliferation of nuclear weapons
• Military strength 
• International milieu

(c) Panchsheel

Guiding principles of Panchsheel are:

• Mutual Respect for each other’s territorial integrity and sovereignty
• Mutual non-interference 
• Equality and co-operation for mutual benefit 
• Peaceful co-existence.

1. Geographical position and size of territory. 

2. Nation’s history, traditions and philosophical basis. 

3. Natural resources 

4. Political stability and structure of Government.

(d) The Five Principles of Co existence

China became republic in 1949.

1. India achieved economic development by ensuring peace and security.

2. By leveraging the nations international partnership, to obtain all that is needed to fuel economic development. 

3. Economic development in the filed of markets, investment, fair global governance and a stable and fair environment conducive for growth. 

4. Currently’ India’s political moves are being influenced by economic imperatives. 

5. Many nations are moving to forge better relationship with India.

Jawaharlal Nehru of India, Tito of Yugoslavia, Nasser of Egypt, Sukarno of Indonesia, and Kwame Nkumarah of Ghana were the founding fathers of NAM.

(d) Singapore