Answer :

We know,

Now,

[∵ tan^{-1}(-x) = -tan^{-1}x]

⇒ -7 + 7x = 2x^{2} – 2x + 2

⇒ 2x^{2} – 9x + 9 = 0

⇒ 2x^{2} – 6x – 3x + 9 = 0

⇒ 2x(x – 3) – 3(x – 3) = 0

⇒ (2x – 3)(x – 3) = 0

**OR**

a*b = ab + 1,

Let p, q ∈ Q,

p*q = pq + 1 ∈ Q

As, P*Q also belong to Q, * defined on Q is binary operation.

Commutative:

Let p, q ∈ Q,

p*q = pq + 1

q*p = qp + 1

As,

pq = qp

⇒ pq + 1 = qp + 1

⇒ p*q = q*p

∴ * satisfies commutative property!

Associativity:

Let p, q, r ∈ Q,

Here,

(p*q)*r = (pq + 1)*r

= (pq + 1)r + 1

= pqr + r + 1

and

p*(q*r) = p*(qr + 1)

= p(qr + 1) + 1

= pqr + p + 1

⇒ (p*q)*r = p*(q*r)

∴ (*) doesn’t satisfy associative property!

Identity:

Let identity be e, then p*e = p

⇒ pe + 1 = p

⇒ pe = p – 1

As, identity is not unique, the binary operation don’t have a identity and as identity is not there, inverse is absurd!

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