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Question
If Δ is an operation such that for integers a and b we have a Δ b = a × b – 2 × a × b + b × b (–a) × b + b × b then find (–7) Δ (–1). Also show that (–7) Δ (–1) ≠ (–1) Δ (–7).
Solution
We have a Δ b = a × b – 2 × a × b + b × (b) (–a) × b + b × b
Now, put a = (–7) and b = (–1)
⇒ (–7) Δ (–1) = (–7) × (–1) –2 × (–7) × (–1) + (–1) × (–1){–(–7)} × (–1) + (–1) × (–1)
= 7 – 14 + 1 × 7 × (–1) + 1
= 7 – 14 – 7 + 1
⇒ –13
Now, put a = (–1) and b = (–7)
⇒ (–1) Δ (–7) = (–1) × (–7)–2 × (–1) × (–7) + (–7) × (–7){–(–1)} × (–7) + (–7) × (–7)
= 7 – 14 + 49(1) × (–7) + 49
= 7 – 14 – 343 + 49
= –301
Clearly, (–7) Δ (–1) ≠ (–1) Δ (–7).
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