If Δ is an operation such that for integers a and b we have a Δb = a

1.	If Δ is an operation such that for integers a and b we have a Δb = a × b – 2 × a × b + b × b (-a) × b + b × b then find(i) 4 Δ (- 3)(ii) (- 7)Δ (- 1)Also show that 4 Δ (- 3) ≠ (- 3) Δ 4 and (-7) Δ (-1) ≠ (-1) Δ (- 7)
Answer» a Δ b = a × b – 2 × a × b + b × b (-a) × b + b × b (i) 4 Δ (-3) = 4 × (-3) – 2 × 4 × (-3) + (-3) × (-3)(-4) × (-3) + (-3) × (-3) = -12 + 24 + 108 + 9 = -12 + 141 = 129 (ii) (-7) Δ (-1) = (-7) × (-1) – 2 × (-7) × (-1) + (-1) × (-1) (7) × (-1) + (-1) × (-1) = 7 -14 - 7 + 1 = 8 - 21 = -13 Now, (-3) Δ 4 = (-3) × 4 – 2 × (-3) ×(4) + 4 × 4(3) × 4 + 4 × 4 = -12 + 24 + 192 + 16 = -12 + 232 = 220 But 4 Δ (-3) = 129 ∴ 4 Δ (-3) ≠ (-3) A 4 And, (-1) Δ (-7) = (-1) × (-7) – 2 × (-1) × (-7) + (-7) × (-7)(1) × (-7) + (-7) × (-7) = 7 -14 – 343 + 49 = 56 – 357 = -301 But (-7) Δ (-1) = -13 ∴ (-7) Δ (-1) ≠ (-1) Δ (-7)

If Δ is an operation such that for integers a and b we have a Δb = a × b – 2 × a × b + b × b (-a) × b + b × b then find(i) 4 Δ (- 3)(ii) (- 7)Δ (- 1)Also show that 4 Δ (- 3) ≠ (- 3) Δ 4 and (-7) Δ (-1) ≠ (-1) Δ (- 7)

Answer»

a Δ b = a × b – 2 × a × b + b × b (-a) × b + b × b

(i) 4 Δ (-3)

= 4 × (-3) – 2 × 4 × (-3) + (-3) × (-3)(-4) × (-3) + (-3) × (-3)

= -12 + 24 + 108 + 9

= -12 + 141

= 129

(ii) (-7) Δ (-1)

= (-7) × (-1) – 2 × (-7) × (-1) + (-1) × (-1) (7) × (-1) + (-1) × (-1)

= 7 -14 - 7 + 1

= 8 - 21

= -13

Now, (-3) Δ 4

= (-3) × 4 – 2 × (-3) ×(4) + 4 × 4(3) × 4 + 4 × 4

= -12 + 24 + 192 + 16

= -12 + 232

= 220

But 4 Δ (-3) = 129

∴ 4 Δ (-3) ≠ (-3) A 4

And, (-1) Δ (-7)

= (-1) × (-7) – 2 × (-1) × (-7) + (-7) × (-7)(1) × (-7) + (-7) × (-7)

= 7 -14 – 343 + 49

= 56 – 357

= -301

But (-7) Δ (-1) = -13

∴ (-7) Δ (-1) ≠ (-1) Δ (-7)