Correct Answer - Option 2 : 52
Calculation:
xy = 19 - x - y
⇒ x + y + xy = 19 ---(1)
yz = 23 - y - z
⇒ y + z + yz = 23 ---(2)
xz = 29 - x - z
⇒ x + z + xz = 29 ---(3)
Adding 1 in equation (1), (2) and (3)
(1 + x) (1 + y) = 20 ---(4)
(1 + y) (1 + z) = 24 ---(5)
(1 + z) (1 + x) = 30 ---(6)
Multiplying equation (4), (5), and (6) together, we get,
(1 + x)2.(1 + y)2.(1 + z) 2 = 20 × 24 × 30
⇒ (1 + x).(1 + y).(1 + z) = √(20 × 24 × 30)
⇒ √(5 × 4 × 6 × 4 × 5 × 6)
⇒ 5 × 4 × 6 = 120
∵ (1 + x) (1 + y) (1 + z) = 120
∴ (1 + x) (1 + y) = 20
∴ (1 + z) = 120/20 = 6
⇒ z = 6 - 1 = 5
∵ (1 + x) (1 + y) (1 + z) = 120
∴ (1 + z) (1 + x) = 30
⇒ (1 + y) = 120/30 = 4
⇒ y = 3
∵ (1 + x) (1 + y) (1 + z) = 120
∴ (1 + x) (1 + y) = 20
⇒ (1 + x) (1 + 3) = 20
⇒ 4(1 + x) = 20/4
⇒ x = 4
∴ 2x2 + 5y2 - z2 = 2 × 42 + 5 × 32 - 52
⇒ 2 × 16 + 5 × 9 - 25
⇒ 32 + 45 - 25
⇒ 52
