? = [5 -[3/4 + {2¹/₂ - (1/2 + 1/6 - 1/7)}]]/2
= [5 - [3/4 + {5/2 -(1/2 + 7 - 6/42)}]]/2
= [5 - [3/4 + {5/2 - (1/2 + 1/42)}]]/2
= [5 - [3/4 + {5/2 - (21 + 1 /42)}]]/2

Correct Option: A

? = [5 -[3/4 + {2¹/₂ - (1/2 + 1/6 - 1/7)}]]/2
= [5 - [3/4 + {5/2 -(1/2 + 7 - 6/42)}]]/2
= [5 - [3/4 + {5/2 - (1/2 + 1/42)}]]/2
= [5 - [3/4 + {5/2 - (21 + 1 /42)}]]/2
= [5 - [3/4 + {5/2 - 22/42}]]/2
= [5 - [3/4 + {105 - 22/42}]]/2
= [5 - [3/4 + 83/42]] / 2
= [5 - [63 + 166/84]]/2
= (191/84)/2
=191/168 = 1²³/₁₆₈

Given expression
[a³ + b³ + c³ - 3abc] / [a² + b² + c² - ab - bc - ca]
= a + b + c

Correct Option: A

Given expression
[a³ + b³ + c³ - 3abc] / [a² + b² + c² - ab - bc - ca]
= a + b + c
= 0.5 + 0.2 + 0.3
where, a = 0.5, b = 0.2, c = 0.3

√7 / (4 + √2) = [7 / (4 + √2)] x [(4 - √2 ) / (4 - √2]
= 7(4 - √2) / (16 - 2)
[∵ (a + b)(a - b) = a² - b²]
= 7(4 - √2 )/14

Correct Option: C

√7 / (4 + √2) = [7 / (4 + √2)] x [(4 - √2 ) / (4 - √2]
= 7(4 - √2) / (16 - 2)
[∵ (a + b)(a - b) = a² - b²]
= 7(4 - √2 )/14
= (4 - √2)/2
= (4 - 1.4142)/2
= 2.5858/2 = 1.2929

(0.96)³ - (0.1)³/ [(0.96)³ + 0.096 + 0.01]
= (0.96)³ - (0.1)³ = (0.96 - 0.1) [(0.96)² + 0.96 x 0.1 + (0.1)²]
[∵ a³ - b³ = (a - b) (a² + ab + b²)]

Correct Option: A

(0.96)³ - (0.1)³/ [(0.96)³ + 0.096 + 0.01]
= (0.96)³ - (0.1)³ = (0.96 - 0.1) [(0.96)² + 0.96 x 0.1 + (0.1)²]
[∵ a³ - b³ = (a - b) (a² + ab + b²)]
= (0.96 - 0.1) [(0.96)² + 0.096 + 0.01]/(0.96)² + 0.096 + 0.01
= 0.96 - 0.1 = 0.86

Use this formula
(a + b + c)² = (a² + b² + c²) + 2(ab + bc + ca)

Correct Option: C

We know that,
(a + b + c)² = (a² + b² + c²) + 2(ab + bc + ca)
⇒ 196 = 96 + 2(ab + bc + ca)
⇒ 2(ab + bc + ca) = 196 - 96 = 100
∴ (ab + bc + ca) = 100/2 = 50