If x + 1/x = 2, then 2x ² + 2/3x ² + 5x + 3 =?

We know that, (a + b)² = a² + b² + 2ab
= 234 + 2 x 108 = 450
(a - b)² = a² + b² - 2ab
= 234 - 2 x 108 = 18
? (a + b)²/(a - b)² = 450/18 = 25
? [(a + b)/(a - b)]² = 25
? (a + b)/(a - b) = ?25 = 5

[725 x 725 x 725 + 371 x 371 x 371] / [725 x 725 - 725 x 371 + 371 x 371]
= 725 + 371 = 1096
[? (a³ + b³) / (a² - ab + b²) = (a + b) (a² - ab + b²)/(a² - ab + b²) = a + b ]

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

(70 ÷ 100) ^{? + 3} = (0.49)⁴ x (0.343)⁴ ÷ (0.2401)⁴
(0.7) ^{? + 3} = (0.49 x 0.343/0.2401)⁴
(0.7)^{? + 3} = (0.7)⁴
On comparing the exponents both sides, we get
? + 3 = 4
? ? = 4 - 3 = 1

(?)^1/2 = (5568 ÷ 87)^1/3 + (72 x 2)^1/2
= (64)^1/3 + (144)^1/2
? ? = (4 + 12)² = 256

? a + (1/b) = 1
ab + 1 = b ......(i)

Also, b + (1/c) = 1
? b = 1 - (1/c) ....(ii)

From Eqs. (i) and (ii), we get
ab + 1 = 1 - (1/c)
? ab = -1/c
? abc = -1

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

Let total score = N.
Then, the highest score = 3N/11.
Remainder = (N - 3N/11) = 8N/11
Next highest score = 3/11 of 8N/11 = 24N/121

According to the question,
(3N/11) - (24N/121) = 18
? 33N - 24N = 18 x 121
? 9N = 18 x 121
? N = (18 x 121)/9 = 242

Since, a + b + c = 0
Then, a + b = - c .....(i)
a + c = - b ........(ii)
b + c = - a ......(iii)
Now,[(a + b)/c + (b + c)/a + (c + a)/b)] x [a/(b + c) + b/(c + a) + c/(a + b)]

Now, putting the value of a + b, b + c and c + a from Eqs (i), (ii) and (iii), we get

Given expression = [(-c)/c + (-a)/a + (-b)/b] [a/-a + b/-b + c/-c]
[(-1) + (-1) + (-1)] [(-1) + (-1) + (-1)]
= (-3) x (-3)
= 9

Given expression
= (2 - 1/3) (2 - 3/5) (2 - 5/7) ... (2 - 997/999)
= ((6 - 1)/3) ((10 - 3)/5) ((14 - 5)/7) ... ((1998 - 997)/999)
= 5/3 x 7/5 x 9/7 x ... x 1001/999
Multiply and solve by using the law of algebra,
= 5/3 x 7/5 x 9/7 x ... x 1001/999
= 1001/3