[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
? = 32/3 + 23/4 + 11/2
= 3 + 2 + 1 + (2/3 + 3/4 + 1/2)
= 6 + (8 + 9 + 6)/12 = 6 + (23/12)
= 6 + (111/12) = 711/12
(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
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
Given expression = 2x2 + 2/3x2 + 3 + 5x
On dividing numerator and denominator by x, we get
= 2(x + 1/x)/3(x + 1/x) + 5 = 2 x 2/3 x 2 + 5 = 4/11
[given x + 1/x = 2]
? 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
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