Suppose Ravi earns Rs. N in each of the 11 months.
Then earning in January = Rs. 2 x N.
? Total annual income = (11 x N + 2 x N) = Rs. 13 x N
Part of total earning in January = (2 x N) / (13 x N) = 2/13
Let there be N members, other than father .
Father's share = 1/4, other's share = 3/4.
Each of other's share = 3/4N
? 3 x 3/4N = 1/4
? N = 9
Hence, the total number of members = N + 1 = 10
Let total score be N.
Then highest score = 3N/11
Remainder = (N - 3N/11) = 8N/11
Next highest score = 3/11 of 8N/11 = 24N/121
Now, ? 3N/11 - 24N/121 = 9
? 9N/121 = 9
?N = 121
? Let the fraction = N
? (9 x N)/7 - (7 x N)/9 = 8/21
? (32 x N)/63 = 8/21
? N = (8/21) x (63/32) = 3/4
? Correct answer = N x 7/9 = 7/9 x 3/4 = 7/12
? 7/4 + 5/2 + 67/12 + 10 + 9/4 = (21 + 30 + 67 + 40 + 27/12) = 185/12
This is nearly greater than 15. Let required fraction be x.
Then, 185/12 - x = 15
? x = (185/12) - 15
= 5/12
? N x N x 1/N = 1826/27
? N3 = 512/27
? N3= (8/3)3
? N = 8/3 = 22/3
(?)2 = ?132 + 28 / 4 - (3)3 + 107
= ?169 + 7 - 27 + 107
= ?256
= ?16
= 4
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
a2 + 1 = a
? a + 1/a = 1
On squaring both sides, we get
a2 + 1/a2 + 2 = 1
On cubing both sides, we get
(a2 + 1/a2)3 = (-1)3
? a6 + 1/a6 + 3a2 x 1/a2 (a2 + 1/a2) = -1
? a6 + 1/a6 + 3 x (-1) = -1
Now, a6 + 1/a6 + 1 = 3
As, a12 + a6 + 1 can be written as a6 + 1/a6 + 1
? a12 + a6 + 1 = 3
? 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 + a = 1 - 1/c
? ab = -1/c
? abc = -1
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
[(-c)/c + (-a)/a + (-b)/b] x [a/-a + b/-b + c/-c]
[(-1) + (-1) + (-1)] [(-1) + (-1) + (-1)]
= (-3) x (-3) = 9
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