? 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
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]
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
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
Let, total number of workers be N .
Then number of women = N/3
Number of men = 2N/3
Number of women having children = 1/3 of 1/2 of N/3 = N/18
Number of men having children = 2/3 of 3/4 of 2N/3 = N/3
Number of workers having children = N/18 + N/3 = 7N/18
Number of workers having no children = N - 7N/18 = 11N/18
=11/18 of all workers
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