Speed of Man = 4.5 kmph
Speed of stream = 1.5 kmph
Speed in DownStream = 6 kmph
Speed in UpStream = 3 kmph
Average Speed = (2 x 6 x 3)/9 = 4 kmph.
1 x 1 + 1 = 2
2 x 2 + 2 = 6
6 x 3 + 3 = 21
21 x 4 + 4 = 88
88 x 5 + 5 = 445
445 x 6 + 6 = 2676 Should come in place of '?'
Let the four parts be (a - 3d), (a - d), (a + d) and (a + 3d).
(a - 3d) + (a - d) + (a +d ) + (a + 3d) = 124
? a = 31
Also, (a - 3d) (a + 3d) = (a - d) (a + d) - 128
? a2 - 9d2 = a2 - d2 - 128
? 8d2 = 128
? d = 4
a = 31, d = 4
So, the four parts are 19, 27, 35, 43.
Let a be the first term and and r be the common ratio of the GP. From the given problem,
a + arn - 1 = 66 ...(i)
Also, ar x arn - 2 = 128 ? a2rn - 1 = 128 ... (ii)
From Eq. (ii),
a.arn - 1 = 128
arn - 1 = 128/ a
On substitution this in Eq.(i), we get
a + 128/a = 66
a2 - 66a + 128 = 0
a = [-b ± ?b2 - 4ac] /2a
= 66 ± ?662 - 4 x 128 x 1)/2 = 64 or 2
Series pattern
The elements of the given series are the numbers formed by joining together consecutive add numbers in order
i.e, 1 and 3, 3 and 5, 5 and 7, 7 and 9, 9 and 11,...
? Missing tern = Number formed by joining 11 and 13 = 1113
In ?ACE
?A + ?C + ?E = 180°
Similarly in ?DFB
?D + ?F + ?B = 180°
? (?A + ?C + ?E) + (?D + ?F + ?B) = 360°
?A + ?B + ?C + ?D + ?E + ?F = 360°
Required answer = [64 log10 2] + 1
= [ 64 x 0.3010 ] + 1
= 19.264 + 1
= 19 + 1
= 20
Given Exp.= log23 x log 32 x log34 x log43
= (log3 / log2) x ( log2 / log3) x (log4 / log3) x (log3 / log4) = 1
Given Exp. = log (a2 / bc) + log (b2 / ac) + log (c2 / ab)
= log [(a2 x b2 x c2) / (a2 x b2 x c2)]
=log 1
=0
logax + loga(1+x) = 0
? logax (x+1) = loga1 (since log 1 = 0)
? x(x +1) = 1
? x2 + x - 1 = 0
? log23 x1 +log22 x1 + log2x = 11
? 1/3log2x + 1/2log2x + log2x = 11
? (1/3 + 1/2 +1)log2x = 11
? 11/6log2x = 11
? log2x = 11 x 6 / 11 =6
? x = 26 = 64
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