Given that,
x = ( ?2 + 1 ) / ( ?2 - 1 )
Expanding given equation by multiplying and dividing with ?2 + 1
we will get,
? x = ( ( ?2 + 1 ) / ( ?2 - 1 ) ) x ( ( ?2 + 1 ) / ( ?2 + 1 ) )
? x = ( ?2 + 1 ) x ( ?2 + 1 ) / ( ?2 - 1 ) x ( ?2 + 1 )
? x = ( ?2 + 1 ) 2 / ( ?2 - 1 ) x ( ?2 + 1 ) [ Use the formula (A + B)(A - B) = A2 - B2 ]
? x = ( ?2 + 1 ) 2 / ( ?22 - 12 ) [ Use the formula (A + B)2 = A2 + 2AB + B2 ]
? x = ( 2 + 1 + 2?2 ) / (2 - 1)
? x = 3 + 2?2 .....................(i)
and given in question x - y = 4 ?2
? y = x - 4?2
Put the value of x [ from Equation (i) ]
? y = 3 + 2?2 - 4?2
? y = 3 - 2?2
Now, x2 + y2 = (3 + 2?2)2 + (3 - 2?2)
Use the Algebra formula and solve the equation.
[ Use the formula (A + B)2 = A2 + 2AB + B2 ]
[ Use the formula (A - B)2 = A2 - 2AB + B2 ]
? x2 + y2 = 9 + 8 + 12?2 + 9 + 8 - 12?2
? x2 + y2 = 34
Given that, 30 % of A = 20 % of B
? A/B = 20/30 = 2/3
? A : B = 2 : 3
Let initial quantity be Q, and final quantity be F
F = Q(1 - 8/Q)
=> Q = 20
? log5[(x2 + x ) / x] = 2
? log10(x + 1) = 2
? x + 1 = 25
? x = 24
∴ Sum of 20 numbers (0 x 20) = 0.
It is quite possible that 19 of these numbers may be positive and if their sum is a then 20th number is (-a).
We know that speed is inversely proportional to time.
Given that, (Speed of A ) : (speed of B ) = 2 : 7
?(Time taken by A ) : (Time taken by B ) = 1/2 : 1/7 = 7 : 2
Let th distance = D
and usual speed = V
According to the question,
D/(3V/4) - D/V = 2
? D/V = 3 x 2 = 6
Time taken to cover the distance with usual speed = 6 h
?2n = 64
? 2n/2 = 26
? n/2 = 6
? n = 12
Distance = 160 km
Relative Speed = 8 + 2 = 10
Time = Distance/Relative speed = 160/10 = 16 h
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