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c. f. stands for cumulative frequency

Its formula is cf( n ) = cf( n - 1 ) + f ( n)

with cf( 1 ) = f(1 )

f (n )stands for corresponding freq for the n th observation

volume if fully immersed = 120/600 m^3 = 0.2 m^3

upward thrust = 200 kg

wt that can be kept = 80 kg

let a = (m/n)

so the period is 2n

a )

because fractional x is a periodic with period 1

So it only requires for sin ax to be periodic with integer period

write sin rx by Imiginary part of exp( irx ). Now sum the geometric series and take the imaginary part .

x >= 1/2

inversely proportional to the sqrtof their corresponding Ys

See , it is a long prob . So hang on

Let the ring is at a distance x above the gnd

So extension in the spring = sqrt( x ^2 + ( 2 R ) ^2 ) - 2R

Tension in the spring = 4mg/ R * extension

Let the angle made by the spring to the horizontal be @

So , The comp Fsin @ adds the accelaration of the ring ( along with mg ) & comp. Fcos @ gives the normal force on the spring . So the frictional force is

F cos @

Now , from geometry

cos @ = 2R / sqrt( x^2 + ( 2R)^2 )

sin @ = x / sqrt( x^2 + ( 2R ) ^2 )

Now the eqn of motion for the ring is

mv dv/dx = mg - Fcos @ + F sin @

after putting the values and on simplification this becomes v dv = g dx - 8 g { dx - 2r dx / sqrt ( x^2 + (2R)^2 } + 4g/R { x dx - 2R x dx/ sqrt ( x^2 + ( 2R)^2 }

integrating the above from 0 to 3 R / 2 and putting v^2 = 3 g R we get

= 1 / { 16 ( 1.5 - ln 4 ) } = 0.549

see, a is a const. qty & z runs from 0 to a

See Cartesian geometry is essentially based on Eucledian geometry . It is another way of looking at the things .

extremely sorry . Have made a very silly mistake .Black-listed is correct.

let y = f(x ) = 3 sin x - cos x + 2

giving , sin( x - /6 ) = ( y- 2 ) / 2

so , x = /6 + arcsin { ( y- 2 ) / 2 }

so inverse is = /6 + arcsin { ( x - 2 ) / 2 }

defined in [ 0, 4 ] [ -pi/3 , 2 pi/ 3 ]

Euclidian means classical Non - Reimannian geometry