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Dear rashmi ur query is well explained and responded by nunoxic.

According to me your answer seems to be ok and logical.

study material on goiit website itself should help u out.

u may use following link

http://www.goiit.com/chapters/tutorial/physics/newtons_law_com.htm

all real numbers are subset of complex numbers.

Thus all real nos. are complex numbers but vice versa may not be true.

eg. 4.256 is a real no. and also complex as in can be expressed in a+ib form

i.e. 4.256 + i0, so here imaginary part is zero.But ofcourse the no. is complex

Work done by gravity will be equal to the difference in final and initial gravitational potential energy of the system.

Actually the work done by gravitational energy is responsible for increase in the level of cylinder and thus we need to consider the work done by gravity only.

The adjacent turns of spring will be N-S poles just like in a bar magnet. So these turns are attracted by each other as the magnetic field will not be uniform as the spacings between the two turns is not zero.

This will cause compression and expansion of the spring.

Total points can be 7 in following cases

(2,2,2,1), (2,2,1,2), (2,1,2,2) and (1,2,2,2)

so total probability = (0.5 x 0.5 x0.5 x0.05) x 4

If I = _{[ 0]}

^{[ 2]} e^{cos x} cosx(sinx) dx

then put, cosx = t

so -sinx dx = dt

or I = I = _{[ 0]}

^{[ 2]} e^t t dt

Now integrate e^t t by parts.

To solve this problem here we follow the simple logic

let y1 + y2 + y3 + y4 + y5 = X

then, if X > 9

N/(y1 + y2 + y3 + y4 + y5) = N/X < 10000

But for X = 9, then to have the ratio N/X maximum N should be maximum which is possible only when N = 90000

so N/X = 10000

similarly when X = 8, 7, 6, 5, 3, 2 or 1

then N/X = 10000 is the maximum value.

Thus in no case N/X exceeds 10000

so maximum value of N/(y1 + y2 + y3 + y4 + y5) = 10000

Angular speed of the reflected ray will be twice to that of angular speed of the mirror that is 2w.

Well there are certain fundamental laws or postulates in physics are taken or granted without caring for its derivation.

Some of these laws are

Newtons laws, Schrodinger equation etc. are certain pillars on which physics and thus the physical world around rests.

The force on moving charged particle in magnetic field is also one of them. so we take the fact into consideration without caring about its origin or reason. But ofcourse the interaction of charged particle in electric and magnetic field is due to the exchange of photons.

mass = 10kg

u = 0

kinetic energy = K = (1/2)mv² = 80 Joule

so, v = 4m/s

then using a = (v - u)/4

or a = 4/4 = 1 m/s²

so force exerted = F = ma = 10 x 1 = 10 Newtons

Ammeter

sin5A = sin (A+4A)

= sinA cos4A + cosA sin4A

= sinA (cos² 2A - sin² 2A) + 2sin2A cos2A cosA

= sinA[(cos²A - sin²A)² -4sin²A cos²A] + 4sinA cos²A(cos²A - sin²A)

=sinA[cos⁴A+sin⁴A - 2sin²A cos²A-4sin²A cos²A]+ 4sinAcos⁴A - 4sin³Acos²A

= sinA[cos⁴A+sin⁴A - 6sin²A cos²A]+ 4sinAcos⁴A - 4sin³A cos²A

= sinA cos⁴A+sin⁵A - 6sin³A cos²A+ 4sinAcos⁴A - 4sin³A cos²A

= 5cos⁴AsinA - 10cos²sin³A + sin⁵A

To evaluate the integral

_[ -2]^{[ 2]} min { x - [x] , - x - [x] } dx, where [x] is the greatest integer function

we have,

f(x) = min { x - [x] , - x - [x] } = x - [x], for x

[-2, 0]

= - x - [x], for x

[0,2]

So we will split the integral accordingly

_[ -2]^{[ 2]} min { x - [x] , - x - [x] } dx

= _[ -2]^{[ 0]} { x - [x]} dx + _[ 0]^{[ 2]} {- x - [x]} dx

= _[ -2]^{[ -1]} {x + 2}dx + _[ -1]^{[ 0]} { x+1}dx+_[ 0]^{[ 1]} {- x-0} dx+_[ 1]^{[ 2]} {-x-1} dx

Now integrate the above expression to obtain the result.