Q. Prove that: 

                       (n-1)         

                 SUMMATION  (n-k)cos 2k   =   - n/2 , where n>=3 is an integer.

                        k=1                      ------

                                                      n  

Q. If cos(x-y), cosx, cos(x+y) are in H.P. then cosx sec(y/2)=
Q.If in a triangle PQR, sinP,sinQ,sinR are in A.P.,then prove that the altitudes are in H.P.

Q. For any odd integer n>=1 , n³- (n-1)³+.........(-1)^n-11³=

Q.A hemispherical bowl of radius r=0.1 m is rotating about its own axis (which is vertical), with an angular velocity . A particle of mass 10^-2 kg on the frictionless inner surface of the bowl is rotating with the same .The particle is at a height h from the bottom of the bowl.

Obtain the relation between h and . Also find the minimum value of  needed in order to have a non-zero value of h.

Q. If A (0,/2) then (x²+x)^1/2 + tan²A       is always greater than or equal to

                                                                (x²+x)^1/2

      1) 2tanA

     2)1

     3)2

     4)sec²A

Q. If a+b+c=0, then prove that the quadratic equation 3ax²+2bx+c=0 has

     at least one root in [0,1].

Q. For every positive integer n,

      prove that (4n+1)^1/2< (n)^1/2 + (n+1)^1/2 < (4n+2)^1/2

        Hence or otherwise p.t.

    [(n)^1/2 + (n+1)^1/2]=[(4n+1)^1/2]

where [x] denotes greatest integer function.

 

Q. There is a rectangular plate of mass M kg of dimensions (a x b). The plate is held in horizontal position by striking n small balls each of mass m per unit area per unit time. These are striking in a half region of the plate s.t.  area being striked=  b x a

             ---

              2

 

The balls are colliding elastically with vel. v. What is v?

n=100,M=3 kg,m=0.01 kg,b=2 m , a= 1m,g=10 m/s²

Q. Three particles of the same mass m are fixed to a uniform circular hoop of mass m and radius R at the corners of an equilateral triangle. The hoop is free to rotate in a vertical plane about a point P on the circumference opposite to one of the masses. Find the equivalent length of a simple pendulum.

 

PLEASE ANSWER THIS QUICKLY AS I HAVE A TEST TOMORROW.

Q. A simple pendulum of length L and mass m has a spring of force constant k connected to it at a distance x below the point of suspension. Find the frequency of vibration of the system.

 

Conserve the mechanical energy to get the answer to the next part.

 

The correct answer is:

Vertical Component       (a²+4b²)mg

                                    -----------------

                                     4(a²+b²)

Horizontal  Component    3abmg

                                    -------------

                                      4(a²+b²)

 

If somebody is able to get the answer,please give me the solution.

Q. A rectangular plate ABCD of mass m and dimensions a x b is supported in

     a vertical plane by two hinges at A & B as shown in the figure. Find the instantaneous reaction of the hinge at A immediately after B is removed.