Tunnel through earth

1. The problem statement, all variables and given/known data
A particle is dropped from a height equal to the radius of the earth into a tunnel dug through the earth along one of the diameters. Which of the following is/are true--
(A) Particle will oscillate through the earth to a height R on both sides
(B) Particle will execute simple harmonic motion
(C) Motion of the particle is periodic
(D) Particle passes through the centre of earth with a speed √ (2GM/R)

2. Relevant equations


3. The attempt at a solution
The answers are (A), (C), (D).

I know A and C will be true, A due to conservation of energy and C because such motions are almost always periodic, but I don't get why (B) is wrong and (D) is true.
Is (B) wrong because field above the earth is GM/r^2 and inside the earth GMr/R^3 and these are different, so it won't be SHM?
In any case, I can't prove (D) to be true (I tried energy conservation)

 

Sorry, got it.