12. A pitcher throws a ball of mass m at an angle such that the ball can travel a maximum distance R. Find the initial kinetic energy K₀ of the ball. The gravitational acceleration is denoted as g. (Ignore the height of the pitcher)
(A) K₀=2mgR
(B) K₀=mgR
(C) K₀=(1/2) mgR
(D) K₀=(1/4) mgR
(E) K₀=(1/8) mgR

58. Consider hydrogen-like atoms (atomic number = Z) that are at excited state with some principal quantum number n. It is found that the emission spectrum has 6 different lines. Furthermore, the most energetic photon in the emission spectrum has an energy of 51 eV. Find the atomic number Z. [Hydrogen atom En = -13.6/n2 (eV)] (A) 5 (B) 4 (C) 3 (D) 2 (E) 1

59. According to the Beer-Lambert Law, how many half-value layers are required to reduce the intensity of X-rays to 1/80 of their original intensity? [log10 2 = 0.301] (A) 4 (B) 5 (C) 6 (D) 7 (E) 8

60. Positron Emission Tomography is a highly advanced imaging technology in modern medicine. PET is primarily used to observe cellular metabolism; this is especially useful for cancer. After the isotope in the radiopharmaceutical enters the body, it migrates to the lesion site and emits a positron. As the positron moves through the tissue, it strikes an electron within the body. When the two collide, annihilation occurs, converting matter into energy and emitting two gamma rays of equal energy. The detectors then use these two rays to calculate the precise location of the lesion. Assume that both the electron and the positron carry a kinetic energy K=0.23m0c2, where m0 is the rest mass of an electron and c is the speed of light. What is the wavelength of the gamma rays produced after a head-on collision? [Compton wavelength is λ0=h/(m0c), where h is Planck's constant] (A) λ = 2.83 λ0 (B) λ = 1.62 λ0 (C) λ = 0.81 λ0 (D) λ = 0.40 λ0 (E) λ = 0.25 λ0

61. If (S)-1-bromo-2-methylbutane undergoes radical bromination at the chiral center, is the obtained major product (or mixture of products) optically active? (A) Yes, (S) (B) Yes, (R) (C) No, racemic (D) No, due to symmetry (E) Depends on temperature

62. How many of the following electron configurations for the species in their ground state are correct? Ca: 1s2 2s2 2p6 3s2 3p6 4s1     Br: 1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p4     V: [Ar] 3s2 3d3 As: [Ar] 4s2 3d10 4p2  P: 1s2 2s2 2p6 3p5 (A) 1 (B) 2 (C) 3 (D) 4 (E) None of the above

63. Which of the following techniques is best for determining the molecular weight? (A) IR (B) UV-Vis (C) Mass Spectrometry (D) NMR (E) CD

64. In the ground state, the diatomic molecule O2 is paramagnetic due to the unpaired electrons in its molecular orbitals. How many unpaired electrons does O2 have in its ground state? (A) 1 (B) 2 (C) 3 (D) 4 (E) None of the above

65. What is the product of the Beckmann Rearrangement? (A) Amine (B) Amide (C) Ketone (D) Carboxylic acid (E) Nitrile

66. Calcium fluoride (CaF2), known as fluorite, is used in dental products (e.g., fluoride treatments and varnishes) to promote enamel remineralization and prevent caries. In its crystal structure (fluorite type), the coordination number for the Ca2+ cations is 8. The coordination number for the F- anions in this crystal is ______. (A) 2 (B) 4 (C) 6 (D) 8 (E) 10

67. What is the rearrangement product of 3,3-dimethyl-1-butene reacting with HCl? (A) 3-chloro-2,2-dimethylbutane (B) 1,2-dichloro-3,3-dimethylbutane (C) 1-chloro-3,3-dimethylbutane (D) 2-chloro-3,3-dimethylbutane (E) 2-chloro-2,3-dimethylbutane

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