Jeev faltered at the very end with a bogey on his closing hole, the ninth. He shot a fine 67 on the first day but carded two-over 72 to be one-under 139 for 36 holes.
Kapur, who shot 71 on first day added a two-under 68 on the second, but it was not enough.
Though the second round was yet to be completed, the cut was likely to fall at two-under and both Indians would miss it by one shot.
Kapur had an eventful start with a birdie followed by a bogey. He again birdied the third, double bogeyed the fourth and birdied the fifth to get to even par. Then he had four pars before a birdie on 10th and 13th to get to one-under for the tournament. He parred the rest of the holes.
Soren Kjeldsen was 12-under with a second successive 64 while Wade Ormsby added a two-under 68 to his first round 61 to get to 11-under alongside David Horsey (66).